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Soliton solutions to KdV equation with spatio-temporal dispersion
Ocean Engineering 114 (2016) 192–203
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Soliton solutions to KdV equation with spatio-temporal dispersion Houria Triki a, Turgut Ak b,n, Seithuti Moshokoa c, Anjan Biswas d,e a
Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria Department of Transportation Engineering, Faculty of Engineering, Yalova University, 77100 Yalova, Turkey Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa d Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA e Faculty of Science, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi-Arabia b c
art ic l e i nf o
a b s t r a c t
Article history: Received 7 September 2015 Accepted 16 January 2016
This paper discusses a soliton solution that is of a different structure than that is usually known. The equation with spatio-temporal dispersion is studied in this paper both analytically and numerically. The governing equations that are addressed in this paper are the Korteweg-de Vries (KdV) equation, modified KdV equation and finally KdV equation with power law nonlinearity. Numerically these three equations are all addressed with quintic B-spline collocation method and the stability analysis is carried out. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Solitons Integrability Numerical simulations
1. Introduction
2. Governing model
The dynamics of shallow water wave flow is studied with several models. KdV equation is the most popular model that is frequently visible in various journals. There are several results that are reported over the past few decades that studied this model (Wazwaz, 2004; Darvishi et al., 2007; Wazwaz, 2008; Antonova and Biswas, 2009; Biswas, 2009a, 2009b; Wazwaz, 2010; Kapoor et al., 2012; Triki et al., 2013; Wazwaz, 2013; Mirzazadeh et al., 2015; Cai et al., 2015). Integrability aspects, perturbation theory, numerical solutions, conservation laws and various other aspects are all addressed. This paper revisits another form of KdV equation that is referred to as improved KdV (IKdV) equation. This contains an additional dispersion term apart from the usual third order dispersion. It is the spatio-temporal dispersion. This paper will retrieve soliton solutions to the IKdV equation, modified IKdV equation as well as IKdV equation with power law nonlinearity. The solitary wave ansatz approach will be applied. That carries a generalized structure. Subsequently numerical simulations will support the analytical solutions that are given. Interaction of two solitary waves is also studied numerically in this paper.
The three types of KdV equation are studied by the aid of the following models: (i) The improved KdV equation: ut þ auux þ b1 uxxx þb2 uxxt ¼ 0; (ii) The improved mKdV equation: ut þ au2 ux þ b1 uxxx þ b2 uxxt ¼ 0;
Corresponding author. E-mail addresses: [email protected] (H. Triki), [email protected] (T. Ak), [email protected] (S. Moshokoa), [email protected] (A. Biswas). http://dx.doi.org/10.1016/j.oceaneng.2016.01.022 0029-8018/& 2016 Elsevier Ltd. All rights reserved.
ð2Þ
(iii) The improved KdV equation with power law nonlinearity: ut þ aun ux þ b1 uxxx þ b2 uxxt ¼ 0;
ð3Þ
where a, b1, b2 are arbitrary real constants. To find exact soliton solutions of Eqs. (1)–(3) we adopt an ansatz of the form uðx; tÞ ¼
A p; ðB þC cosh τ þ D sinh τ Þ
ð4Þ
where
τ ¼ μðx vtÞ n
ð1Þ
ð5Þ
Here in Eqs. (4) and (5), A is the amplitude of the soliton, μ is the inverse width and v represents the velocity of the soliton. Also B, C and D are constants to be determined, while the exponent p are unknown and will be determined later.
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Obviously, the form of ansatz (4) is more general than the ansatz of Yu (2010) because of the appearance of the arbitrary exponent p and parameter C related to the cosh function; and thus it provides a possible way to find novel solitonic solutions for nonlinear wave equations possessing all kinds of dispersive and nonlinear terms. The solution structure given by (4) includes the well-known solitary waves as well as the singular solitary waves. If B ¼ D ¼ 0, one recovers the usual solitary waves. However, for B ¼ C ¼ 0, one obtains singular solitary waves. Both of these solution structures are well known in the literature. This paper will secure solutions to IKdV equation and its hierarchies with all of these parameters being non-zero.
3. The improved KdV equation Let us first consider the improved KdV equation (1), ð6Þ
and from ansatz Eq. (4), substituting values of u, ut, ux, uxxx and uxxt into Eq. (6), we have ApvμðC sinh τ þ D cosh τ Þ ðB þ C cosh τ þ D sinh τÞ
pþ1
aA2 pμðC sinh τ þ D cosh τÞ 2p þ 1
ðB þC cosh τ þ D sinh τ Þ
ðb1 b2 vÞAμ3 p3 ðC sinh τ þ D cosh τÞ
þ
ðb1 b2 vÞABμ3 pðp þ 1Þð2p þ 1ÞðC sinh τ þ D cosh τÞ
ðB þ C cosh τ þ D sinh τÞ
ð15Þ BCD 2Av þ 2ðb1 b2 vÞAμ2 p2 Aðp þ 1Þð2p þ 1Þμ2 ðb1 b2 vÞ ¼ 0; ð16Þ Solving the above equations, we get B ¼0 and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 v ; μ¼ 2 b1 b2 v A¼
3vðC 2 D2 Þ ; a
ð18Þ
with D a 7 C. Hence the improved KdV equation (1) has the solitary wave solution 3vðC 2 D2 Þ=a ðC cosh τ þD sinh τÞ
2
;
ð19Þ
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 v ðx vtÞ: τ¼ 2 b1 b2 v
ð20Þ
It is very interesting to see that in the particular case where C ¼1 and b2 ¼ 0 in the expression (19), the solitary wave solution can be reduced to a simple form 3vð1 D2 Þ=a rffiffiffiffiffiffi rffiffiffiffiffiffi 2 : 1 v 1 v C cosh ξ þ D sinh ξ 2 b1 2 b1
pþ1
pþ2
ðB þC cosh τ þ D sinh τÞ
ðb1 b2 vÞAμ3 pðp þ 1Þðp þ 2Þ B2 C 2 þ D2 ðC sinh τ þ D cosh τÞ ðB þ C cosh τ þ D sinh τ Þ
ð17Þ
uðx; tÞ ¼
pþ3
ð21Þ
where ξ ¼ x vt, which is exactly the solitary wave solution of the celebrated KdV equation studied in Yu (2010).
pμBCD 4Av 4ðb1 b2 vÞAμ2 p2 2Aðp þ 1Þð2p þ 1Þμ2 ðb1 b2 vÞ ¼ 0;
uðx; tÞ ¼
ut þ auux þ b1 uxxx þ b2 uxxt ¼ 0;
193
¼ 0;
ð7Þ Then to determine the exponent p, we usually balance the linear terms of highest order in the resulting Eq. (7) with the highest order nonlinear terms. Thus, from matching the exponents of 2p þ 1 pþ3 1=ðB þ C cosh τ þ D sinh τÞ and 1=ðB þC cosh τ þ D sinh τÞ terms in Eq. (7), we obtain 2p þ 1 ¼ p þ 3;
ð8Þ
so that p ¼ 2:
ð9Þ
ut þ au2 ux þ b1 uxxx þ b2 uxxt ¼ 0;
ð22Þ
and from ansatz, substituting values of u, ut, ux, uxxx and uxxt into Eq. (22), we have ApvμðC sinh τ þ D cosh τÞ
aA3 pμðC sinh τ þ D cosh τÞ
ðB þ C cosh τ þ D sinh τÞ ðB þ C cosh τ þ D sinh τÞ ðb1 b2 vÞAμ3 p3 ðC sinh τ þ D cosh τÞ pþ1 ðB þ C cosh τ þ D sinh τÞ þ
ðb1 b2 vÞABμ3 pðp þ 1Þð2p þ1ÞðC sinh τ þ D cosh τÞ ðB þ C cosh τ þD sinh τÞ
pþ2
ðb1 b2 vÞAμ pðp þ 1Þðp þ 2Þ B C þ D ðC sinh τ þ D cosh τ Þ 3
3p þ 1
2
2
2
ðB þ C cosh τ þ D sinh τÞ
pþ3
¼ 0;
ð23Þ ð11Þ
Apvμ 2C 2 D þ D3 B2 D aA2 pμD ðb1 b2 vÞAμ3 p h Dðp þ 1Þðp þ 2Þ B2 C 2 þD2 þ p2 B2 D D3 2C 2 D þ ðp þ 1Þð2p þ 1ÞBD ¼ 0;
Let us now consider the improved mKdV equation (2):
pþ1
If we put p ¼2 in Eq. (7), we can determine the soliton parameters n by setting the corresponding coefficients of cosh τ (n ¼ 0; 1; 3), n sinh τ (n ¼ 1; 3) and cosh τ sinh τ, respectively, to zero. Hence we find ð10Þ Apvμ D3 þ 3C 2 D ðb1 b2 vÞAμ3 p3 3C 2 D þ D3 ¼ 0; Apvμ 3D2 C þ C 3 ðb1 b2 vÞAμ3 p3 3D2 C þ C 3 ¼ 0;
4. The improved mKdV equation
Now, from (23) equating the exponents 3p þ1 and p þ3 yields p ¼ 1;
ð24Þ pþ1
ð12Þ
Apvμ 2CD2 þ C 3 þ B2 C aA2 pμC ðb1 b2 vÞAμ3 p h p2 2CD2 þ C 3 þ B2 C þCðp þ 1Þðp þ 2Þ B2 C 2 þ D2 D i ðp þ 1Þð2p þ 1ÞB2 C ¼ 0;
ð13Þ
pμBðC 2 þ D2 Þ Av ðb1 b2 vÞAμ2 p2 þ Aðpþ 1Þð2p þ 1Þμ2 ðb1 b2 vÞ ¼ 0;
ð14Þ
, Now, noting that the functions 1=ðB þ C cosh τ þ D sinh τÞ pþ2 pþ3 and 1=ðB þ C cosh τ þ D sinh τÞ , in 1=ðB þ C cosh τ þ D sinh τÞ (23), are linearly independent, setting their respective coefficients in (23) to zero, yields the following set of relations, respectively rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v μ¼ ; ð25Þ b1 b2 v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6vðC 2 D2 Þ ; A¼ a B ¼ 0:
ð26Þ ð27Þ
194
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
and D a 7C. Hence we get the solitary wave solution of Eq. (22), when the above parameters p, μ, A and B are substituted in Eq. (4) as n o1=2 6vðC 2 D2 Þ=a ; ð28Þ uðx; tÞ ¼ C cosh τ þ D sinh τ where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ðx vtÞ: τ¼ b1 b2 v
τ¼
n 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ðx vtÞ: b1 b2 v
and the amplitude A is related to the velocity v and the constants C and D as given by Eq. (34).
6. Numerical simulation of the improved KdV type equations ð29Þ
5. The improved KdV equation with power law nonlinearity
6.1. The application of the quintic B-spline collocation method Let us take into consideration the improved KdV equation with power law nonlinearity (3) by the following boundary conditions: Uða; tÞ ¼ 0;
It will be very interesting to assess the soliton solutions for the improved KdV-type models with power law nonlinearity. For pursuing such a goal, we consider the improved KdV equation with power law nonlinearity (3): ut þaun ux þb1 uxxx þ b2 uxxt ¼ 0;
ð30Þ
Choosing the new solitary wave ansatz (4) and substituting values of u, ut, ux, uxxx and uxxt into Eq. (30), we have ApvμðC sinh τ þ D cosh τÞ pþ1
aAn þ 1 pμðC sinh τ þ D cosh τÞ
ðB þ C cosh τ þ D sinh τÞ ðB þC cosh τ þ D sinh τÞ ðb1 b2 vÞAμ3 p3 ðC sinh τ þ D cosh τÞ pþ1 ðB þ C cosh τ þ D sinh τÞ þ
pðn þ 1Þ þ 1
ðb1 b2 vÞABμ3 pðp þ1Þð2p þ 1ÞðC sinh τ þ D cosh τÞ ðB þC cosh τ þ D sinh τÞ
ðB þ C cosh τ þ D sinh τ Þ
pþ3
Uðb; tÞ ¼ 0;
U x ða; tÞ ¼ 0;
U x ðb; tÞ ¼ 0;
Uðx; 0Þ ¼ f ðxÞa rx rb;
¼ 0;
Proceeding as before, we obtain
n 2
A¼
ð32Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ; b1 b2 v
ð33Þ !1=n
vðn þ 1Þðn þ 2ÞðC 2 D2 Þ 2a
;
B ¼ 0:
ð34Þ
ð35Þ
and D a 7 C. Thus, finally, the novel solitary wave solution of Eq. (30) is given by uðx; tÞ ¼ where
A ðC cosh τ þ D sinh τÞ
2=n
;
ð39Þ
For the numerical calculation, the solution domain of the problem is limited over a range a r x r b. The range is separated into uniformly sized finite elements of length h by the knots xm such that a ¼ x0 ox1 o ⋯ o xN ¼ b. The set of quintic B-spline functions
ϕ 2 ðxÞ; ϕ 1 ðxÞ; …; ϕN þ 1 ðxÞ; ϕN þ 2 ðxÞ forms a basis over the problem domain ½a; b. We search the numerical solution U N ðx; tÞ to the exact solution Uðx; tÞ in the form N þ2 X
ϕj ðxÞδj ðtÞ
ð40Þ
where δj ðtÞ are time dependent parameters to be determined from the boundary and collocation conditions. Quintic B-splines ϕm ðxÞ, (m ¼ 2ð1ÞN þ 2), at the knots xm are determined over the range ½a; b by Prenter (1975):
8 > ðx xm 3 Þ5 ; > > > > > ðx xm 3 Þ5 6ðx xm 2 Þ5 ; > > > > > ðx > xm 3 Þ5 6ðx xm 2 Þ5 þ 15ðx xm 1 Þ5 ; 1 < ϕm ðxÞ ¼ 5 ðx xm 3 Þ5 6ðx xm 2 Þ5 þ 15ðx xm 1 Þ5 20ðx xm Þ5 ; h > > > ðx xm 3 Þ5 6ðx xm 2 Þ5 þ 15ðx xm 1 Þ5 20ðx xm Þ5 þ 15ðx xm þ 1 Þ5 > > > > > ðx xm 3 Þ5 6ðx xm 2 Þ5 þ 15ðx xm 1 Þ5 20ðx xm Þ5 þ 15ðx xm þ 1 Þ5 6ðx xm þ 2 Þ5 > > > > : 0;
μ¼
ð38Þ
j ¼ 2
ð31Þ
2 ; n
t 40:
and the initial condition.
