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Perturbation of dispersive shallow water waves
Ocean Engineering 63 (2013) 1–7
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Perturbation of dispersive shallow water waves Polina Razborova a, Houria Triki b, Anjan Biswas a,n a b
Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria
a r t i c l e i n f o
abstract
Article history: Received 23 September 2012 Accepted 12 January 2013 Available online 1 March 2013
This paper addresses the dynamics of dispersive shallow water wave that is governed by the Rosenau– KdV equation with power law nonlinearity. The singular 1-soliton solution is derived by the ansatz method. Subsequently, the soliton perturbation theory is applied to obtain the adiabatic parameter dynamics of the water waves. Finally, the integration of the perturbed Rosenau–KdV equation is obtained by the ansatz method as well as the semi-inverse variational principle. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Solitons Singular solitons Dispersion Integrability
1. Introduction The theory of shallow water waves is one of the very important areas of research in ocean dynamics. In this discipline, the study of solitary waves receives a lot of attention. Therefore, it is important to focus on solitary waves in a detailed manner from a mathematical point of view. There are several models that describe the dynamics of shallow water waves such as the Korteweg–de Vries (KdV) equation, Boussinesq equation, Benjamin–Bona–Mahoney equation, Kawahara equation, Peregrine equation or the regularized long wave equation and many others (Antonova and Biswas, 2009; Biswas et al., 2011; Chen et al., 2007; Drazin and Johnson, 1989; Ebadi et al., 2013; El-Ganaini, 2012a,b; El-Sabbagh and El-Ganaini, 2012; Esfahani, 2011; Girgis and Biswas, 2010; Girgis et al., 2010; Girgis and Biswas, 2011; Krishnan and Khan, 2001; Labidi and Biswas, 2011; Pan and Zhang, 2012; Wang et al., 2011; Wazwaz, 2012; Zhang, 2007; Zuo, 2009; Zuo et al., 2010). Besides these, there are the coupled vector KdV equation, coupled vector Boussinesq equation, Gear–Grimshaw model or the Bona–Chen equation that all model two-layered shallow water waves. These kind of scenarios occur, for example, in an oil spill along an ocean shore like the Exxon Valdez spill at Prince William Sound in Alaska. This paper is, however, going to focus on the dispersive shallow water waves. Again, there are several models that describe these waves. A few of them are the Rosenau–Kawahara equation, sixth order Boussinesq equation, Rosenau–KdV (R–KdV) equation. Similar to R–KdV equation, a couple of authors studied
n
Corresponding author. Tel.: þ302 857 7913; fax: þ 302 857 7054. E-mail address: [email protected] (A. Biswas).
0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.01.014
the Rosenau-RLW equation that also models the dynamics of dispersive shallow water waves, but with the equal width factor taken into account (Pan and Zhang, 2012; Zuo et al., 2010). This paper is going to shine some light on the R–KdV equation with power law nonlinearity in the presence of several perturbation terms that arise due to numerous natural phenomena along ocean shores and beaches. The integrability aspect of this equation will be addressed along with the conservation laws and soliton perturbation theory. The integration phenomena will lead to several constraint conditions that will naturally fall out during the course of derivation of the soliton solution.
2. Governing equation The dimensionless form of the R–KdV equation that is going to be studied in this paper is given by Esfahani (2011) qt þaqx þ bqxxx þcqxxxxt þkðqn Þx ¼ 0
ð1Þ
Here in (1), the dependent variable qðx,tÞ represents the shallow water wave profile while the independent variables x and t represent the spatial and temporal variables, respectively. The coefficient of a represents the drifting term, the coefficient of b is the third order dispersion and the coefficient of c represents the higher order dispersion term. Finally, the last term with k is the nonlinear term where n is the power law nonlinearity parameter and it represents the strength of nonlinearity. However, the restriction is that n a 0,1
ð2Þ
From a historical standpoint, this equation was studied earlier by a couple of authors (Esfahani, 2011; Zuo, 2009). The solitary wave solutions were obtained earlier both with regular nonlinearity
2
P. Razborova et al. / Ocean Engineering 63 (2013) 1–7
and with power law nonlinearity. Unfortunately, in 2011, a complete picture of the solitary wave parameters was not presented (Esfahani, 2011). This paper first completes the job of giving a full scenario of the soliton parameters and subsequently proceeds to venture into further detailed analysis of this equation, in the presence of perturbation terms, that is important in the context of oceanography. The topological soliton solution or shock wave solutions of this equation was studied earlier (Ebadi et al., 2013). In this paper, therefore, two types of soliton solutions will be studied. The first one is the solitary wave. Subsequently, the singular solitary wave solution will be derived using the ansatz method. These two are detailed in the next two sections.
2.1.1. Conservation laws The two conserved quantities that (1) possess are given by Esfahani (2011) 1 2 Z 1 G G A 2 n1 M¼ ð13Þ q dx ¼ 1 2 B 1 þ G 2 n1 and E¼
Z
1
ðq2 þ cq2xx Þ dx
1
1 4 G A fðn1Þ ðn þ 7Þð3n þ5Þ þ 256ðn þ 2ÞcB g 2 n1 ¼ 1 4 Bðn1Þ2 ðn þ 7Þð3n þ 5Þ þ G 2 n1 2
2.1. Solitary wave solution The solitary wave solution or 1-soliton solution of the R–KdV equation is given by Esfahani (2011) and Zuo (2009) qðx,tÞ ¼ A sech
4=ðn1Þ
½BðxvtÞ
ð4Þ
2.2. Singular soliton
and the velocity (v) of the soliton is 2
aðn1Þ4 þ 16bB ðn1Þ2
ð6Þ
ðn1Þ þ 256cB4 4
or v¼
bðn1Þ2
ð7Þ
2
4cB ðn2 þ 2n þ5Þ
ð14Þ which represent the momentum and energy of the shallow water wave, respectively. The conserved quantities are computed by using the 1-soliton solution that is given by (3).
The relation between the soliton amplitude (A) and the inverse width (B) is given by " # 2 1=ðn1Þ 2ðn þ 1Þðn þ 3Þð3n þ 1ÞbB ð5Þ A¼ kðn1Þ2 ðn2 þ 2n þ 5Þ
v¼
The singular solitary wave solution of the R–KdV equation will now be the focus in this section. It is the same method that was employed to derive the solitary wave solution earlier in 2011 (Chen et al., 2007). Singular solitons are another kind of solitary waves that appear with a singularity, typically infinite discontinuity (Wazwaz, 2012). Singular solitons can be connected to solitary waves when the center position of the solitary wave is imaginary (Drazin and Johnson, 1989). Therefore it is not out of place to address the issue of singular solitons in this paper. This paper will, however, obtain the singular soliton solution to R–KdV equation by the ansatz method (Esfahani, 2011). The starting hypothesis is therefore given by p
qðx,tÞ ¼ A csch t
Upon equating the two expressions of the velocity of the soliton leads to the constraint condition 2
bðn1Þ4 ¼ 4cB2 f16ðn þ1Þ2 bB þ aðn1Þ2 ðn2 þ2n þ5Þg
ð8Þ
and bk 40
ð15Þ
where
t ¼ BðxvtÞ
ð16Þ
and p 40
ð9Þ
for odd values of n. Now, from (8), the soliton width (B) can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31=2 2 2 2 2 2 n1 4acðn þ 2n þ 5Þ þ a2 c2 ðn2 þ 2n þ 5Þ þ16b cðn þ1Þ 5 B¼ n þ1 32bc
ð17Þ
for solitons to exist. Here, in (15) and (16), A and B are free parameters, while v is the velocity of the wave. The unknown exponent p will be determined in terms of n during the process of the derivation of the soliton solution to (1). Substituting (15) into (1), leads to 2
p
ðvabp B2 þ cvp4 B4 Þcsch t 2
þ B ðp þ1Þðp þ 2Þf2cvB2 ðp2 þ 2p þ 2Þbgcsch
ð10Þ
4
Substituting (10) into (5) leads to
pþ4
þ cvB ðp þ1Þðp þ 2Þðp þ3Þðp þ 4Þcsch
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 31=ðn1Þ 2 acðn2 þ 2nþ 5Þ þ a2 c2 ðn2 þ 2n þ 5Þ2 þ 16b cðnþ 1Þ2 ðnþ 3Þð3n þ 1Þ 6 7 7 A¼6 4 5 16ckðn þ 1Þðn2 þ 2nþ 5Þ
ð11Þ which is the amplitude of the soliton. It is easily observed from (10) and (11) that the soliton amplitude and the width are guaranteed to exist provided c40
G
ð3Þ
where n41
4
2
ð12Þ
It needs to be noted that these exact values of the soliton width and amplitude, as given by (10) and (11), respectively, were not reported in 2011 and is thus being reported for the first time, in this paper (Esfahani, 2011).