U N ðx; tÞ ¼
pþ2
ðb1 b2 vÞAμ3 pðp þ 1Þðp þ 2Þ B2 C 2 þ D2 ðC sinh τ þD cosh τ Þ
p¼
ð37Þ
ð36Þ
½xm 3 ; xm 2 ½xm 2 ; xm 1 ½xm 1 ; xm ½xm ; xm þ 1
ð41Þ
½xm þ 1 ; xm þ 2 ½xm þ 2 ; xm þ 3 otherwise:
Each quintic B-spline covers six elements so that each element ½xm ; xm þ 1 is covered by six B-splines. A typical finite interval ½xm ; xm þ 1 is mapped to the interval ½0; 1 by a local coordinate transformation described by hξ ¼ x xm , 0 r ξ r 1. Thus, quintic B-splines (41) in terms of ξ over ½0; 1 can be given as the following:
ϕm 2 ¼ 1 5ξ þ 10ξ2 10ξ3 þ 5ξ4 ξ5 ; ϕm 1 ¼ 26 50ξ þ 20ξ2 þ 20ξ3 20ξ4 þ 5ξ5 ; ϕm ¼ 66 60ξ2 þ 30ξ4 10ξ5 ; ϕm þ 1 ¼ 26 þ 50ξ þ 20ξ2 20ξ3 20ξ4 þ 10ξ5 ; ϕm þ 2 ¼ 1 þ 5ξ þ 10ξ2 þ 10ξ3 þ 5ξ4 5ξ5 ; ϕm þ 3 ¼ ξ5 :
ð42Þ
Substituting trial function (41) into Eq. (40), the nodal values of U; U 0 ; U 00 and U 000 at the knots xm are obtained in terms of the element parameters δm by U N ðxm ; tÞ ¼ U m ¼ δm 2 þ 26δm 1 þ 66δm þ 26δm þ 1 þ δm þ 2
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Table 1 The conserved quantities and the error norms for single solitary wave with v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼ 0, h ¼0.05 and Δt ¼ 0:05, 30 r x r 30. t
M
P
E
L2 104
L1 104
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.1117547944 1.1117547682 1.1117547791 1.1117548341 1.1117549899 1.1117554084 1.1117562922 1.1117582646 1.1117626659 1.1117725365 1.1117945631
0.1118231860 0.1118231860 0.1118231860 0.1118231860 0.1118231860 0.1118231860 0.1118231860 0.1118231860 0.1118231863 0.1118231878 0.1118231952
0.0202058914 0.0202058914 0.0202058914 0.0202058914 0.0202058914 0.0202058913 0.0202058913 0.0202058909 0.0202058889 0.0202058789 0.0202058291
0.000000 0.002588 0.004125 0.005498 0.007433 0.011890 0.023075 0.049577 0.109721 0.244394 0.544889
0.000000 0.001100 0.001876 0.004941 0.011951 0.027855 0.062402 0.139195 0.310501 0.692404 1.54381
5 ð δm 2 10δm 1 þ 10δm þ 1 þ δm þ 2 Þ; h 20 U ″m ¼ 2 ðδm 2 þ 2δm 1 6δm þ 2δm þ 1 þ δm þ 2 Þ; h 60 000 U m ¼ 3 ð δm 2 þ 2δm 1 2δm þ 1 þ δm þ 2 Þ; h U 0m ¼
ð43Þ
where the symbols 0 ;00 and 000 symbolize differentiation according to x, respectively. The splines ϕm ðxÞ and its four principle derivatives vanish outside the interval ½xm 3 ; xm þ 3 . Now we identify the collocation points with the knots and use Eq. (43) to evaluate Um, its necessary space derivatives and substitute into Eq. (3) to obtain the set of the coupled ordinary differential equations. For the linearization technique we get the following equation:
Z m ¼ U nm ¼ ðδm 2 þ 26δm 1 þ66δm þ 26δm þ 1 þ δm þ 2 Þn
M
P
E
3.589248 3.588690 3.588560 3.588745 3.589231 3.590057 3.591301 3.593080 3.595552 3.598923 3.603466
0.331677 0.331692 0.331703 0.331722 0.331755 0.331809 0.331899 0.332050 0.332310 0.332759 0.333533
0.059987 0.059938 0.059908 0.059857 0.059772 0.059633 0.059403 0.059012 0.058340 0.057175 0.055158
Fig. 2. Interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 58, x2 ¼ 23, C ¼1, D¼ 0, h ¼ 0.05 and Δt ¼ 0:1, 0 r x r 80, t ¼2.
and indicates derivative with respect to t. If time parameters δi's and its time derivatives δ_ i 's in Eq. (44) are discretized by the Crank–Nicolson formula and usual finite difference approximation, respectively: 1 2
δ_ i ¼
δi ¼ ðδn þ δn þ 1 Þ;
δn þ 1 δn Δt
ð45Þ
we obtain a recurrence relationship between two time levels n and nþ1 n þ1 relating two unknown parameters δi , δin for i ¼ m 2; m 1; …; m þ 1; m þ 2
γ 1 δnmþ12 þ γ 2 δnmþ11 þ γ 3 δnmþ 1 þ γ 4 δnmþþ11 þ γ 5 δnmþþ12 ¼ γ 5 δnmþ12 þ γ 4 δm 1 þ γ 3 δm þ γ 2 δm þ 1 þ γ 1 δm þ 2 n
n
n
ð46Þ
where
þ
where
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
n
δ_ m 2 þ 26δ_ m 1 þ 66δ_ m þ 26δ_ m þ 1 þ 2δ_ m þ 2 5aZ m ð δm 2 10δm 1 þ 10δm þ 1 þ δm þ 2 Þ h 60b1 þ 3 ð δm 2 þ2δm 1 2δm þ 1 þ δm þ 2 Þ h 20b2 _ þ 2 ðδ m 2 þ 2δ_ m 1 6δ_ m þ 2δ_ m þ 1 þ δ_ m þ 2 Þ ¼ 0 h
Table 2 The conserved quantities for interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a ¼6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 58, x2 ¼ 23, C¼1, D ¼ 0, h ¼0.05 and Δt ¼ 0:1, 0r x r 80. t
Fig. 1. Single solitary wave with amplitude ¼ 0:15000, v¼.3, a ¼6, b1 ¼ 1, b2 ¼ 0:1, C ¼1, D¼ 0, h ¼ 0.05 and Δt ¼ 0:05, 30r x r 30, 0 r t r 5.
195
ð44Þ
γ 1 ¼ ½1 EZ m M þ K ; γ 2 ¼ ½26 10EZ m þ 2M þ 2K ; γ 3 ¼ ½66 6K ; γ 4 ¼ ½26 þ 10EZ m 2M þ 2K ; γ 5 ¼ ½1 þEZ m þ M þ K ; m ¼ 0; 1; …; N;
E¼
5 aΔt; 2h
M¼
30 h
3
b1 Δt;
K¼
120 4
h
b2 :
ð47Þ
196
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Table 3 The invariants due to Maxwellian initial condition for IKdV equation with v ¼3, a ¼0.005, b1 ¼ 0:0005, h ¼0.05, Δt ¼ 0:1 and x0 ¼ 40, 0 r x r 80. b2
t
M
P
E
0.0001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817638 1.772480896817636 1.772480896817637 1.772480896817637 1.772480896817638 1.772480896817637 1.772480896817635 1.772480896817637 1.772480896817639 1.772480896817639
1.253458591164609 1.253458591164613 1.253458591164625 1.253458591164643 1.253458591164669 1.253458591164703 1.253458591164746 1.253458591164794 1.253458591164852 1.253458591164916 1.253458591164987
0.001078923031966 0.001078923031956 0.001078923031929 0.001078923031883 0.001078923031819 0.001078923031737 0.001078923031637 0.001078923031518 0.001078923031382 0.001078923031228 0.001078923031057
0.001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817638 1.772480896817638 1.772480896817637 1.772480896817638 1.772480896817639 1.772480896817638 1.772480896817638 1.772480896817638 1.772480896817638 1.772480896817638
1.254586556675805 1.254586556675802 1.254586556675798 1.254586556675789 1.254586556675777 1.254586556675764 1.254586556675744 1.254586556675722 1.254586556675697 1.254586556675669 1.254586556675637
0.001078923031966 0.001078923031957 0.001078923031931 0.001078923031887 0.001078923031826 0.001078923031747 0.001078923031652 0.001078923031539 0.001078923031409 0.001078923031262 0.001078923031098
0.01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817637 1.772480896817638 1.772480896817638 1.772480896817640 1.772480896817636 1.772480896817639 1.772480896817637 1.772480896817638 1.772480896817638 1.772480896817638
1.265866211787758 1.265866211787721 1.265866211787614 1.265866211787433 1.265866211787180 1.265866211786855 1.265866211786459 1.265866211785991 1.265866211785451 1.265866211784839 1.265866211784158
0.001078923031966 0.001078923031960 0.001078923031943 0.001078923031914 0.001078923031874 0.001078923031822 0.001078923031759 0.001078923031684 0.001078923031598 0.001078923031501 0.001078923031392
0.1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817638 1.772480896817637 1.772480896817638 1.772480896817637 1.772480896817639 1.772480896817640 1.772480896817641 1.772480896817639 1.772480896817642 1.772480896817638
1.378662762907293 1.378662762907257 1.378662762907151 1.378662762906977 1.378662762906733 1.378662762906411 1.378662762906025 1.378662762905563 1.378662762905034 1.378662762904435 1.378662762903758
0.001078923031966 0.001078923031965 0.001078923031963 0.001078923031961 0.001078923031957 0.001078923031952 0.001078923031947 0.001078923031940 0.001078923031932 0.001078923031923 0.001078923031913
Table 4 The conserved quantities and the error norms for solitary waves with v¼ 0.3, a ¼6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼0, h ¼ 0.05 and Δt ¼ 0:05, 30 r x r 30. t
M
P
E
L2 104
L1 104
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
3.1883675645 3.1883675166 3.1883675365 3.1883676369 3.1883679213 3.1883686854 3.1883702982 3.1883739167 3.1883819310 3.1883999694 3.1884400770
1.1225493997 1.1225493998 1.1225493998 1.1225493998 1.1225493998 1.1225493998 1.1225493998 1.1225494000 1.1225494010 1.1225494059 1.1225494305
0.1144137462 0.1144137462 0.1144137462 0.1144137462 0.1144137462 0.1144137461 0.1144137459 0.1144137446 0.1144137380 0.1144137047 0.1144135385
0.000000 0.013355 0.022385 0.030586 0.038860 0.048981 0.066504 0.108080 0.211139 0.452393 0.998233
0.000000 0.008080 0.011098 0.016509 0.021820 0.050856 0.113930 0.254135 0.566894 1.26415 2.81862
For the linearization technique, the term Un in non-linear term U n U x is taken as Z m ¼ U nm ¼ ðδm 2 þ 26δm 1 þ 66δm þ 26δm þ 1 þ δm þ 2 Þn :
ð48Þ
The system (46) consists of (N þ1) linear equations including (N þ 5) unknown parameters ðδ 2 ; δ 1 ; …; δN þ 1 ; δN þ 2 ÞT . To obtain
a unique solution to this system, we need four additional constraints. These are got from the boundary conditions and can be used to eliminate δ 2 ; δ 1 and δN þ 1 , δN þ 2 from the system (46) which becomes later a matrix equation for the N þ1 unknowns d ¼ ðδ0 ; δ1 ; …; δN ÞT of the form Ad
nþ1
n
¼ Bd :
ð49Þ
The matrices A and B are pentagonal ðN þ 1Þ ðN þ 1Þ matrices and so are easily solved. However, two or three inner iterations are nn n n n1 Þ at each time step to applied to the term δ ¼ δ þ 12 ðδ δ cope with the non-linearity caused by Zm. Before the solution 0 process begins iteratively, the initial vector d ¼ ðδ0 ; δ1 ; …; δN 1 ; δN Þ must be determined by using the initial condition and the following derivatives at the boundary conditions: U N ðx; 0Þ ¼ U ðxm ; 0Þ; ðU N Þx ða; 0Þ ¼ 0; ðU N Þxx ða; 0Þ ¼ 0;
m ¼ 0; 1; 2; …; N
ðU N Þx ðb; 0Þ ¼ 0; ðU N Þxx ðb; 0Þ ¼ 0;
ð50Þ 0
So we have the following matrix form of the initial vector d : 0
Wd ¼ B
ð51Þ
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Fig. 3. Solitary waves with amplitude ¼ 0:54772, v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼ 0, h ¼0.05 and Δt ¼ 0:05, 30 r x r 30, 0r t r 5. Table 5 The conserved quantities for interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 45, x2 ¼ 23, C¼ 1, D¼ 0, h ¼0.05 and Δt ¼ 0:1, 0 r x r 80. t 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
M
P
E
3.597679 3.597655 3.597687 3.597750 3.597834 3.597935 3.598054 3.598199 3.598376 3.598598 3.598877
0.348712 0.348713 0.348713 0.348713 0.348714 0.348714 0.348715 0.348716 0.348717 0.348719 0.348721
0.009807 0.009809 0.009810 0.009810 0.009810 0.009811 0.009811 0.009812 0.009814 0.009817 0.009822
where 2
54
6 25:25 6 6 6 1 6 6 6 W ¼6 6 6 6 6 6 4
6
67:50
26:25
1
26
66
26
1
1
26
66
26
1
⋱ 1
26
66
26
1
26:25
67:50
6
60
7 7 7 7 7 7 7 7 7 7 1 7 7 7 25:25 5 54
a ib ; a þ ib
ð53Þ
where a ¼ γ 3 þ ðγ 2 þ γ 4 Þ cos ½hk þ γ 1 þ γ 5 cos ½2hk; b ¼ ðγ 4 γ 2 Þ sin ½hk þ ðγ 5 γ 1 Þ sin ½2hk:
ð54Þ
The modulus of j gj is 1, for this reason the linearized scheme is unconditionally stable.
and the error norm L1
6.2. Stability analysis
L1 ¼ J U exact U N J 1 Cmax U exact ðU N Þj ; j j
ð52Þ
j ¼ 1; 2; …; N 1:
ð56Þ
the improved KdV equation with power law nonlinearity satisfies three conservation laws given by mass (M), momentum (P), and energy (E) (Sanchez, 2015): Z
The stability analysis is based on the von Neumann approach in which the growth factor g of the error in a typical mode of n amplitude δ^ ,
δnj ¼ δ^ eijkh
g¼
J¼1
n
where k is the mode number and h is the element size, is determined from a linearization of the numerical scheme. In order to apply the stability analysis, the generalized improved KdV equation can be linearized by supposing that the quantity Un in the nonlinear term U n U x is locally constant. Replacing mode (52) into (46) gives the growth factor g of the form
In this section, we take into consideration the motion of single solitary wave solution for three test problems. Accuracy and efficiency of the method is reckoned by the error norm L2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N
2 u X exact
exact ð55Þ L2 ¼ J U U N J 2 C th ðU N Þj ;
U j
d ¼ δ0 ; δ1 ; …; δN 1 ; δN and B ¼ ½U ðx0 ; 0Þ; U ðx1 ; 0Þ; …; U ðxN 1 ; 0Þ; U ðxN ; 0ÞT . This matrix system can be solved efficiently by using a variant of Thomas algorithm. 0
Fig. 4. Interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 45, x2 ¼ 23, C ¼1, D¼ 0, h ¼ 0.05 and Δt ¼ 0:1, 0 r x r 80, t ¼2.
6.3. Numerical experiments and discussion
3
60
197
M¼
Z
b
U dx; a
b
P¼ a
½U 2 þ b2 ðU x Þ2 dx;
Z E¼ a
b
(
) 2aU n þ 2 b1 ðU x Þ2 dx ðn þ 1Þðn þ 2Þ
ð57Þ In the simulation of solitary wave motion, the invariants M, P and E are observed to check the conversation of the numerical algorithm.
198
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Table 6 The invariants due to Maxwellian initial condition for ImKdV equation with v¼3, a¼ 0.005, b1 ¼ 0:0005, h ¼ 0.05, Δt ¼ 0:1 and x0 ¼ 40, 0 r x r 80. b2
t
M
P
E
0.0001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817643 1.772480896817656 1.772480896817676 1.772480896817707 1.772480896817746 1.772480896817794 1.772480896817850 1.772480896817914 1.772480896817987 1.772480896818067
1.253458591164609 1.253458591164632 1.253458591164704 1.253458591164821 1.253458591164985 1.253458591165195 1.253458591165453 1.253458591165754 1.253458591166099 1.253458591166490 1.253458591166923
0.000111886200788 0.000111886200752 0.000111886200646 0.000111886200468 0.000111886200220 0.000111886199902 0.000111886199515 0.000111886199060 0.000111886198537 0.000111886197948 0.000111886197295
0.001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817642 1.772480896817652 1.772480896817675 1.772480896817708 1.772480896817745 1.772480896817789 1.772480896817843 1.772480896817909 1.772480896817978 1.772480896818058
1.254586556675805 1.254586556675806 1.254586556675812 1.254586556675822 1.254586556675835 1.254586556675852 1.254586556675875 1.254586556675899 1.254586556675928 1.254586556675960 1.254586556675996
0.000111886200788 0.000111886200754 0.000111886200653 0.000111886200486 0.000111886200252 0.000111886199951 0.000111886199585 0.000111886199155 0.000111886198660 0.000111886198102 0.000111886197482
0.01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817641 1.772480896817652 1.772480896817668 1.772480896817694 1.772480896817723 1.772480896817760 1.772480896817804 1.772480896817856 1.772480896817913 1.772480896817976
1.265866211787758 1.265866211787634 1.265866211787260 1.265866211786638 1.265866211785767 1.265866211784650 1.265866211783285 1.265866211781676 1.265866211779823 1.265866211777727 1.265866211775392
0.000111886200788 0.000111886200767 0.000111886200705 0.000111886200601 0.000111886200455 0.000111886200269 0.000111886200041 0.000111886199772 0.000111886199462 0.000111886199112 0.000111886198722
0.1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817638 1.772480896817642 1.772480896817644 1.772480896817654 1.772480896817661 1.772480896817678 1.772480896817696 1.772480896817713 1.772480896817733 1.772480896817752
1.378662762907293 1.378662762907139 1.378662762906681 1.378662762905905 1.378662762904825 1.378662762903437 1.378662762901747 1.378662762899748 1.378662762897442 1.378662762894822 1.378662762891891
0.000111886200788 0.000111886200785 0.000111886200778 0.000111886200767 0.000111886200751 0.000111886200730 0.000111886200704 0.000111886200674 0.000111886200639 0.000111886200600 0.000111886200556
Table 7 The conserved quantities and the error norms for solitary waves with v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C¼1, D ¼0, h ¼ 0.05 and Δt ¼ 0:05 for n¼ 3 and 5, 30 r x r 30. n
t
M
P
E
L2 104
L1 104
n ¼3
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
4.1242880435 4.1242879885 4.1242880113 4.1242881269 4.1242884540 4.1242893328 4.1242911943 4.1242954877 4.1243045021 4.1243254144 4.1243711541
2.0383679294 2.0383679293 2.0383679293 2.0383679293 2.0383679293 2.0383679293 2.0383679294 2.0383679296 2.0383679309 2.0383679374 2.0383679699
0.0938208891 0.0938208890 0.0938208890 0.0938208890 0.0938208890 0.0938208890 0.0938208886 0.0938208869 0.0938208782 0.0938208341 0.0938206144
0.000000 0.043633 0.072262 0.102453 0.133682 0.166084 0.201782 0.250370 0.347399 0.590793 1.18948
0.000000 0.026164 0.043693 0.062457 0.079513 0.095912 0.131036 0.292292 0.652010 1.45396 3.24186
n ¼5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
4.6916975075 4.6916974397 4.6916974543 4.6916975670 4.6916979034 4.6916988233 4.6917008058 4.6917058112 4.6917144690 4.6917372083 4.6917850253
2.8255684288 2.8255684122 2.8255683956 2.8255683790 2.8255683624 2.8255683458 2.8255683293 2.8255683130 2.8255682979 2.8255682886 2.8255683083
0.0791877278 0.0791877330 0.0791877380 0.0791877431 0.0791877482 0.0791877533 0.0791877588 0.0791877658 0.0791877808 0.0791878351 0.0791880861
0.000000 0.176929 0.341153 0.515784 0.699270 0.892441 1.09653 1.31435 1.55559 1.85599 2.35948
0.000000 0.125603 0.223654 0.323375 0.425967 0.530855 0.644002 0.760441 0.884985 1.53766 3.42844
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Fig. 5. Solitary waves with amplitude ¼ 0:79370, v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼ 0, h ¼0.05 and Δt ¼ 0:05 for n¼ 3, 30 r x r 30, 0 r t r 5.
6.4. The improved KdV equation 6.4.1. The motion of single solitary wave For this problem, the improved KdV Eq. (1) is considered for the boundary conditions U-0 as x- 7 1 and the initial condition
Fig. 6. Solitary waves with amplitude ¼ 1:00981, v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼0, h ¼ 0.05 and Δt ¼ 0:05 for n ¼ 5, 30 r x r 30, 0 r t r 5.
6.4.2. The interaction of two solitary waves In this section, we have studied the interaction of two well separated solitary waves by using the following initial condition: Uðx; 0Þ ¼
3vðC 2 D2 Þ=a Uðx; 0Þ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : 1 v 1 v x þ D sinh x C cosh 2 b1 b2 v 2 b1 b2 v ð58Þ Note that the solitary wave solution of the equation can be written as 3vðC 2 D2 Þ=a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : 1 v 1 v C cosh ðx vtÞ þ D sinh ðx vtÞ 2 b1 b2 v 2 b1 b2 v
uðx; tÞ ¼
ð59Þ We have used the parameters v¼ 0.3, a ¼6, b1 ¼ 1, b2 ¼ 0:1, C ¼1, D ¼0, h ¼0.05 and Δt ¼ 0:05 over the interval ½ 30; 30 to coincide with solitary wave solution of the improved KdV equation. So, the solitary wave has an amplitude 0.15000 and the computations are done up to time t ¼5 to obtain the invariants and error norms L2 and L1 at various times. Error norms L2, L1 and three invariants of the improved KdV equation are listed in Table 1. It is seen from the table that the error norms are found to be small enough. The percentage of the relative error of the invariants M, P and E are calculated with respect to the conserved quantities at t ¼0. Percentage of relative changes of M, P and E are found to be 3:58 10 3 %, 8:24 10 6 % and 3:08 10 4 %, respectively. Thus, the invariants remain almost constant as the time progress. Also, Table 1 exhibits a comparison of the values of the invariants and error norms obtained by the present method. Fig. 1 shows the motion of solitary wave at various time levels. It is observed that the soliton moves to the right at a constant speed and preserves its amplitude and shape with the increasing time, as expected.
199
2 X i ¼ 1 ½C
Ai coshðκ i ðx xi ÞÞ þ D sinhðκ i ðx xi ÞÞ2
where Ai ¼ 3vi ðC 2 D2 Þ=a, κ i ¼ 12
:
ð60Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi vi , vi, xi, i¼ 1,2 are arbitrary b1 b2 vi
constants. Eq. (56) represents two solitary waves having different amplitudes at the same direction. We have considered the parameters v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 58, x2 ¼ 23, C ¼1, D ¼0, h¼ 0.05 and Δt ¼ 0:1 over the interval ½0; 80. The simulation is done up to t¼ 2 and the values of the conserved quantities M, P and E are listed in Table 2. It is clear that the obtained values of invariants are almost constant during the computer run. Fig. 2 illustrates the interaction of two solitary waves at t ¼2.
6.4.3. Birth of solitons We consider the improved KdV Eq. (1) with the following Maxwellian initial condition: uðx; 0Þ ¼ expð ðx 40Þ2 Þ:
ð61Þ
In this case, the behavior of the solution depends on the values of b2 . Therefore, we chose the values of b2 ¼ 0:0001, b2 ¼ 0:001, b2 ¼ 0:01 and b2 ¼ 0:1. The other parameters are v ¼3, a ¼0.005, b1 ¼ 0:0005, h¼ 0.05, Δt ¼ 0:1 and x0 ¼ 40 over the interval ½0; 80. The numerical computations are done up to t ¼5. The values of the conserved quantities of motion for different b2 are presented in Table 3. The maximum relative changes of the conserved quantities from their initial values are less than 5:01 10 14 %, 2:84 10 10 % and 8:43 10 8 %, respectively.
200
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Table 8 The conserved quantities for interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a ¼6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 38, x2 ¼ 20, C¼ 1, D ¼ 0, h ¼ 0.05 and Δt ¼ 0:1 for n ¼3 and 5, 0 r x r 80. t
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
n¼ 3
n¼ 5
M
P
E
M
P
E
3.595651 3.595656 3.595759 3.595909 3.596078 3.596251 3.596423 3.596595 3.596770 3.596952 3.597144
0.369841 0.369847 0.369849 0.369851 0.369854 0.369857 0.369860 0.369863 0.369867 0.369870 0.369874
0.013433 0.013451 0.013454 0.013455 0.013455 0.013455 0.013455 0.013456 0.013456 0.013456 0.013456
3.595651 3.595656 3.595759 3.595909 3.596078 3.596251 3.596423 3.596595 3.596770 3.596952 3.597144
0.369841 0.369847 0.369849 0.369851 0.369854 0.369857 0.369860 0.369863 0.369867 0.369870 0.369874
0.013981 0.013999 0.014002 0.014003 0.014003 0.014003 0.014004 0.014004 0.014004 0.014004 0.014004
Fig. 8. Interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 38, x2 ¼ 20, C¼ 1, D ¼0, h ¼ 0.05 and Δt ¼ 0 for n¼ 5, 0 r x r 80, t ¼2.
wave solution of the improved mKdV equation. Hence, the solitary wave has amplitude 0.54772 and the simulations are run up to time t ¼5 to obtain the invariants and the error norms at several times. Error norms L2 and L1 and conserved quantities are reported in Table 4. It can be easily seen from the table that the obtained error norms are small. The agreement between numerical and analytic solution is excellent. Percentage of relative changes of M, P and E are found to be 2:27 10 3 %, 2:73 10 6 % and 1:81 10 4 %, respectively. Perspective views of the traveling solitons are graphed at diverse time levels in Fig. 3. 6.5.2. The interaction of two solitary waves To study interaction of two solitary waves we use the boundary conditions U-0 as x- 71 and the initial condition
Fig. 7. Interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 38, x2 ¼ 20, C¼ 1, D ¼ 0, h ¼ 0.05 and Δt ¼ 0:1 for n¼ 3, 0 r x r 80, t ¼2.