pþ2
tkA
t
n1
np
n csch t ¼ 0: ð18Þ
Now, from (18), by the aid of balancing principle, equating the exponents pn and p þ4 implies pn ¼ p þ4
ð19Þ
so that p¼
4 n1
ð20Þ
which shows that (4) must hold in order for solitons to exist. Again from (18) the linearly independent functions are pþj csch t for j ¼0,2,4. Hence, setting their respective coefficients to zero gives the same value of the soliton velocity as in (6) and (7). The relation between the soliton parameters A and B is the
P. Razborova et al. / Ocean Engineering 63 (2013) 1–7
same as in (8). Then the free parameters are the same as given by (10) and (11) with their respective constraint conditions. Thus, finally, the singular 1-soliton solution for (1) is given by qðx,tÞ ¼ A csch
4=ðn1Þ
½BðxvtÞ
ð21Þ
3
and Girgis and Biswas (2010) 1 2 G G dM EaA 2 n1 ¼ EaM ¼ 1 2 dt B þ G 2 n1
ð29Þ
which shows that MðtÞ ¼ M0 eEat
3. Soliton perturbation theory In the presence of perturbation terms, the R–KdV equation is given by Antonova and Biswas (2009) and Girgis and Biswas (2010) qt þ aqx þbqxxx þ cqxxxxt þ kðqn Þx ¼ ER
ð22Þ
where E represents the perturbation parameter and R represents the perturbation terms. When these perturbation terms are turned on, the modified conservation laws are given by Z 1 dM ¼E R dx ð23Þ dt 1 dE ¼ 2E dt
Z
1
qR dx
ð24Þ
1
which respectively represent the adiabatic variation of the soliton momentum and the soliton energy. The slow change in the soliton velocity is given by Z 2 aðn1Þ4 þ 16bB ðn1Þ2 E 1 þ xR dx ð25Þ v¼ M 1 ðn1Þ4 þ 256cB4
ð30Þ
where M0 is the initial momentum of the soliton. Hence, in the limiting case lim MðtÞ ¼ 0
ð31Þ
t-1
for a o0. This shows that the momentum adiabatically dissipates with time, in the presence of shoaling since this is a dissipative perturbation term. The adiabatic change of the energy is given by Antonova and Biswas (2009) and Girgis and Biswas (2010) ( )G 1 G 4 3 dE a 16bB 256rB ðn þ 2Þ 2 n1 ¼ 2EA2 þ 1 4 dt B ðn1Þðn þ 7Þ ðn1Þ2 ðn þ 7Þð3n þ 5Þ G þ 2 n1 ð32Þ This means that when dE=dt ¼ 0, the solitons will travel with constant energy for a stable fixed value of the width given by " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#1=2 bðn1Þð3n þ 5Þ 7 ðn1Þ ð3n þ 5ÞD B¼ ð33Þ 32rðn þ 2Þ where the discriminant D is given by 2
D ¼ b ð3n þ 5Þ4arðn þ 2Þðn þ7Þ
or v¼
bðn1Þ2 4cB
2
ðn2 þ 2n þ5Þ
þ
E M
Z
ð34Þ
whenever
1
xR dx
ð26Þ
1
b2 ð3n þ 5Þ 4 4arðn þ 2Þðn þ 7Þ
ð35Þ
and by 2
D ¼ 4arðn þ 2Þðn þ 7Þb ð3n þ5Þ 3.1. Perturbation terms In this paper, the specific perturbation terms that will be taken into consideration are given by Antonova and Biswas (2009) and Girgis and Biswas (2010) R ¼ aq þ bqxx þ gqx qxx þ dqm qx þ lqqxxx þ nqqx qxx þ sq3x þ xqx qxxxx þ Zqxx qxxx þ rqxxxx þ cqxxxxx þ kqqxxxxx
ð27Þ
So, the perturbed R–KdV equation that is going to be considered in this paper is qt þ aqx þbqxxx þ cqxxxxt þ kðqn Þx
b2 ð3n þ 5Þ o 4arðn þ 2Þðn þ 7Þ
ð28Þ
Amongst these perturbations, a represents the shoaling coefficient (Chen et al., 2007) and b is coefficient of dissipation (Chen et al., 2007). The coefficient of d is the higher order nonlinear dispersion with 1 rm r 4 (Antonova and Biswas, 2009; Girgis and Biswas, 2010) while the coefficient of c represents the fifth order spatial dispersion (Antonova and Biswas, 2009; Girgis and Biswas, 2010). The coefficient of r is due to higher order stabilizing term (Antonova and Biswas, 2009; Biswas et al., 2011; Girgis and Biswas, 2010). The remaining coefficients appear in the context of Whitham hierarchy of shallow water wave equation (Antonova and Biswas, 2009; Drazin and Johnson, 1989). In the presence of these perturbation terms, the adiabatic change of the momentum is given by Antonova and Biswas (2009)
ð37Þ
If, however, D ¼0, namely
b2 ð3n þ 5Þ ¼ 4arðn þ 2Þðn þ7Þ the fixed value of the soliton width is given by bðn1Þð3n þ5Þ 1=2 B¼ 32rðn þ 2Þ
ð38Þ
ð39Þ
in which case the constraint condition
br 4 0
¼ Eðaqþ bqxx þ gqx qxx þ dqm qx þ lqqxxx þ nqqx qxx þ sq3x þ xqx qxxxx þ Zqxx qxxx þ rqxxxx þ cqxxxxx þ kqqxxxxx Þ
ð36Þ
for
ð40Þ
is valid. Also, the corresponding fixed value of the amplitude, in this case by virtue of (5), is n 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio31=ðn1Þ ðn þ 1Þðn þ 3Þð3n þ 1Þb bð3n þ 5Þ þ ð3n þ 5ÞD 5 A ¼4 ð41Þ 16krðn2 þ 2n þ 5Þðn1Þðn þ2Þ where the same analysis as in (34)–(37) holds. If, however, the discriminant vanishes by virtue of (38), the fixed value of the amplitude is then given by 1=ðn1Þ ðn þ 1Þðn þ3Þð3n þ 1Þð3n þ 5Þbb A¼ ð42Þ 16krðn2 þ 2n þ5Þðn1Þðn þ 2Þ in which case the constraint condition reduces to bkrb 4 0 for odd values of n.
ð43Þ
4
P. Razborova et al. / Ocean Engineering 63 (2013) 1–7
For these specific perturbation terms, given by (27), the slow change in the velocity of the soliton is
For solitary wave solution to (46), the starting hypothesis is given by
2
v¼
aðn1Þ4 þ 16bB ðn1Þ2 ðn1Þ4 þ 256cB4 2 2m þ 2 G m 6 dA 16A2 B2 fnðn1Þ2 þ sðn þ 17Þg n1 þ EB6 4m þ 1 2m þ2 3ðn1Þðn þ5Þðn þ 11Þ þ 12 G n1 6 G n1 6 1 þ G n1 2 þ
4.1. Solitary waves
p
qðx,tÞ ¼ A sech ½BðxvtÞ
where p is the unknown exponent that will be determined by the aid of the balancing principle. Then, by the ansatz method, the amplitude (A), width (B) and velocity (v) will be determined. Substituting (47) into (46) leads to 2
ðn1Þ2 ðn þ7Þð3n þ 5Þ 4 2 1 þ G G 7 n1 n1 2 7 4 1 5 2 þ G G n1 2 n1
4
n1
2 2
ð44Þ
4cB2 ðn2 þ 2n þ5Þ 2 2mþ 2 G 6 dAm 16A2 B2 fnðn1Þ2 þ sðn þ17Þg n1 EB6 4m þ 1 2m þ2 1 þ 3ðn1Þðn þ 5Þðn þ 11Þ þ G n1 2 6 G n1 6 1 þ G n1 2
t
t
np
t þðg þ l þ xp2 B2 þ Zp2 B2 2p þ kp B Þp AB sech t 2 ðp þ 1ÞAB ½gp þ ðp þ 2Þl þ 2p2 ðp þ 1ÞZB2 2p þ 2 þ 2ðp2 þ 2p þ 2Þfxp þ ðp þ2ÞkgB2 sech t 4 þ ðp þ1Þðp þ 2ÞAB fpðp þ 3Þx þpðp þ1ÞZ 2p þ 4 þ ðp þ3Þðp þ 4Þkg sech t 3p 3p þ 2 2 2 2 þ p A B ðs þ nÞ sech tpA2 B2 fps þ ðp þ1Þng sech t¼0 knA
3
sech 2
2
ð48Þ Then, from (48) equating the exponents np and p þ 4, by the aid of balancing principle, leads to the same situation as in (19) and (20). Again, as before, from the linearly independent functions, the amplitude–width relation is still given by (5) while the velocity of the soliton is given by 2
v¼
8AB2 fðn1Þð3n þ 5Þðg3lÞ16ðn þ 2Þð3xZ5kÞB2 g
ðn1Þ2 ðn þ7Þð3n þ 5Þ 3 4 2 1 þ G G 7 n1 n1 2 7 4 1 5 2 þ G G n1 2 n1
pþ4
þ ðp þ1Þðp þ 2Þðp þ3Þðp þ 4Þðcv þ cÞB sech
bðn1Þ2
þ
pþ2
þ ðp þ1Þðp þ 2Þ½b2ðp þ2p þ 2Þðcv þ cÞB2 B2 sech 2
or v¼
p
ðvabp B2 þ cvp4 B4 þ cp4 B4 Þsech t
8AB2 fðn1Þð3n þ 5Þðg3lÞ16ðn þ 2Þð3xZ5kÞB2 g
ð47Þ
aðn1Þ4 þ 16bB ðn1Þ2 256cB4
ð49Þ
ðn1Þ þ 256cB4 4
or v¼ ð45Þ
bðn1Þ2 4cðn2 þ 2n þ5ÞB2
ð50Þ
4cB2 ðn2 þ2n þ 5Þ
Then equating these two expression for velocity leads to the relation 2
2
by virtue of (25) and (26).