6.5. The improved mKdV equation 6.5.1. The motion of single solitary wave For this problem, the improved mKdV Eq. (2) is considered for the boundary conditions U-0 as x- 7 1 and the initial condition n
o1=2 6vðC 2 D2 Þ=a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Uðx; 0Þ ¼ v v C cosh x þ D sinh x b1 b2 v b1 b2 v
ð62Þ
Note that the solitary wave solution of the equation can be written as n o1=2 6vðC 2 D2 Þ=a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : uðx; tÞ ¼ v v C cosh ðx vtÞ þ D sinh ðx vtÞ b1 b2 v b1 b2 v ð63Þ The parameters v ¼0.3, a ¼6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼0, h¼0.05 and Δt ¼ 0:05 with interval ½ 30; 30 are taken to compare the results obtained by the present method with solitary
2 X
Ai : ð64Þ C cosh½ κ ðx x Þ þ D sinh ½κ i ðx xi Þ i i i¼1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Ai ¼ ½6vi ðC 2 D2 Þ=a1=2 , κ i ¼ b1 vib2 vi , vi, xi, i¼1,2 are arbiUðx; 0Þ ¼
trary constants. The initial condition (59) represents two solitary waves, one with amplitude A1 placed initially at x ¼ x1 and the second with amplitude A2 placed at x ¼ x2 . For the numerical simulation, we choose the parameters v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 45, x2 ¼ 23, C ¼1, D ¼0, h ¼0.05 and Δt ¼ 0:1 over the interval ½0; 80. The experiments are run from t¼0 to t¼2 and the values of the invariant quantities M, P and E are reported in Table 5. It is observed that the numerical values of invariants remain almost constant during the computer run. Fig. 4 illustrates the interaction of two solitary waves at t ¼2. 6.5.3. Birth of solitons The improved mKdV Eq. (2) with development of the Maxwellian initial condition uðx; 0Þ ¼ expð ðx 40Þ2 Þ;
ð65Þ
into a train of solitary waves is considered. For the Maxwellian initial condition, behavior of the solution depends on the values of b2. So, we take b2 ¼ 0:0001, b2 ¼ 0:001, b2 ¼ 0:01 and b2 ¼ 0:1. The other parameters are v ¼ 3, a ¼0.005, b1 ¼ 0:0005, h¼0.05, Δt ¼ 0:1 and x0 ¼ 40 over the interval ½0; 80. The
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
201
Table 9 The invariants due to Maxwellian initial condition for IKdV equation with power law nonlinearity v ¼3, a¼ 0.005, b1 ¼ 0:0005, h ¼ 0.05, Δt ¼ 0:1 and x0 ¼ 40 for n¼ 3, 0 r x r 80. b2
t
n ¼3 M
P
E
0.0001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817657 1.772480896817705 1.772480896817790 1.772480896817912 1.772480896818065 1.772480896818252 1.772480896818469 1.772480896818719 1.772480896818997 1.772480896819305
1.253458591164609 1.253458591164670 1.253458591164856 1.253458591165162 1.253458591165591 1.253458591166138 1.253458591166802 1.253458591167577 1.253458591168463 1.253458591169454 1.253458591170547
0.000230308728809 0.000230308728877 0.000230308729080 0.000230308729417 0.000230308729887 0.000230308730488 0.000230308731216 0.000230308732068 0.000230308733041 0.000230308734130 0.000230308735330
0.001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817654 1.772480896817704 1.772480896817788 1.772480896817907 1.772480896818052 1.772480896818231 1.772480896818445 1.772480896818690 1.772480896818959 1.772480896819259
1.254586556675805 1.254586556675824 1.254586556675883 1.254586556675982 1.254586556676119 1.254586556676297 1.254586556676511 1.254586556676761 1.254586556677047 1.254586556677368 1.254586556677722
0.000230308728809 0.000230308728872 0.000230308729062 0.000230308729376 0.000230308729815 0.000230308730376 0.000230308731057 0.000230308731855 0.000230308732767 0.000230308733790 0.000230308734919
0.01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817651 1.772480896817689 1.772480896817755 1.772480896817845 1.772480896817958 1.772480896818100 1.772480896818264 1.772480896818455 1.772480896818669 1.772480896818908
1.265866211787758 1.265866211787555 1.265866211786946 1.265866211785932 1.265866211784513 1.265866211782693 1.265866211780473 1.265866211777859 1.265866211774853 1.265866211771459 1.265866211767683
0.000230308728809 0.000230308728845 0.000230308728953 0.000230308729132 0.000230308729383 0.000230308729705 0.000230308730097 0.000230308730559 0.000230308731091 0.000230308731691 0.000230308732359
0.1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817654 1.772480896817704 1.772480896817788 1.772480896817907 1.772480896818052 1.772480896818231 1.772480896818445 1.772480896818690 1.772480896818959 1.772480896819259
1.254586556675805 1.254586556675824 1.254586556675883 1.254586556675982 1.254586556676119 1.254586556676297 1.254586556676511 1.254586556676761 1.254586556677047 1.254586556677368 1.254586556677722
0.000230308728809 0.000230308728872 0.000230308729062 0.000230308729376 0.000230308729815 0.000230308730376 0.000230308731057 0.000230308731855 0.000230308732767 0.000230308733790 0.000230308734919
numerical computations are done up to t¼ 5. The numerical values of the conserved quantities during the simulations are given in Table 6 for different values of b2 . The maximum relative changes of the conserved quantities from their initial values are less than 2:42 10 11 %, 1:12 10 9 % and 3:12 10 6 %, respectively.
Note that the solitary wave solution of the equation can be written as uðx; tÞ ¼
ð2aÞ1=n
½vðn þ 1Þðn þ 2ÞðC 2 D2 Þ1=n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=n : n v n v C cosh ðx vtÞ þ D sinh ðx vtÞ 2 b1 b2 v 2 b1 b2 v
ð67Þ Finally, the parameters v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C ¼ 1, D¼ 0, h¼ 0.05 and Δt ¼ 0:05 with interval ½ 30; 30 are chosen to compare the results obtained by the present method with solitary 6.6.1. The motion of single solitary wave wave solution of the improved KdV equation with power law For this problem, the improved mKdV equation with power law nonlinearity for n ¼3,5. Therefore, the solitary waves have amplinonlinearity (3) is considered for the boundary conditions U-0 as tudes 0:79370, 1.00981 for n ¼3,5 respectively. The experiments x- 71 and the initial condition are run from the time t ¼0 to the time t ¼5 to obtain the invariants and the error norms L2 and L1 at different times. Error norms L2 ½vðn þ 1Þðn þ 2ÞðC 2 D2 Þ1=n Uðx; 0Þ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=n : and L1 and conserved quantities are listed in Table 7 for n ¼3,5. It n v n v is clearly seen from the table that the error norms obtained by the x þ D sinh x ð2aÞ1=n C cosh 2 b1 b2 v 2 b1 b2 v present method are small enough. The agreement between ð66Þ numerical and analytic solutions are perfect. Percentage of relative 6.6. The improved KdV equation with power law nonlinearity
202
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Table 10 The invariants due to Maxwellian initial condition for IKdV equation with power law nonlinearity v¼ 3, a¼ 0.005, b1 ¼ 0:0005, h ¼ 0.05, Δt ¼ 0:1 and x0 ¼ 40 for n ¼ 5, 0 r x r 80. b2
t
n¼ 5 M
P
E
0.0001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817713 1.772480896817928 1.772480896818291 1.772480896818793 1.772480896819427 1.772480896820184 1.772480896821053 1.772480896822032 1.772480896823103 1.772480896824264
1.253458591164609 1.253458591164811 1.253458591165414 1.253458591166407 1.253458591167776 1.253458591169500 1.253458591171556 1.253458591173916 1.253458591176550 1.253458591179429 1.253458591182526
0.000467139218580 0.000467139218723 0.000467139219148 0.000467139219850 0.000467139220817 0.000467139222036 0.000467139223489 0.000467139225158 0.000467139227023 0.000467139229064 0.000467139231259
0.001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817709 1.772480896817918 1.772480896818265 1.772480896818747 1.772480896819355 1.772480896820083 1.772480896820929 1.772480896821881 1.772480896822928 1.772480896824064
1.254586556675805 1.254586556675914 1.254586556676240 1.254586556676780 1.254586556677527 1.254586556678469 1.254586556679598 1.254586556680896 1.254586556682354 1.254586556683954 1.254586556685682
0.000467139218580 0.000467139218710 0.000467139219096 0.000467139219734 0.000467139220617 0.000467139221733 0.000467139223071 0.000467139224614 0.000467139226347 0.000467139228253 0.000467139230315
0.01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817685 1.772480896817835 1.772480896818081 1.772480896818426 1.772480896818865 1.772480896819402 1.772480896820028 1.772480896820749 1.772480896821554 1.772480896822450 1.772480896817638 1.772480896817650 1.772480896817690 1.772480896817753 1.772480896817851 1.772480896817972 1.772480896818118 1.772480896818292 1.772480896818496 1.772480896818722 1.772480896818980
1.265866211787758 1.265866211787452 1.265866211786538 1.265866211785015 1.265866211782889 1.265866211780170 1.265866211776864 1.265866211772981 1.265866211768534 1.265866211763535 1.265866211758000 1.378662762907293 1.378662762906997 1.378662762906126 1.378662762904673 1.378662762902643 1.378662762900027 1.378662762896830 1.378662762893052 1.378662762888696 1.378662762883758 1.378662762878247
0.000467139218580 0.000467139218643 0.000467139218830 0.000467139219142 0.000467139219577 0.000467139220134 0.000467139220810 0.000467139221605 0.000467139222515 0.000467139223538 0.000467139224669 0.000467139218580 0.000467139218585 0.000467139218599 0.000467139218622 0.000467139218655 0.000467139218697 0.000467139218748 0.000467139218808 0.000467139218878 0.000467139218957 0.000467139219045
0.1
changes of M, P and E are found to be 2:02 10 3 %, 2:93 10 4 % for n ¼3, 1:87 10 3 %, 1:99 10 6 %, 4:26 10 6 %, 4:52 10 4 % for n ¼5, respectively. The profiles of the solitary wave at different time levels are shown in Figs. 5 and 6 for n ¼3,5. 6.6.2. The interaction of two solitary waves We consider the improved KdV equation with power law nonlinearity (3) for the boundary conditions U-0 as x- 7 1 and the initial condition 2 X
Ai : ð68Þ ½C coshð κ ðx x ÞÞ þ D sinhðκ i ðx xi ÞÞ2=n i i i¼1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=n where Ai ¼ f½vi ðn þ 1Þðn þ 2ÞðC 2 D2 Þ=2ag , κ i ¼ n2 b1 vib2 vi , vi, xi , Uðx; 0Þ ¼
i¼1,2 are arbitrary constants. Eq. (62) represents two solitary waves having different amplitudes at the same direction. We have chosen x1 ¼ 45, x2 ¼ 23 for n ¼3, x1 ¼ 38, x2 ¼ 20 for n¼ 5. The other parameters are v1 ¼ 0:17, v2 ¼ 0:34, a ¼6, b1 ¼ 3, b2 ¼ 1, C ¼1, D¼ 0, h ¼0.05 and Δt ¼ 0:1 over the interval ½0; 80. The
computer program has run to time t¼ 2. To record the conservative quantities of the invariants M, P and E, the calculated values are given in Table 8 for n¼ 3,5. It is clear that the obtained values of invariants are almost constant during the computer run. Figs. 7 and 8 illustrate the interaction of two solitary waves at t¼2 for n ¼ 3; 5. 6.6.3. Birth of solitons Finally, we have studied the improved KdV equation with power law nonlinearity (3) with the development of the Maxwellian initial condition uðx; 0Þ ¼ expð ðx 40Þ2 Þ;
ð69Þ
into a train of solitary waves. As it is known, with the Maxwellian initial condition (63), the behavior of the solution depends on the values of b2 . We study each of the following cases: b2 ¼ 0:0001, b2 ¼ 0:001, b2 ¼ 0:01 and b2 ¼ 0:1. The other parameters are v¼ 3, a ¼0.005, b1 ¼ 0:0005, h¼ 0.05, Δt ¼ 0:1 and x0 ¼ 40 over the interval ½0; 80. The numerical computations are done up to t¼ 5. The obtained numerical values of the invariants of motion for
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
different b2 are given in Tables 9 and 10, for n ¼3, 5 respectively. The maximum relative changes of the conserved quantities from their initial values are less than 9:41 10 11 %, 1:59 10 9 % and 2:83 10 6 % for n ¼3, 3:74 10 10 %, 2:35 10 9 % and 2:71 10 6 % for n ¼5, respectively. Consequently, as seen from the four cases for n ¼1, 2, 3 and 5, the changes of the invariants are reasonably small.
7. Conclusion This paper studied shallow water waves with IKdV equation, mIKdV equation and finally the IKdV equation with power law nonlinearity. Solitary wave solutions are obtained by ansatz method that is not of the usual norm. There are several numerical simulations that are obtained in this paper along with the interaction of solitary waves. The numerical simulations employed quintic B-spline collocation method. The results of this paper come with a lot of encouragement for further future investigations in this paper. Later soliton solutions of this equation, belonging to the type discussed in this paper, will be computed for the perturbed KdV equation. There are several perturbation terms that are studied in this context. These perturbation terms will be included. The solution structure will be considered for vector coupled KdV equation as well that is studied in the context of two or three layered shallow water wave dynamics. The results of that research are awaited at this time. This is just a tip of the iceberg.
Acknowledgment The author, Turgut Ak, is grateful to The Scientific and Technological Research Council of Turkey for granting scholarship for Ph.D. studies.
203
References Antonova, M., Biswas, A., 2009. Adiabatic parameter dynamics of perturbed solitary waves. Commun. Nonlinear Sci. Numer. Simul. 14, 734–748. Biswas, A., 2009a. Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion. Commun. Nonlinear Sci. Numer. Simul. 14, 3503–3506. Biswas, A., 2009b. Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients. Nonlinear Dyn. 58 (1–2), 345–348. Cai, S., Xu, J., Liu, J., Chen, Z., Xie, J., Li, J., He, Y., 2015. Retrieval of the maximum horizontal current speed induced by ocean internal solitary waves from low resolution time series mooring data based on the KdV theory. Ocean Eng. 94, 88–93. Darvishi, M.T., Kheybari, S., Khani, F., 2007. A numerical solution of the Lax's 7thorder KdV equation by pseudospectral method and Darvishi's preconditioning. Int. J. Contemp. Math. Sci. 2 (22), 1097–1106. Kapoor, S., Rawat, S., Dhawan, S., 2012. Numerical investigation of separated solitary waves solution for KdV equation through finite element technique. Int. J. Comput. Appl. 40 (14), 27–33. Mirzazadeh, M., Eslami, M., Biswas, A., 2015. 1-Soliton solution of KdV6 equation. Nonlinear Dyn. 80 (1–2), 387–396. Prenter, P.M., 1975. Splines and Variational Methods. John Wiley, New York. Sanchez, P., 2015. Cnoidal solutions, shock waves, and solitary wave solutions of the improved Korteweg-de Vries equation. Rom. J. Phys. 60 (3–4), 379–400. Triki, H., Milovic, D., Biswas, A., 2013. Solitary waves and shock waves of the KdV6 equation. Ocean Eng. 73, 119–125. Wazwaz, A.-M., 2004. Compact and noncompact solutions for nonlinear dispersive variants of the generalized KdV equation. Appl. Math. Comput. 159, 577–588. Wazwaz, A.-M., 2008. The integrable KdV6 equations: multiple soliton solutions and multiple singular soliton solutions. Appl. Math. Comput. 204, 963–972. Wazwaz, A.-M., 2010. A study on KdV and Gardner equations with time-dependent coefficients and forcing terms. Appl. Math. Comput. 217, 2277–2281. Wazwaz, A.-M., 2013. Multiple soliton solutions and rational solutions for the (2þ 1)-dimensional dispersive long water-wave system. Ocean Eng. 60, 95–98. Yu, L., 2010. New type soliton solutions to Korteweg-de Vries and Benjamin–Bona– Mahony equations. Chin. Phys. Lett. 27 (9).