bðn1Þ4 ¼ 4B2 ½ðn2 þ 2n þ 5Þfðn1Þ2 ðac þ cÞ þ 16bcB g64bcB
4. Exact soliton solution of the perturbed R–KdV equation
and solving this bi-quadratic equation for the width B leads to
ð51Þ
The exact 1-soliton solution to the perturbed R–KdV equation for strong perturbation terms will be now obtained by the aid of the ansatz method. The perturbed R–KdV equation that is going to be studied in this section is given by Antonova and Biswas (2009) and Girgis and Biswas (2010)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31=2 2 2 2 2 2 2 n1 4ðac þ cÞðn þ 2n þ 5Þ þ ðac þ cÞ ðn2 þ 2n þ 5Þ þ 16b cðn þ 1Þ 5 B¼ 32bc nþ1
ð52Þ Substituting (52) into (5) leads to
qt þaqx þbqxxx þ cqxxxxt þ kðqn Þx 2
2 6 ðac þ cÞðn þ 2n þ 5Þ þ A¼6 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=ðn1Þ 2 ðac þ cÞ2 ðn2 þ 2n þ 5Þ2 þ 16b cðn þ 1Þ2 ðn þ 3Þð3n þ1Þ7 7 5 16ckðn þ1Þðn2 þ 2n þ 5Þ
¼ gqx qxx þ lqqxxx þ nqqx qxx þ sq3x þ xqx qxxxx þ Zqxx qxxx þ cqxxxxx þ kqqxxxxx
ð46Þ
where certain perturbation terms due to shoaling, stabilization and second order dispersion are removed. These removed terms are dissipative, as seen earlier, and do not permit integrability of the perturbed R–KdV equation. Eq. (46) will now be integrated by the ansatz method. The study is going to be split into the following two sections for solitary waves and singular solitons.
ð53Þ
It needs to be noted that, when c ¼ 0, the relations (51)–(53) collapse to (8), (10) and (11), respectively. The additional linearly independent functions from (48) leads to a set of constraint relations given by
sþn ¼ 0
ð54Þ
4s þðn þ 3Þn ¼ 0
ð55Þ
P. Razborova et al. / Ocean Engineering 63 (2013) 1–7
This linear system of equations given by (54) and (55) implies the unique solution s 0 ¼ ð56Þ n 0 since the determinant of the coefficient matrix is given by D ¼ n1
ð57Þ
and consequently D a0 since n a 1 as indicated in (2). Furthermore, there is another set of constraint relations that must remain valid for the integrability of (46) by the ansatz method. These are given by ðn1Þ2 f2g þðn þ 1Þlg þ2½8ðn þ 3ÞZ þ ðn2 þ 2n þ 5Þfðn þ3Þx þ 2ðn þ 1ÞkgB2 ¼ 0 ð3n þ 1Þx þðn þ 3ÞZ þ nð3n þ 1Þk ¼ 0
ð59Þ
ðn1Þ ðg þ lÞ þ16ðx þ Z þ kÞB ¼ 0
ð60Þ
This set of linear equations, given by (58)–(60) gives the solution 2 3 2 3 2 2 2 2 ðn 4n9Þ ðn þ 8n þ 3Þ 2 3 8B2ðn1Þ 8B2 ðn1Þ 3 þ 17n2 þ 29n1Þ 3 þ 17n2 þ 29n1Þ ð3n ð3n x 6 7 6 7 7 6 ðn1Þðn þ 2Þð3n þ 1Þðn2 þ 1Þ 7 6Z7 6 þ 1Þðn3 þ 4n2 þ 9n6Þ 7 6 7 ðn1Þð3n 6 7 6 2 2 3 þ 17n2 þ 29n1Þ 3 þ 17n2 þ 29n1Þ 7 6 7 6 16B 16B ð3n ð3n 6 7 6 6 7 6 k 7 ¼ 6 ðn1Þðn3 þ 9n2 þ 11n þ 11Þ 7 3 2 g þ 7 6 7l ðn1Þðn þ 9n þ 27n5Þ 6 7 6 6 16B 7 2 3 2 6 7 6 16B2 ð3n3 þ 17n2 þ 29n1Þ 7 ð3n þ 17n þ 29n1Þ 7 7 6 4g5 6 7 6 7 1 0 4 5 4 5 l 0 1 ð61Þ Thus, the system of linear equations (58)–(60) for the perturbation coefficients lead to the conclusion that the five perturbation coefficients, x, Z, k, g and l can be solved in terms of two linearly independent parameters, namely g and l only. This is because the matrix coefficient of the system of linear equations as indicated by (58)–(60) has rank 2. Hence, their solution matrix given by (61) has dimension 2. Therefore, in principle, Eq. (46) can be solved by the ansatz method provided the perturbation coefficients satisfy relations (56) and (61). 4.2. Singular solitons For singular solitons of the perturbed R–KdV equation given by (46), the starting hypothesis is still taken to be (15). Then substituting (15) into (46) gives 2