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Soliton solutions to KdV equation with spatio-temporal dispersion Houria Triki a, Turgut Ak b,n, Seithuti Moshokoa c, Anjan Biswas d,e a
Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria Department of Transportation Engineering, Faculty of Engineering, Yalova University, 77100 Yalova, Turkey Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa d Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA e Faculty of Science, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi-Arabia b c
art ic l e i nf o
a b s t r a c t
Article history: Received 7 September 2015 Accepted 16 January 2016
This paper discusses a soliton solution that is of a different structure than that is usually known. The equation with spatio-temporal dispersion is studied in this paper both analytically and numerically. The governing equations that are addressed in this paper are the Korteweg-de Vries (KdV) equation, modified KdV equation and finally KdV equation with power law nonlinearity. Numerically these three equations are all addressed with quintic B-spline collocation method and the stability analysis is carried out. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Solitons Integrability Numerical simulations
1. Introduction
2. Governing model
The dynamics of shallow water wave flow is studied with several models. KdV equation is the most popular model that is frequently visible in various journals. There are several results that are reported over the past few decades that studied this model (Wazwaz, 2004; Darvishi et al., 2007; Wazwaz, 2008; Antonova and Biswas, 2009; Biswas, 2009a, 2009b; Wazwaz, 2010; Kapoor et al., 2012; Triki et al., 2013; Wazwaz, 2013; Mirzazadeh et al., 2015; Cai et al., 2015). Integrability aspects, perturbation theory, numerical solutions, conservation laws and various other aspects are all addressed. This paper revisits another form of KdV equation that is referred to as improved KdV (IKdV) equation. This contains an additional dispersion term apart from the usual third order dispersion. It is the spatio-temporal dispersion. This paper will retrieve soliton solutions to the IKdV equation, modified IKdV equation as well as IKdV equation with power law nonlinearity. The solitary wave ansatz approach will be applied. That carries a generalized structure. Subsequently numerical simulations will support the analytical solutions that are given. Interaction of two solitary waves is also studied numerically in this paper.
The three types of KdV equation are studied by the aid of the following models: (i) The improved KdV equation: ut þ auux þ b1 uxxx þb2 uxxt ¼ 0; (ii) The improved mKdV equation: ut þ au2 ux þ b1 uxxx þ b2 uxxt ¼ 0;
Corresponding author. E-mail addresses: [email protected] (H. Triki), [email protected] (T. Ak), [email protected] (S. Moshokoa), [email protected] (A. Biswas). http://dx.doi.org/10.1016/j.oceaneng.2016.01.022 0029-8018/& 2016 Elsevier Ltd. All rights reserved.
ð2Þ
(iii) The improved KdV equation with power law nonlinearity: ut þ aun ux þ b1 uxxx þ b2 uxxt ¼ 0;
ð3Þ
where a, b1, b2 are arbitrary real constants. To find exact soliton solutions of Eqs. (1)–(3) we adopt an ansatz of the form uðx; tÞ ¼
A p; ðB þC cosh τ þ D sinh τ Þ
ð4Þ
where
τ ¼ μðx vtÞ n
ð1Þ
ð5Þ
Here in Eqs. (4) and (5), A is the amplitude of the soliton, μ is the inverse width and v represents the velocity of the soliton. Also B, C and D are constants to be determined, while the exponent p are unknown and will be determined later.
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Obviously, the form of ansatz (4) is more general than the ansatz of Yu (2010) because of the appearance of the arbitrary exponent p and parameter C related to the cosh function; and thus it provides a possible way to find novel solitonic solutions for nonlinear wave equations possessing all kinds of dispersive and nonlinear terms. The solution structure given by (4) includes the well-known solitary waves as well as the singular solitary waves. If B ¼ D ¼ 0, one recovers the usual solitary waves. However, for B ¼ C ¼ 0, one obtains singular solitary waves. Both of these solution structures are well known in the literature. This paper will secure solutions to IKdV equation and its hierarchies with all of these parameters being non-zero.
3. The improved KdV equation Let us first consider the improved KdV equation (1), ð6Þ
and from ansatz Eq. (4), substituting values of u, ut, ux, uxxx and uxxt into Eq. (6), we have ApvμðC sinh τ þ D cosh τ Þ ðB þ C cosh τ þ D sinh τÞ
pþ1
aA2 pμðC sinh τ þ D cosh τÞ 2p þ 1
ðB þC cosh τ þ D sinh τ Þ
ðb1 b2 vÞAμ3 p3 ðC sinh τ þ D cosh τÞ
þ
ðb1 b2 vÞABμ3 pðp þ 1Þð2p þ 1ÞðC sinh τ þ D cosh τÞ
ðB þ C cosh τ þ D sinh τÞ
ð15Þ BCD 2Av þ 2ðb1 b2 vÞAμ2 p2 Aðp þ 1Þð2p þ 1Þμ2 ðb1 b2 vÞ ¼ 0; ð16Þ Solving the above equations, we get B ¼0 and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 v ; μ¼ 2 b1 b2 v A¼
3vðC 2 D2 Þ ; a
ð18Þ
with D a 7 C. Hence the improved KdV equation (1) has the solitary wave solution 3vðC 2 D2 Þ=a ðC cosh τ þD sinh τÞ
2
;
ð19Þ
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 v ðx vtÞ: τ¼ 2 b1 b2 v
ð20Þ
It is very interesting to see that in the particular case where C ¼1 and b2 ¼ 0 in the expression (19), the solitary wave solution can be reduced to a simple form 3vð1 D2 Þ=a rffiffiffiffiffiffi rffiffiffiffiffiffi 2 : 1 v 1 v C cosh ξ þ D sinh ξ 2 b1 2 b1
pþ1
pþ2
ðB þC cosh τ þ D sinh τÞ
ðb1 b2 vÞAμ3 pðp þ 1Þðp þ 2Þ B2 C 2 þ D2 ðC sinh τ þ D cosh τÞ ðB þ C cosh τ þ D sinh τ Þ
ð17Þ
uðx; tÞ ¼
pþ3
ð21Þ
where ξ ¼ x vt, which is exactly the solitary wave solution of the celebrated KdV equation studied in Yu (2010).
pμBCD 4Av 4ðb1 b2 vÞAμ2 p2 2Aðp þ 1Þð2p þ 1Þμ2 ðb1 b2 vÞ ¼ 0;
uðx; tÞ ¼
ut þ auux þ b1 uxxx þ b2 uxxt ¼ 0;
193
¼ 0;
ð7Þ Then to determine the exponent p, we usually balance the linear terms of highest order in the resulting Eq. (7) with the highest order nonlinear terms. Thus, from matching the exponents of 2p þ 1 pþ3 1=ðB þ C cosh τ þ D sinh τÞ and 1=ðB þC cosh τ þ D sinh τÞ terms in Eq. (7), we obtain 2p þ 1 ¼ p þ 3;
ð8Þ
so that p ¼ 2:
ð9Þ
ut þ au2 ux þ b1 uxxx þ b2 uxxt ¼ 0;
ð22Þ
and from ansatz, substituting values of u, ut, ux, uxxx and uxxt into Eq. (22), we have ApvμðC sinh τ þ D cosh τÞ
aA3 pμðC sinh τ þ D cosh τÞ
ðB þ C cosh τ þ D sinh τÞ ðB þ C cosh τ þ D sinh τÞ ðb1 b2 vÞAμ3 p3 ðC sinh τ þ D cosh τÞ pþ1 ðB þ C cosh τ þ D sinh τÞ þ
ðb1 b2 vÞABμ3 pðp þ 1Þð2p þ1ÞðC sinh τ þ D cosh τÞ ðB þ C cosh τ þD sinh τÞ
pþ2
ðb1 b2 vÞAμ pðp þ 1Þðp þ 2Þ B C þ D ðC sinh τ þ D cosh τ Þ 3
3p þ 1
2
2
2
ðB þ C cosh τ þ D sinh τÞ
pþ3
¼ 0;
ð23Þ ð11Þ
Apvμ 2C 2 D þ D3 B2 D aA2 pμD ðb1 b2 vÞAμ3 p h Dðp þ 1Þðp þ 2Þ B2 C 2 þD2 þ p2 B2 D D3 2C 2 D þ ðp þ 1Þð2p þ 1ÞBD ¼ 0;
Let us now consider the improved mKdV equation (2):
pþ1
If we put p ¼2 in Eq. (7), we can determine the soliton parameters n by setting the corresponding coefficients of cosh τ (n ¼ 0; 1; 3), n sinh τ (n ¼ 1; 3) and cosh τ sinh τ, respectively, to zero. Hence we find ð10Þ Apvμ D3 þ 3C 2 D ðb1 b2 vÞAμ3 p3 3C 2 D þ D3 ¼ 0; Apvμ 3D2 C þ C 3 ðb1 b2 vÞAμ3 p3 3D2 C þ C 3 ¼ 0;
4. The improved mKdV equation
Now, from (23) equating the exponents 3p þ1 and p þ3 yields p ¼ 1;
ð24Þ pþ1
ð12Þ
Apvμ 2CD2 þ C 3 þ B2 C aA2 pμC ðb1 b2 vÞAμ3 p h p2 2CD2 þ C 3 þ B2 C þCðp þ 1Þðp þ 2Þ B2 C 2 þ D2 D i ðp þ 1Þð2p þ 1ÞB2 C ¼ 0;
ð13Þ
pμBðC 2 þ D2 Þ Av ðb1 b2 vÞAμ2 p2 þ Aðpþ 1Þð2p þ 1Þμ2 ðb1 b2 vÞ ¼ 0;
ð14Þ
, Now, noting that the functions 1=ðB þ C cosh τ þ D sinh τÞ pþ2 pþ3 and 1=ðB þ C cosh τ þ D sinh τÞ , in 1=ðB þ C cosh τ þ D sinh τÞ (23), are linearly independent, setting their respective coefficients in (23) to zero, yields the following set of relations, respectively rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v μ¼ ; ð25Þ b1 b2 v sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6vðC 2 D2 Þ ; A¼ a B ¼ 0:
ð26Þ ð27Þ
194
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
and D a 7C. Hence we get the solitary wave solution of Eq. (22), when the above parameters p, μ, A and B are substituted in Eq. (4) as n o1=2 6vðC 2 D2 Þ=a ; ð28Þ uðx; tÞ ¼ C cosh τ þ D sinh τ where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ðx vtÞ: τ¼ b1 b2 v
τ¼
n 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ðx vtÞ: b1 b2 v
and the amplitude A is related to the velocity v and the constants C and D as given by Eq. (34).
6. Numerical simulation of the improved KdV type equations ð29Þ
5. The improved KdV equation with power law nonlinearity
6.1. The application of the quintic B-spline collocation method Let us take into consideration the improved KdV equation with power law nonlinearity (3) by the following boundary conditions: Uða; tÞ ¼ 0;
It will be very interesting to assess the soliton solutions for the improved KdV-type models with power law nonlinearity. For pursuing such a goal, we consider the improved KdV equation with power law nonlinearity (3): ut þaun ux þb1 uxxx þ b2 uxxt ¼ 0;
ð30Þ
Choosing the new solitary wave ansatz (4) and substituting values of u, ut, ux, uxxx and uxxt into Eq. (30), we have ApvμðC sinh τ þ D cosh τÞ pþ1
aAn þ 1 pμðC sinh τ þ D cosh τÞ
ðB þ C cosh τ þ D sinh τÞ ðB þC cosh τ þ D sinh τÞ ðb1 b2 vÞAμ3 p3 ðC sinh τ þ D cosh τÞ pþ1 ðB þ C cosh τ þ D sinh τÞ þ
pðn þ 1Þ þ 1
ðb1 b2 vÞABμ3 pðp þ1Þð2p þ 1ÞðC sinh τ þ D cosh τÞ ðB þC cosh τ þ D sinh τÞ
ðB þ C cosh τ þ D sinh τ Þ
pþ3
Uðb; tÞ ¼ 0;
U x ða; tÞ ¼ 0;
U x ðb; tÞ ¼ 0;
Uðx; 0Þ ¼ f ðxÞa rx rb;
¼ 0;
Proceeding as before, we obtain
n 2
A¼
ð32Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ; b1 b2 v
ð33Þ !1=n
vðn þ 1Þðn þ 2ÞðC 2 D2 Þ 2a
;
B ¼ 0:
ð34Þ
ð35Þ
and D a 7 C. Thus, finally, the novel solitary wave solution of Eq. (30) is given by uðx; tÞ ¼ where
A ðC cosh τ þ D sinh τÞ
2=n
;
ð39Þ
For the numerical calculation, the solution domain of the problem is limited over a range a r x r b. The range is separated into uniformly sized finite elements of length h by the knots xm such that a ¼ x0 ox1 o ⋯ o xN ¼ b. The set of quintic B-spline functions
ϕ 2 ðxÞ; ϕ 1 ðxÞ; …; ϕN þ 1 ðxÞ; ϕN þ 2 ðxÞ forms a basis over the problem domain ½a; b. We search the numerical solution U N ðx; tÞ to the exact solution Uðx; tÞ in the form N þ2 X
ϕj ðxÞδj ðtÞ
ð40Þ
where δj ðtÞ are time dependent parameters to be determined from the boundary and collocation conditions. Quintic B-splines ϕm ðxÞ, (m ¼ 2ð1ÞN þ 2), at the knots xm are determined over the range ½a; b by Prenter (1975):
8 > ðx xm 3 Þ5 ; > > > > > ðx xm 3 Þ5 6ðx xm 2 Þ5 ; > > > > > ðx > xm 3 Þ5 6ðx xm 2 Þ5 þ 15ðx xm 1 Þ5 ; 1 < ϕm ðxÞ ¼ 5 ðx xm 3 Þ5 6ðx xm 2 Þ5 þ 15ðx xm 1 Þ5 20ðx xm Þ5 ; h > > > ðx xm 3 Þ5 6ðx xm 2 Þ5 þ 15ðx xm 1 Þ5 20ðx xm Þ5 þ 15ðx xm þ 1 Þ5 > > > > > ðx xm 3 Þ5 6ðx xm 2 Þ5 þ 15ðx xm 1 Þ5 20ðx xm Þ5 þ 15ðx xm þ 1 Þ5 6ðx xm þ 2 Þ5 > > > > : 0;
μ¼
ð38Þ
j ¼ 2
ð31Þ
2 ; n
t 40:
and the initial condition.