p
ðvabp B2 þcvp4 B4 þ cp4 B4 Þcsch t
4
þðp þ 1Þðp þ2Þðp þ 3Þðp þ 4Þðcvþ cÞB csch n1
pþ4
t
t
np
2p
þ kp2 B2 Þp2 AB2 csch t þðp þ 1ÞAB2 ½gp þ ðp þ2Þl þ 2p2 ðp þ1ÞZB2 þ2ðp2 þ2p þ2Þfxp þðp þ 2ÞkgB2 csch
2p þ 2
t
4
þðp þ 1Þðp þ2ÞAB fpðp þ3Þx þ pðp þ 1ÞZ 2p þ 4
2 2 2
3p
ð63Þ
As seen from (63), it is only the nonlinear dispersion and the fifth order dispersion terms that permit integrability of the perturbed R–KdV equation. The first step is to consider the traveling wave hypothesis that is taken to be qðx,tÞ ¼ gðxvtÞ ¼ gðsÞ
ð64Þ
where s ¼ xvt
ð65Þ
Substituting this traveling wave assumption into (63) and integrating once while taking the integration constant to be zero gives
d
n
00
ðvaÞgbg þ ðcvþ cÞg 0000 kg þ 2
mþ1
gm þ 1 ¼ 0
2
4
ð66Þ 4
where the notations g 00 ¼ d g=ds and g 0000 ¼ d g=ds are adopted. Now, multiplying both sides of (66) by g 0 ¼ dg=ds and integrating leads to ðvaÞg 2 bðg 0 Þ2 þðcv þ cÞf2g 0 g 000 ðg 00 Þ2 g 2k
gn þ 1 gm þ 2 þ 2d ¼K n þ1 ðm þ 1Þðm þ 2Þ
ð67Þ
where K is the integration constant. The stationary integral is then defined as Z 1 Z 1 J¼ K ds ¼ ½ðvaÞg 2 bðg 0 Þ2 þ ðcv þ cÞf2g 0 g 000 ðg 00 Þ2 g 1 1 gn þ 1 gm þ 2 þ2d ds ð68Þ 2k n þ1 ðm þ1Þðm þ2Þ
gðsÞ ¼ A sech
csch t þðg þ l þ xp2 B2 þ Zp2 B2
þðp þ 3Þðp þ4Þkgcsch
The previous section carried out the integration of the perturbed R–KdV equation by the ansatz method. In this section, there is another analytical approach that will be adopted to integrate the perturbed R–KdV equation. This is the semiinverse variational principle (SVP) (El-Ganaini, 2012a,b; El-Sabbagh and El-Ganaini, 2012; Girgis et al., 2010; Girgis and Biswas, 2011; Labidi and Biswas, 2011; Zhang, 2007). This technique, however, does not give an exact solution, unlike the ansatz method as discussed in the previous section, although this is a purely analytical approach to solve the equation. Therefore, there are fewer perturbation terms that can be incorporated in order to be able to apply the SVP. Thus, the perturbed R–KdV equation that will be considered in this section is given by Girgis et al. (2010), Girgis and Biswas (2011) and Labidi and Biswas (2011)
Now, choosing pþ2
ðp þ 1Þðp þ 2Þ½b2ðp2 þ 2p þ 2Þðcv þ cÞB2 B2 csch
knA
5. Semi-inverse variational principle
qt þaqx þ bqxxx þcqxxxxt þkðqn Þx ¼ dqm qx þ cqxxxxx
2
2
ð58Þ
5
t 3p þ 2
þp A B ðs þ nÞcsch t þ pA2 B2 fps þ ðp þ 1Þngcsch
t¼0 ð62Þ
Thus, following the same procedure as in the previous section for solitary waves, this analysis leads to exactly the same results as given by (54)–(61). The free parameters A and B are still given by (53) and (52), respectively.
4=ðn1Þ
ðBsÞ
ð69Þ
as a solution hypothesis for (63), where A and B are still the amplitude and inverse width of the soliton, respectively, the stationary integral J reduces to 2m þ 4 1 G G 2m þ 2 2dA n1 2 J¼ 2m þ4 1 ðm þ1Þðm þ2ÞB þ G n1 2 " 2 ðvaÞA2 16bA B 768ðn þ 2Þðcv þ cÞA2 B3 þ ðn1Þðn þ7Þ B ðn1Þ2 ðn þ 7Þð3n þ 5Þ #G 4 G 1 nþ1 32ðn þ 3ÞKA n1 2 ð70Þ 4 1 ðn þ 1Þðn þ 7Þð3n þ 5ÞB þ G n1 2 Then, the SVP states that the amplitude (A) and the width (B) of the perturbed R–KdV equation (63) is given by the solution of the
6
P. Razborova et al. / Ocean Engineering 63 (2013) 1–7
coupled system of equations (Girgis et al., 2010; Girgis and Biswas, 2011; Labidi and Biswas, 2011) @J ¼0 @A
ð71Þ
and @J ¼0 @B
ð72Þ
From (70), Eqs. (71) and (72) are, respectively, given by 2
16bB 768ðn þ 2Þðcv þ cÞB4 ðn1Þðn þ7Þ ðn1Þ2 ðn þ7Þð3n þ 5Þ 2mþ 4 4 1 þ G m G dA 16kðn þ 3ÞAn1 n1 n1 2 ¼0 þ 2mþ 4 1 4 m þ1 ðn þ 7Þð3n þ 5Þ þ G G n1 2 n1
va
ð73Þ
and 2
16bB 2304ðn þ2Þðcvþ cÞB4 þ ðn1Þðn þ 7Þ ðn1Þ2 ðn þ 7Þð3n þ 5Þ 2m þ4 4 1 þ G G m 2dA n1 n1 2 þ 2m þ4 1 4 ðm þ 1Þðm þ 2Þ þ G G n1 2 n1
va þ
32kðn þ 3ÞAn1 ¼0 ðn þ 1Þðn þ 7Þð3n þ5Þ
ð74Þ
after simplification. Subtracting (73) from (74) leads to the biquadratic equation for the width of the soliton as 2
256NB4 þ 32ðn1Þðn þ 7ÞbB Mðn1Þ2 ðn þ 7Þ2 ¼ 0
ð75Þ
where
2m þ4 4 1 þ G mdA 16kðn1Þðn þ3ÞAn1 n1 n1 2 M¼ 2m þ4 1 4 ðm þ 1Þðm þ 2Þ ðn þ 1Þðn þ 7Þð3n þ 5Þ þ G G n1 2 n1 ð76Þ m
G
and N¼
12ðn þ 2Þðn þ 7Þðcv þ cÞ 3n þ 5
The solution of (75) is then n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio31=2 2 2 1 4ðn1Þðn þ 7Þ b þ b þ NM 5 B¼ N 4
ð77Þ
ð78Þ
for MN 4 0
ð79Þ
and Nðb þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b þNM Þ 4 0
ð80Þ
Finally, the amplitude can be obtained from the polynomial equation (73) or (74). Also, the velocity of the soliton can also be obtained directly from either (73) or (74). Once these parameters are obtained, the soliton solution of the perturbed R–KdV equation can be written as (69), which is the analytical soliton solution of the perturbed R–KdV equation (63).
6. Conclusions This paper addresses the dynamics of dispersive solitary waves that is modeled by the R–KdV equation with power law
nonlinearity in the presence of perturbation terms. The 1soliton solution is obtained by the ansatz method for solitary waves and singular solitons. Then, the soliton perturbation theory is implemented in order to describe the adiabatic dynamics of the perturbed soliton. It was observed that the momentum dissipates in the presence of shoaling, while the solitons travel with a fixed value of the energy when the amplitude and the width of the soliton get locked to a definitive value. Finally, the integration of the perturbed R–KdV equation is carried out by the aid of two integration tools. They are the ansatz method as well as the SVP. These results lead to a set of constraint conditions that must hold in order for the soliton dynamics to hold. In this context, the future of this research holds pretty strong. These results will be extended further. The soliton solutions will be obtained with stochastic perturbation terms after integrating the corresponding Langevin equation. This will lead to a mean free value of the soliton parameters in the presence of such perturbation terms. The quasi-stationary soliton solution will also be obtained by the aid of multiple-scale perturbation theory, and finally the quasi-particle theory of soliton–soliton interaction will be developed. These just form the tip of the iceberg. From a different perspective, the R–KdV equation with power law nonlinearity can be explored further along. The cnoidal waves and other doubly periodic function solutions will be obtained by the aid of mapping method and the ansatz method. Additionally, the F-expansion method will be applied in order to reveal further spectrum of solutions. Moreover the Lie symmetry analysis will be applied to give an additional display of solutions. Finally, the exp-function method as well as the G0 =G-expansion method will imply another set of solutions. These solutions will then be connected to each other. From the numerical standpoint, this equation will be addressed by the variational iteration method and those results will be compared with the analytical solutions that are obtained by the above methods. These results will be reported in future publications.