U N ðx; tÞ ¼
pþ2
ðb1 b2 vÞAμ3 pðp þ 1Þðp þ 2Þ B2 C 2 þ D2 ðC sinh τ þD cosh τ Þ
p¼
ð37Þ
ð36Þ
½xm 3 ; xm 2 ½xm 2 ; xm 1 ½xm 1 ; xm ½xm ; xm þ 1
ð41Þ
½xm þ 1 ; xm þ 2 ½xm þ 2 ; xm þ 3 otherwise:
Each quintic B-spline covers six elements so that each element ½xm ; xm þ 1 is covered by six B-splines. A typical finite interval ½xm ; xm þ 1 is mapped to the interval ½0; 1 by a local coordinate transformation described by hξ ¼ x xm , 0 r ξ r 1. Thus, quintic B-splines (41) in terms of ξ over ½0; 1 can be given as the following:
ϕm 2 ¼ 1 5ξ þ 10ξ2 10ξ3 þ 5ξ4 ξ5 ; ϕm 1 ¼ 26 50ξ þ 20ξ2 þ 20ξ3 20ξ4 þ 5ξ5 ; ϕm ¼ 66 60ξ2 þ 30ξ4 10ξ5 ; ϕm þ 1 ¼ 26 þ 50ξ þ 20ξ2 20ξ3 20ξ4 þ 10ξ5 ; ϕm þ 2 ¼ 1 þ 5ξ þ 10ξ2 þ 10ξ3 þ 5ξ4 5ξ5 ; ϕm þ 3 ¼ ξ5 :
ð42Þ
Substituting trial function (41) into Eq. (40), the nodal values of U; U 0 ; U 00 and U 000 at the knots xm are obtained in terms of the element parameters δm by U N ðxm ; tÞ ¼ U m ¼ δm 2 þ 26δm 1 þ 66δm þ 26δm þ 1 þ δm þ 2
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Table 1 The conserved quantities and the error norms for single solitary wave with v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼ 0, h ¼0.05 and Δt ¼ 0:05, 30 r x r 30. t
M
P
E
L2 104
L1 104
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.1117547944 1.1117547682 1.1117547791 1.1117548341 1.1117549899 1.1117554084 1.1117562922 1.1117582646 1.1117626659 1.1117725365 1.1117945631
0.1118231860 0.1118231860 0.1118231860 0.1118231860 0.1118231860 0.1118231860 0.1118231860 0.1118231860 0.1118231863 0.1118231878 0.1118231952
0.0202058914 0.0202058914 0.0202058914 0.0202058914 0.0202058914 0.0202058913 0.0202058913 0.0202058909 0.0202058889 0.0202058789 0.0202058291
0.000000 0.002588 0.004125 0.005498 0.007433 0.011890 0.023075 0.049577 0.109721 0.244394 0.544889
0.000000 0.001100 0.001876 0.004941 0.011951 0.027855 0.062402 0.139195 0.310501 0.692404 1.54381
5 ð δm 2 10δm 1 þ 10δm þ 1 þ δm þ 2 Þ; h 20 U ″m ¼ 2 ðδm 2 þ 2δm 1 6δm þ 2δm þ 1 þ δm þ 2 Þ; h 60 000 U m ¼ 3 ð δm 2 þ 2δm 1 2δm þ 1 þ δm þ 2 Þ; h U 0m ¼
ð43Þ
where the symbols 0 ;00 and 000 symbolize differentiation according to x, respectively. The splines ϕm ðxÞ and its four principle derivatives vanish outside the interval ½xm 3 ; xm þ 3 . Now we identify the collocation points with the knots and use Eq. (43) to evaluate Um, its necessary space derivatives and substitute into Eq. (3) to obtain the set of the coupled ordinary differential equations. For the linearization technique we get the following equation:
Z m ¼ U nm ¼ ðδm 2 þ 26δm 1 þ66δm þ 26δm þ 1 þ δm þ 2 Þn
M
P
E
3.589248 3.588690 3.588560 3.588745 3.589231 3.590057 3.591301 3.593080 3.595552 3.598923 3.603466
0.331677 0.331692 0.331703 0.331722 0.331755 0.331809 0.331899 0.332050 0.332310 0.332759 0.333533
0.059987 0.059938 0.059908 0.059857 0.059772 0.059633 0.059403 0.059012 0.058340 0.057175 0.055158
Fig. 2. Interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 58, x2 ¼ 23, C ¼1, D¼ 0, h ¼ 0.05 and Δt ¼ 0:1, 0 r x r 80, t ¼2.
and indicates derivative with respect to t. If time parameters δi's and its time derivatives δ_ i 's in Eq. (44) are discretized by the Crank–Nicolson formula and usual finite difference approximation, respectively: 1 2
δ_ i ¼
δi ¼ ðδn þ δn þ 1 Þ;
δn þ 1 δn Δt
ð45Þ
we obtain a recurrence relationship between two time levels n and nþ1 n þ1 relating two unknown parameters δi , δin for i ¼ m 2; m 1; …; m þ 1; m þ 2
γ 1 δnmþ12 þ γ 2 δnmþ11 þ γ 3 δnmþ 1 þ γ 4 δnmþþ11 þ γ 5 δnmþþ12 ¼ γ 5 δnmþ12 þ γ 4 δm 1 þ γ 3 δm þ γ 2 δm þ 1 þ γ 1 δm þ 2 n
n
n
ð46Þ
where
þ
where
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
n
δ_ m 2 þ 26δ_ m 1 þ 66δ_ m þ 26δ_ m þ 1 þ 2δ_ m þ 2 5aZ m ð δm 2 10δm 1 þ 10δm þ 1 þ δm þ 2 Þ h 60b1 þ 3 ð δm 2 þ2δm 1 2δm þ 1 þ δm þ 2 Þ h 20b2 _ þ 2 ðδ m 2 þ 2δ_ m 1 6δ_ m þ 2δ_ m þ 1 þ δ_ m þ 2 Þ ¼ 0 h
Table 2 The conserved quantities for interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a ¼6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 58, x2 ¼ 23, C¼1, D ¼ 0, h ¼0.05 and Δt ¼ 0:1, 0r x r 80. t
Fig. 1. Single solitary wave with amplitude ¼ 0:15000, v¼.3, a ¼6, b1 ¼ 1, b2 ¼ 0:1, C ¼1, D¼ 0, h ¼ 0.05 and Δt ¼ 0:05, 30r x r 30, 0 r t r 5.
195
ð44Þ
γ 1 ¼ ½1 EZ m M þ K ; γ 2 ¼ ½26 10EZ m þ 2M þ 2K ; γ 3 ¼ ½66 6K ; γ 4 ¼ ½26 þ 10EZ m 2M þ 2K ; γ 5 ¼ ½1 þEZ m þ M þ K ; m ¼ 0; 1; …; N;
E¼
5 aΔt; 2h
M¼
30 h
3
b1 Δt;
K¼
120 4
h
b2 :
ð47Þ
196
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Table 3 The invariants due to Maxwellian initial condition for IKdV equation with v ¼3, a ¼0.005, b1 ¼ 0:0005, h ¼0.05, Δt ¼ 0:1 and x0 ¼ 40, 0 r x r 80. b2
t
M
P
E
0.0001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817638 1.772480896817636 1.772480896817637 1.772480896817637 1.772480896817638 1.772480896817637 1.772480896817635 1.772480896817637 1.772480896817639 1.772480896817639
1.253458591164609 1.253458591164613 1.253458591164625 1.253458591164643 1.253458591164669 1.253458591164703 1.253458591164746 1.253458591164794 1.253458591164852 1.253458591164916 1.253458591164987
0.001078923031966 0.001078923031956 0.001078923031929 0.001078923031883 0.001078923031819 0.001078923031737 0.001078923031637 0.001078923031518 0.001078923031382 0.001078923031228 0.001078923031057
0.001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817638 1.772480896817638 1.772480896817637 1.772480896817638 1.772480896817639 1.772480896817638 1.772480896817638 1.772480896817638 1.772480896817638 1.772480896817638
1.254586556675805 1.254586556675802 1.254586556675798 1.254586556675789 1.254586556675777 1.254586556675764 1.254586556675744 1.254586556675722 1.254586556675697 1.254586556675669 1.254586556675637
0.001078923031966 0.001078923031957 0.001078923031931 0.001078923031887 0.001078923031826 0.001078923031747 0.001078923031652 0.001078923031539 0.001078923031409 0.001078923031262 0.001078923031098
0.01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817637 1.772480896817638 1.772480896817638 1.772480896817640 1.772480896817636 1.772480896817639 1.772480896817637 1.772480896817638 1.772480896817638 1.772480896817638
1.265866211787758 1.265866211787721 1.265866211787614 1.265866211787433 1.265866211787180 1.265866211786855 1.265866211786459 1.265866211785991 1.265866211785451 1.265866211784839 1.265866211784158
0.001078923031966 0.001078923031960 0.001078923031943 0.001078923031914 0.001078923031874 0.001078923031822 0.001078923031759 0.001078923031684 0.001078923031598 0.001078923031501 0.001078923031392
0.1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817638 1.772480896817637 1.772480896817638 1.772480896817637 1.772480896817639 1.772480896817640 1.772480896817641 1.772480896817639 1.772480896817642 1.772480896817638
1.378662762907293 1.378662762907257 1.378662762907151 1.378662762906977 1.378662762906733 1.378662762906411 1.378662762906025 1.378662762905563 1.378662762905034 1.378662762904435 1.378662762903758
0.001078923031966 0.001078923031965 0.001078923031963 0.001078923031961 0.001078923031957 0.001078923031952 0.001078923031947 0.001078923031940 0.001078923031932 0.001078923031923 0.001078923031913
Table 4 The conserved quantities and the error norms for solitary waves with v¼ 0.3, a ¼6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼0, h ¼ 0.05 and Δt ¼ 0:05, 30 r x r 30. t
M
P
E
L2 104
L1 104
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
3.1883675645 3.1883675166 3.1883675365 3.1883676369 3.1883679213 3.1883686854 3.1883702982 3.1883739167 3.1883819310 3.1883999694 3.1884400770
1.1225493997 1.1225493998 1.1225493998 1.1225493998 1.1225493998 1.1225493998 1.1225493998 1.1225494000 1.1225494010 1.1225494059 1.1225494305
0.1144137462 0.1144137462 0.1144137462 0.1144137462 0.1144137462 0.1144137461 0.1144137459 0.1144137446 0.1144137380 0.1144137047 0.1144135385
0.000000 0.013355 0.022385 0.030586 0.038860 0.048981 0.066504 0.108080 0.211139 0.452393 0.998233
0.000000 0.008080 0.011098 0.016509 0.021820 0.050856 0.113930 0.254135 0.566894 1.26415 2.81862
For the linearization technique, the term Un in non-linear term U n U x is taken as Z m ¼ U nm ¼ ðδm 2 þ 26δm 1 þ 66δm þ 26δm þ 1 þ δm þ 2 Þn :
ð48Þ
The system (46) consists of (N þ1) linear equations including (N þ 5) unknown parameters ðδ 2 ; δ 1 ; …; δN þ 1 ; δN þ 2 ÞT . To obtain
a unique solution to this system, we need four additional constraints. These are got from the boundary conditions and can be used to eliminate δ 2 ; δ 1 and δN þ 1 , δN þ 2 from the system (46) which becomes later a matrix equation for the N þ1 unknowns d ¼ ðδ0 ; δ1 ; …; δN ÞT of the form Ad
nþ1
n
¼ Bd :
ð49Þ
The matrices A and B are pentagonal ðN þ 1Þ ðN þ 1Þ matrices and so are easily solved. However, two or three inner iterations are nn n n n1 Þ at each time step to applied to the term δ ¼ δ þ 12 ðδ δ cope with the non-linearity caused by Zm. Before the solution 0 process begins iteratively, the initial vector d ¼ ðδ0 ; δ1 ; …; δN 1 ; δN Þ must be determined by using the initial condition and the following derivatives at the boundary conditions: U N ðx; 0Þ ¼ U ðxm ; 0Þ; ðU N Þx ða; 0Þ ¼ 0; ðU N Þxx ða; 0Þ ¼ 0;
m ¼ 0; 1; 2; …; N
ðU N Þx ðb; 0Þ ¼ 0; ðU N Þxx ðb; 0Þ ¼ 0;
ð50Þ 0
So we have the following matrix form of the initial vector d : 0
Wd ¼ B
ð51Þ
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Fig. 3. Solitary waves with amplitude ¼ 0:54772, v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼ 0, h ¼0.05 and Δt ¼ 0:05, 30 r x r 30, 0r t r 5. Table 5 The conserved quantities for interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 45, x2 ¼ 23, C¼ 1, D¼ 0, h ¼0.05 and Δt ¼ 0:1, 0 r x r 80. t 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
M
P
E
3.597679 3.597655 3.597687 3.597750 3.597834 3.597935 3.598054 3.598199 3.598376 3.598598 3.598877
0.348712 0.348713 0.348713 0.348713 0.348714 0.348714 0.348715 0.348716 0.348717 0.348719 0.348721
0.009807 0.009809 0.009810 0.009810 0.009810 0.009811 0.009811 0.009812 0.009814 0.009817 0.009822
where 2
54
6 25:25 6 6 6 1 6 6 6 W ¼6 6 6 6 6 6 4
6
67:50
26:25
1
26
66
26
1
1
26
66
26
1
⋱ 1
26
66
26
1
26:25
67:50
6
60
7 7 7 7 7 7 7 7 7 7 1 7 7 7 25:25 5 54
a ib ; a þ ib
ð53Þ
where a ¼ γ 3 þ ðγ 2 þ γ 4 Þ cos ½hk þ γ 1 þ γ 5 cos ½2hk; b ¼ ðγ 4 γ 2 Þ sin ½hk þ ðγ 5 γ 1 Þ sin ½2hk:
ð54Þ
The modulus of j gj is 1, for this reason the linearized scheme is unconditionally stable.
and the error norm L1
6.2. Stability analysis
L1 ¼ J U exact U N J 1 Cmax U exact ðU N Þj ; j j
ð52Þ
j ¼ 1; 2; …; N 1:
ð56Þ
the improved KdV equation with power law nonlinearity satisfies three conservation laws given by mass (M), momentum (P), and energy (E) (Sanchez, 2015): Z
The stability analysis is based on the von Neumann approach in which the growth factor g of the error in a typical mode of n amplitude δ^ ,
δnj ¼ δ^ eijkh
g¼
J¼1
n
where k is the mode number and h is the element size, is determined from a linearization of the numerical scheme. In order to apply the stability analysis, the generalized improved KdV equation can be linearized by supposing that the quantity Un in the nonlinear term U n U x is locally constant. Replacing mode (52) into (46) gives the growth factor g of the form
In this section, we take into consideration the motion of single solitary wave solution for three test problems. Accuracy and efficiency of the method is reckoned by the error norm L2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N
2 u X exact
exact ð55Þ L2 ¼ J U U N J 2 C th ðU N Þj ;
U j
d ¼ δ0 ; δ1 ; …; δN 1 ; δN and B ¼ ½U ðx0 ; 0Þ; U ðx1 ; 0Þ; …; U ðxN 1 ; 0Þ; U ðxN ; 0ÞT . This matrix system can be solved efficiently by using a variant of Thomas algorithm. 0
Fig. 4. Interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 45, x2 ¼ 23, C ¼1, D¼ 0, h ¼ 0.05 and Δt ¼ 0:1, 0 r x r 80, t ¼2.
6.3. Numerical experiments and discussion
3
60
197
M¼
Z
b
U dx; a
b
P¼ a
½U 2 þ b2 ðU x Þ2 dx;
Z E¼ a
b
(
) 2aU n þ 2 b1 ðU x Þ2 dx ðn þ 1Þðn þ 2Þ
ð57Þ In the simulation of solitary wave motion, the invariants M, P and E are observed to check the conversation of the numerical algorithm.
198
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Table 6 The invariants due to Maxwellian initial condition for ImKdV equation with v¼3, a¼ 0.005, b1 ¼ 0:0005, h ¼ 0.05, Δt ¼ 0:1 and x0 ¼ 40, 0 r x r 80. b2
t
M
P
E
0.0001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817643 1.772480896817656 1.772480896817676 1.772480896817707 1.772480896817746 1.772480896817794 1.772480896817850 1.772480896817914 1.772480896817987 1.772480896818067
1.253458591164609 1.253458591164632 1.253458591164704 1.253458591164821 1.253458591164985 1.253458591165195 1.253458591165453 1.253458591165754 1.253458591166099 1.253458591166490 1.253458591166923
0.000111886200788 0.000111886200752 0.000111886200646 0.000111886200468 0.000111886200220 0.000111886199902 0.000111886199515 0.000111886199060 0.000111886198537 0.000111886197948 0.000111886197295
0.001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817642 1.772480896817652 1.772480896817675 1.772480896817708 1.772480896817745 1.772480896817789 1.772480896817843 1.772480896817909 1.772480896817978 1.772480896818058
1.254586556675805 1.254586556675806 1.254586556675812 1.254586556675822 1.254586556675835 1.254586556675852 1.254586556675875 1.254586556675899 1.254586556675928 1.254586556675960 1.254586556675996
0.000111886200788 0.000111886200754 0.000111886200653 0.000111886200486 0.000111886200252 0.000111886199951 0.000111886199585 0.000111886199155 0.000111886198660 0.000111886198102 0.000111886197482
0.01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817641 1.772480896817652 1.772480896817668 1.772480896817694 1.772480896817723 1.772480896817760 1.772480896817804 1.772480896817856 1.772480896817913 1.772480896817976
1.265866211787758 1.265866211787634 1.265866211787260 1.265866211786638 1.265866211785767 1.265866211784650 1.265866211783285 1.265866211781676 1.265866211779823 1.265866211777727 1.265866211775392
0.000111886200788 0.000111886200767 0.000111886200705 0.000111886200601 0.000111886200455 0.000111886200269 0.000111886200041 0.000111886199772 0.000111886199462 0.000111886199112 0.000111886198722
0.1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817638 1.772480896817642 1.772480896817644 1.772480896817654 1.772480896817661 1.772480896817678 1.772480896817696 1.772480896817713 1.772480896817733 1.772480896817752
1.378662762907293 1.378662762907139 1.378662762906681 1.378662762905905 1.378662762904825 1.378662762903437 1.378662762901747 1.378662762899748 1.378662762897442 1.378662762894822 1.378662762891891
0.000111886200788 0.000111886200785 0.000111886200778 0.000111886200767 0.000111886200751 0.000111886200730 0.000111886200704 0.000111886200674 0.000111886200639 0.000111886200600 0.000111886200556
Table 7 The conserved quantities and the error norms for solitary waves with v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C¼1, D ¼0, h ¼ 0.05 and Δt ¼ 0:05 for n¼ 3 and 5, 30 r x r 30. n
t
M
P
E
L2 104
L1 104
n ¼3
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
4.1242880435 4.1242879885 4.1242880113 4.1242881269 4.1242884540 4.1242893328 4.1242911943 4.1242954877 4.1243045021 4.1243254144 4.1243711541
2.0383679294 2.0383679293 2.0383679293 2.0383679293 2.0383679293 2.0383679293 2.0383679294 2.0383679296 2.0383679309 2.0383679374 2.0383679699
0.0938208891 0.0938208890 0.0938208890 0.0938208890 0.0938208890 0.0938208890 0.0938208886 0.0938208869 0.0938208782 0.0938208341 0.0938206144
0.000000 0.043633 0.072262 0.102453 0.133682 0.166084 0.201782 0.250370 0.347399 0.590793 1.18948
0.000000 0.026164 0.043693 0.062457 0.079513 0.095912 0.131036 0.292292 0.652010 1.45396 3.24186
n ¼5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
4.6916975075 4.6916974397 4.6916974543 4.6916975670 4.6916979034 4.6916988233 4.6917008058 4.6917058112 4.6917144690 4.6917372083 4.6917850253
2.8255684288 2.8255684122 2.8255683956 2.8255683790 2.8255683624 2.8255683458 2.8255683293 2.8255683130 2.8255682979 2.8255682886 2.8255683083
0.0791877278 0.0791877330 0.0791877380 0.0791877431 0.0791877482 0.0791877533 0.0791877588 0.0791877658 0.0791877808 0.0791878351 0.0791880861
0.000000 0.176929 0.341153 0.515784 0.699270 0.892441 1.09653 1.31435 1.55559 1.85599 2.35948
0.000000 0.125603 0.223654 0.323375 0.425967 0.530855 0.644002 0.760441 0.884985 1.53766 3.42844
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Fig. 5. Solitary waves with amplitude ¼ 0:79370, v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼ 0, h ¼0.05 and Δt ¼ 0:05 for n¼ 3, 30 r x r 30, 0 r t r 5.
6.4. The improved KdV equation 6.4.1. The motion of single solitary wave For this problem, the improved KdV Eq. (1) is considered for the boundary conditions U-0 as x- 7 1 and the initial condition
Fig. 6. Solitary waves with amplitude ¼ 1:00981, v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼0, h ¼ 0.05 and Δt ¼ 0:05 for n ¼ 5, 30 r x r 30, 0 r t r 5.
6.4.2. The interaction of two solitary waves In this section, we have studied the interaction of two well separated solitary waves by using the following initial condition: Uðx; 0Þ ¼
3vðC 2 D2 Þ=a Uðx; 0Þ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : 1 v 1 v x þ D sinh x C cosh 2 b1 b2 v 2 b1 b2 v ð58Þ Note that the solitary wave solution of the equation can be written as 3vðC 2 D2 Þ=a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : 1 v 1 v C cosh ðx vtÞ þ D sinh ðx vtÞ 2 b1 b2 v 2 b1 b2 v
uðx; tÞ ¼
ð59Þ We have used the parameters v¼ 0.3, a ¼6, b1 ¼ 1, b2 ¼ 0:1, C ¼1, D ¼0, h ¼0.05 and Δt ¼ 0:05 over the interval ½ 30; 30 to coincide with solitary wave solution of the improved KdV equation. So, the solitary wave has an amplitude 0.15000 and the computations are done up to time t ¼5 to obtain the invariants and error norms L2 and L1 at various times. Error norms L2, L1 and three invariants of the improved KdV equation are listed in Table 1. It is seen from the table that the error norms are found to be small enough. The percentage of the relative error of the invariants M, P and E are calculated with respect to the conserved quantities at t ¼0. Percentage of relative changes of M, P and E are found to be 3:58 10 3 %, 8:24 10 6 % and 3:08 10 4 %, respectively. Thus, the invariants remain almost constant as the time progress. Also, Table 1 exhibits a comparison of the values of the invariants and error norms obtained by the present method. Fig. 1 shows the motion of solitary wave at various time levels. It is observed that the soliton moves to the right at a constant speed and preserves its amplitude and shape with the increasing time, as expected.
199
2 X i ¼ 1 ½C
Ai coshðκ i ðx xi ÞÞ þ D sinhðκ i ðx xi ÞÞ2
where Ai ¼ 3vi ðC 2 D2 Þ=a, κ i ¼ 12
:
ð60Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi vi , vi, xi, i¼ 1,2 are arbitrary b1 b2 vi
constants. Eq. (56) represents two solitary waves having different amplitudes at the same direction. We have considered the parameters v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 58, x2 ¼ 23, C ¼1, D ¼0, h¼ 0.05 and Δt ¼ 0:1 over the interval ½0; 80. The simulation is done up to t¼ 2 and the values of the conserved quantities M, P and E are listed in Table 2. It is clear that the obtained values of invariants are almost constant during the computer run. Fig. 2 illustrates the interaction of two solitary waves at t ¼2.
6.4.3. Birth of solitons We consider the improved KdV Eq. (1) with the following Maxwellian initial condition: uðx; 0Þ ¼ expð ðx 40Þ2 Þ:
ð61Þ
In this case, the behavior of the solution depends on the values of b2 . Therefore, we chose the values of b2 ¼ 0:0001, b2 ¼ 0:001, b2 ¼ 0:01 and b2 ¼ 0:1. The other parameters are v ¼3, a ¼0.005, b1 ¼ 0:0005, h¼ 0.05, Δt ¼ 0:1 and x0 ¼ 40 over the interval ½0; 80. The numerical computations are done up to t ¼5. The values of the conserved quantities of motion for different b2 are presented in Table 3. The maximum relative changes of the conserved quantities from their initial values are less than 5:01 10 14 %, 2:84 10 10 % and 8:43 10 8 %, respectively.
200
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
Table 8 The conserved quantities for interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a ¼6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 38, x2 ¼ 20, C¼ 1, D ¼ 0, h ¼ 0.05 and Δt ¼ 0:1 for n ¼3 and 5, 0 r x r 80. t
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
n¼ 3
n¼ 5
M
P
E
M
P
E
3.595651 3.595656 3.595759 3.595909 3.596078 3.596251 3.596423 3.596595 3.596770 3.596952 3.597144
0.369841 0.369847 0.369849 0.369851 0.369854 0.369857 0.369860 0.369863 0.369867 0.369870 0.369874
0.013433 0.013451 0.013454 0.013455 0.013455 0.013455 0.013455 0.013456 0.013456 0.013456 0.013456
3.595651 3.595656 3.595759 3.595909 3.596078 3.596251 3.596423 3.596595 3.596770 3.596952 3.597144
0.369841 0.369847 0.369849 0.369851 0.369854 0.369857 0.369860 0.369863 0.369867 0.369870 0.369874
0.013981 0.013999 0.014002 0.014003 0.014003 0.014003 0.014004 0.014004 0.014004 0.014004 0.014004
Fig. 8. Interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 38, x2 ¼ 20, C¼ 1, D ¼0, h ¼ 0.05 and Δt ¼ 0 for n¼ 5, 0 r x r 80, t ¼2.
wave solution of the improved mKdV equation. Hence, the solitary wave has amplitude 0.54772 and the simulations are run up to time t ¼5 to obtain the invariants and the error norms at several times. Error norms L2 and L1 and conserved quantities are reported in Table 4. It can be easily seen from the table that the obtained error norms are small. The agreement between numerical and analytic solution is excellent. Percentage of relative changes of M, P and E are found to be 2:27 10 3 %, 2:73 10 6 % and 1:81 10 4 %, respectively. Perspective views of the traveling solitons are graphed at diverse time levels in Fig. 3. 6.5.2. The interaction of two solitary waves To study interaction of two solitary waves we use the boundary conditions U-0 as x- 71 and the initial condition
Fig. 7. Interaction of two solitary waves with v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 38, x2 ¼ 20, C¼ 1, D ¼ 0, h ¼ 0.05 and Δt ¼ 0:1 for n¼ 3, 0 r x r 80, t ¼2.
6.5. The improved mKdV equation 6.5.1. The motion of single solitary wave For this problem, the improved mKdV Eq. (2) is considered for the boundary conditions U-0 as x- 7 1 and the initial condition n
o1=2 6vðC 2 D2 Þ=a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Uðx; 0Þ ¼ v v C cosh x þ D sinh x b1 b2 v b1 b2 v
ð62Þ
Note that the solitary wave solution of the equation can be written as n o1=2 6vðC 2 D2 Þ=a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : uðx; tÞ ¼ v v C cosh ðx vtÞ þ D sinh ðx vtÞ b1 b2 v b1 b2 v ð63Þ The parameters v ¼0.3, a ¼6, b1 ¼ 1, b2 ¼ 0:1, C¼ 1, D ¼0, h¼0.05 and Δt ¼ 0:05 with interval ½ 30; 30 are taken to compare the results obtained by the present method with solitary
2 X
Ai : ð64Þ C cosh½ κ ðx x Þ þ D sinh ½κ i ðx xi Þ i i i¼1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Ai ¼ ½6vi ðC 2 D2 Þ=a1=2 , κ i ¼ b1 vib2 vi , vi, xi, i¼1,2 are arbiUðx; 0Þ ¼
trary constants. The initial condition (59) represents two solitary waves, one with amplitude A1 placed initially at x ¼ x1 and the second with amplitude A2 placed at x ¼ x2 . For the numerical simulation, we choose the parameters v1 ¼ 0:17, v2 ¼ 0:34, a¼ 6, b1 ¼ 3, b2 ¼ 1, x1 ¼ 45, x2 ¼ 23, C ¼1, D ¼0, h ¼0.05 and Δt ¼ 0:1 over the interval ½0; 80. The experiments are run from t¼0 to t¼2 and the values of the invariant quantities M, P and E are reported in Table 5. It is observed that the numerical values of invariants remain almost constant during the computer run. Fig. 4 illustrates the interaction of two solitary waves at t ¼2. 6.5.3. Birth of solitons The improved mKdV Eq. (2) with development of the Maxwellian initial condition uðx; 0Þ ¼ expð ðx 40Þ2 Þ;
ð65Þ
into a train of solitary waves is considered. For the Maxwellian initial condition, behavior of the solution depends on the values of b2. So, we take b2 ¼ 0:0001, b2 ¼ 0:001, b2 ¼ 0:01 and b2 ¼ 0:1. The other parameters are v ¼ 3, a ¼0.005, b1 ¼ 0:0005, h¼0.05, Δt ¼ 0:1 and x0 ¼ 40 over the interval ½0; 80. The
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
201
Table 9 The invariants due to Maxwellian initial condition for IKdV equation with power law nonlinearity v ¼3, a¼ 0.005, b1 ¼ 0:0005, h ¼ 0.05, Δt ¼ 0:1 and x0 ¼ 40 for n¼ 3, 0 r x r 80. b2
t
n ¼3 M
P
E
0.0001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817657 1.772480896817705 1.772480896817790 1.772480896817912 1.772480896818065 1.772480896818252 1.772480896818469 1.772480896818719 1.772480896818997 1.772480896819305
1.253458591164609 1.253458591164670 1.253458591164856 1.253458591165162 1.253458591165591 1.253458591166138 1.253458591166802 1.253458591167577 1.253458591168463 1.253458591169454 1.253458591170547
0.000230308728809 0.000230308728877 0.000230308729080 0.000230308729417 0.000230308729887 0.000230308730488 0.000230308731216 0.000230308732068 0.000230308733041 0.000230308734130 0.000230308735330
0.001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817654 1.772480896817704 1.772480896817788 1.772480896817907 1.772480896818052 1.772480896818231 1.772480896818445 1.772480896818690 1.772480896818959 1.772480896819259
1.254586556675805 1.254586556675824 1.254586556675883 1.254586556675982 1.254586556676119 1.254586556676297 1.254586556676511 1.254586556676761 1.254586556677047 1.254586556677368 1.254586556677722
0.000230308728809 0.000230308728872 0.000230308729062 0.000230308729376 0.000230308729815 0.000230308730376 0.000230308731057 0.000230308731855 0.000230308732767 0.000230308733790 0.000230308734919
0.01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817651 1.772480896817689 1.772480896817755 1.772480896817845 1.772480896817958 1.772480896818100 1.772480896818264 1.772480896818455 1.772480896818669 1.772480896818908
1.265866211787758 1.265866211787555 1.265866211786946 1.265866211785932 1.265866211784513 1.265866211782693 1.265866211780473 1.265866211777859 1.265866211774853 1.265866211771459 1.265866211767683
0.000230308728809 0.000230308728845 0.000230308728953 0.000230308729132 0.000230308729383 0.000230308729705 0.000230308730097 0.000230308730559 0.000230308731091 0.000230308731691 0.000230308732359
0.1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817654 1.772480896817704 1.772480896817788 1.772480896817907 1.772480896818052 1.772480896818231 1.772480896818445 1.772480896818690 1.772480896818959 1.772480896819259
1.254586556675805 1.254586556675824 1.254586556675883 1.254586556675982 1.254586556676119 1.254586556676297 1.254586556676511 1.254586556676761 1.254586556677047 1.254586556677368 1.254586556677722
0.000230308728809 0.000230308728872 0.000230308729062 0.000230308729376 0.000230308729815 0.000230308730376 0.000230308731057 0.000230308731855 0.000230308732767 0.000230308733790 0.000230308734919
numerical computations are done up to t¼ 5. The numerical values of the conserved quantities during the simulations are given in Table 6 for different values of b2 . The maximum relative changes of the conserved quantities from their initial values are less than 2:42 10 11 %, 1:12 10 9 % and 3:12 10 6 %, respectively.
Note that the solitary wave solution of the equation can be written as uðx; tÞ ¼
ð2aÞ1=n
½vðn þ 1Þðn þ 2ÞðC 2 D2 Þ1=n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=n : n v n v C cosh ðx vtÞ þ D sinh ðx vtÞ 2 b1 b2 v 2 b1 b2 v
ð67Þ Finally, the parameters v¼ 0.3, a¼ 6, b1 ¼ 1, b2 ¼ 0:1, C ¼ 1, D¼ 0, h¼ 0.05 and Δt ¼ 0:05 with interval ½ 30; 30 are chosen to compare the results obtained by the present method with solitary 6.6.1. The motion of single solitary wave wave solution of the improved KdV equation with power law For this problem, the improved mKdV equation with power law nonlinearity for n ¼3,5. Therefore, the solitary waves have amplinonlinearity (3) is considered for the boundary conditions U-0 as tudes 0:79370, 1.00981 for n ¼3,5 respectively. The experiments x- 71 and the initial condition are run from the time t ¼0 to the time t ¼5 to obtain the invariants and the error norms L2 and L1 at different times. Error norms L2 ½vðn þ 1Þðn þ 2ÞðC 2 D2 Þ1=n Uðx; 0Þ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=n : and L1 and conserved quantities are listed in Table 7 for n ¼3,5. It n v n v is clearly seen from the table that the error norms obtained by the x þ D sinh x ð2aÞ1=n C cosh 2 b1 b2 v 2 b1 b2 v present method are small enough. The agreement between ð66Þ numerical and analytic solutions are perfect. Percentage of relative 6.6. The improved KdV equation with power law nonlinearity
202
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Table 10 The invariants due to Maxwellian initial condition for IKdV equation with power law nonlinearity v¼ 3, a¼ 0.005, b1 ¼ 0:0005, h ¼ 0.05, Δt ¼ 0:1 and x0 ¼ 40 for n ¼ 5, 0 r x r 80. b2
t
n¼ 5 M
P
E
0.0001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817713 1.772480896817928 1.772480896818291 1.772480896818793 1.772480896819427 1.772480896820184 1.772480896821053 1.772480896822032 1.772480896823103 1.772480896824264
1.253458591164609 1.253458591164811 1.253458591165414 1.253458591166407 1.253458591167776 1.253458591169500 1.253458591171556 1.253458591173916 1.253458591176550 1.253458591179429 1.253458591182526
0.000467139218580 0.000467139218723 0.000467139219148 0.000467139219850 0.000467139220817 0.000467139222036 0.000467139223489 0.000467139225158 0.000467139227023 0.000467139229064 0.000467139231259
0.001
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817709 1.772480896817918 1.772480896818265 1.772480896818747 1.772480896819355 1.772480896820083 1.772480896820929 1.772480896821881 1.772480896822928 1.772480896824064
1.254586556675805 1.254586556675914 1.254586556676240 1.254586556676780 1.254586556677527 1.254586556678469 1.254586556679598 1.254586556680896 1.254586556682354 1.254586556683954 1.254586556685682
0.000467139218580 0.000467139218710 0.000467139219096 0.000467139219734 0.000467139220617 0.000467139221733 0.000467139223071 0.000467139224614 0.000467139226347 0.000467139228253 0.000467139230315
0.01
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.772480896817638 1.772480896817685 1.772480896817835 1.772480896818081 1.772480896818426 1.772480896818865 1.772480896819402 1.772480896820028 1.772480896820749 1.772480896821554 1.772480896822450 1.772480896817638 1.772480896817650 1.772480896817690 1.772480896817753 1.772480896817851 1.772480896817972 1.772480896818118 1.772480896818292 1.772480896818496 1.772480896818722 1.772480896818980
1.265866211787758 1.265866211787452 1.265866211786538 1.265866211785015 1.265866211782889 1.265866211780170 1.265866211776864 1.265866211772981 1.265866211768534 1.265866211763535 1.265866211758000 1.378662762907293 1.378662762906997 1.378662762906126 1.378662762904673 1.378662762902643 1.378662762900027 1.378662762896830 1.378662762893052 1.378662762888696 1.378662762883758 1.378662762878247
0.000467139218580 0.000467139218643 0.000467139218830 0.000467139219142 0.000467139219577 0.000467139220134 0.000467139220810 0.000467139221605 0.000467139222515 0.000467139223538 0.000467139224669 0.000467139218580 0.000467139218585 0.000467139218599 0.000467139218622 0.000467139218655 0.000467139218697 0.000467139218748 0.000467139218808 0.000467139218878 0.000467139218957 0.000467139219045
0.1
changes of M, P and E are found to be 2:02 10 3 %, 2:93 10 4 % for n ¼3, 1:87 10 3 %, 1:99 10 6 %, 4:26 10 6 %, 4:52 10 4 % for n ¼5, respectively. The profiles of the solitary wave at different time levels are shown in Figs. 5 and 6 for n ¼3,5. 6.6.2. The interaction of two solitary waves We consider the improved KdV equation with power law nonlinearity (3) for the boundary conditions U-0 as x- 7 1 and the initial condition 2 X
Ai : ð68Þ ½C coshð κ ðx x ÞÞ þ D sinhðκ i ðx xi ÞÞ2=n i i i¼1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=n where Ai ¼ f½vi ðn þ 1Þðn þ 2ÞðC 2 D2 Þ=2ag , κ i ¼ n2 b1 vib2 vi , vi, xi , Uðx; 0Þ ¼
i¼1,2 are arbitrary constants. Eq. (62) represents two solitary waves having different amplitudes at the same direction. We have chosen x1 ¼ 45, x2 ¼ 23 for n ¼3, x1 ¼ 38, x2 ¼ 20 for n¼ 5. The other parameters are v1 ¼ 0:17, v2 ¼ 0:34, a ¼6, b1 ¼ 3, b2 ¼ 1, C ¼1, D¼ 0, h ¼0.05 and Δt ¼ 0:1 over the interval ½0; 80. The
computer program has run to time t¼ 2. To record the conservative quantities of the invariants M, P and E, the calculated values are given in Table 8 for n¼ 3,5. It is clear that the obtained values of invariants are almost constant during the computer run. Figs. 7 and 8 illustrate the interaction of two solitary waves at t¼2 for n ¼ 3; 5. 6.6.3. Birth of solitons Finally, we have studied the improved KdV equation with power law nonlinearity (3) with the development of the Maxwellian initial condition uðx; 0Þ ¼ expð ðx 40Þ2 Þ;
ð69Þ
into a train of solitary waves. As it is known, with the Maxwellian initial condition (63), the behavior of the solution depends on the values of b2 . We study each of the following cases: b2 ¼ 0:0001, b2 ¼ 0:001, b2 ¼ 0:01 and b2 ¼ 0:1. The other parameters are v¼ 3, a ¼0.005, b1 ¼ 0:0005, h¼ 0.05, Δt ¼ 0:1 and x0 ¼ 40 over the interval ½0; 80. The numerical computations are done up to t¼ 5. The obtained numerical values of the invariants of motion for
H. Triki et al. / Ocean Engineering 114 (2016) 192–203
different b2 are given in Tables 9 and 10, for n ¼3, 5 respectively. The maximum relative changes of the conserved quantities from their initial values are less than 9:41 10 11 %, 1:59 10 9 % and 2:83 10 6 % for n ¼3, 3:74 10 10 %, 2:35 10 9 % and 2:71 10 6 % for n ¼5, respectively. Consequently, as seen from the four cases for n ¼1, 2, 3 and 5, the changes of the invariants are reasonably small.
7. Conclusion This paper studied shallow water waves with IKdV equation, mIKdV equation and finally the IKdV equation with power law nonlinearity. Solitary wave solutions are obtained by ansatz method that is not of the usual norm. There are several numerical simulations that are obtained in this paper along with the interaction of solitary waves. The numerical simulations employed quintic B-spline collocation method. The results of this paper come with a lot of encouragement for further future investigations in this paper. Later soliton solutions of this equation, belonging to the type discussed in this paper, will be computed for the perturbed KdV equation. There are several perturbation terms that are studied in this context. These perturbation terms will be included. The solution structure will be considered for vector coupled KdV equation as well that is studied in the context of two or three layered shallow water wave dynamics. The results of that research are awaited at this time. This is just a tip of the iceberg.
Acknowledgment The author, Turgut Ak, is grateful to The Scientific and Technological Research Council of Turkey for granting scholarship for Ph.D. studies.
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