References Antonova, M., Biswas, A., 2009. Adiabatic parameter dynamics of perturbed solitary waves. Commun. Nonlinear Sci. Numer. Simulation 14 (3), 734–748. Biswas, A., Triki, H., Labidi, M., 2011. Bright and dark solitons of the Rosenau– Kawahara equation with power law nonlinearity. Phys. Wave Phenom. 19 (1), 24–29. Chen, C.Y., Hsu, J.R., Cheng, M.H., Chen, H.H., Kuo, C.F., 2007. An investigation on internal solitary waves in a two layer fluid: propagation and reflection from steep slopes. Ocean Eng. 34 (1), 171–184. Drazin, P.G., Johnson, R.S., 1989. Solitons: An Introduction. Cambridge University Press. Ebadi, G., Mojaver, A., Triki, H., Yildirim, A., Biswas, A., 2013. Topological solitons and other solutions of the Rosenau–KdV equation with power law nonlinearity, Rom. J. Phys. 58 (1–2), 1–10. El-Ganaini, S., 2012a. Applications of He’s variational principle and the first integral method to the Gardner equation. Appl. Math. Sci. 6 (85–88), 4249–4260. El-Ganaini, S.I., 2012b. New soliton solutions of some important nonlinear systems via He’s variational principle. Int. J. Math. Anal. 6 (45–48), 2277–2286. El-Sabbagh, M.F., El-Ganaini, S.I., 2012. The He’s variational principle to the Broer– kaup (BK) and Whitham Broer–Kaup (WBK) systems. Int. Math. Forum 7 (41– 44), 2131–2141. Esfahani, A., 2011. Solitary wave solutions for generalized Rosenau–KdV equation. Commun. Theor. Phys. 55 (3), 396–398. Girgis, L., Biswas, A., 2010. Soliton perturbation theory for nonlinear wave equations. Appl. Math. Comput. 216 (7), 2226–2231. Girgis, L., Zerrad, E., Biswas, A., 2010. Solitary wave solutions of the Peregrine equation. Int. J. Oceans Oceanogr. 4 (1), 45–54. Girgis, L., Biswas, A., 2011. A study of solitary waves by He’s variational principle. Waves Random Complex Media 21 (1), 96–104. Krishnan, E.V., Khan, Q.J.A., 2001. Higher-order KdV-type equations and their stability. Int. J. Math. Math. Sci. 27 (4), 215–220. Labidi, M., Biswas, A., 2011. Application of He’s principles to Rosenau-Kawahara equation. Math. Eng. Sci. Aerospace 2 (2), 183–197. Pan, X., Zhang, L., 2012. Numerical simulation for general Rosenau-RLW equation: an average linearized conservative scheme. Math. Probl. Eng. 2012, 517818.
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Wang, M., Li, D., Cui, P., 2011. A conservative finite difference scheme for the generalized Rosenau equation. Int. J. Pure Appl. Math. 71 (4), 539–549. Wazwaz, A.M., 2012. Solitons and singular solitons for a variety of Boussinesq-like equations. Ocean Eng. 53, 1–5. Zhang, J., 2007. Variational approach to solitary wave solution of the generalized Zakharov equation. Comput. Math. Appl. 54 (7–8), 1043–1046.
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Zuo, J.-M., 2009. Solitons and periodic solutions for the Rosenau–KdV and Rosenau–Kawahara equations. Appl. Math. Comput. 215 (2), 835–840. Zuo, J.-M., Zhang, Y.-M., Zhang, T.-D., Chang, F., 2010. A new conservative difference scheme for the generalized Rosenau-RLW equation. Boundary Value Probl. 2010, Article 516260.
Contents lists available at SciVerse ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Perturbation of dispersive shallow water waves Polina Razborova a, Houria Triki b, Anjan Biswas a,n a b
Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria
a r t i c l e i n f o
abstract
Article history: Received 23 September 2012 Accepted 12 January 2013 Available online 1 March 2013
This paper addresses the dynamics of dispersive shallow water wave that is governed by the Rosenau– KdV equation with power law nonlinearity. The singular 1-soliton solution is derived by the ansatz method. Subsequently, the soliton perturbation theory is applied to obtain the adiabatic parameter dynamics of the water waves. Finally, the integration of the perturbed Rosenau–KdV equation is obtained by the ansatz method as well as the semi-inverse variational principle. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Solitons Singular solitons Dispersion Integrability
1. Introduction The theory of shallow water waves is one of the very important areas of research in ocean dynamics. In this discipline, the study of solitary waves receives a lot of attention. Therefore, it is important to focus on solitary waves in a detailed manner from a mathematical point of view. There are several models that describe the dynamics of shallow water waves such as the Korteweg–de Vries (KdV) equation, Boussinesq equation, Benjamin–Bona–Mahoney equation, Kawahara equation, Peregrine equation or the regularized long wave equation and many others (Antonova and Biswas, 2009; Biswas et al., 2011; Chen et al., 2007; Drazin and Johnson, 1989; Ebadi et al., 2013; El-Ganaini, 2012a,b; El-Sabbagh and El-Ganaini, 2012; Esfahani, 2011; Girgis and Biswas, 2010; Girgis et al., 2010; Girgis and Biswas, 2011; Krishnan and Khan, 2001; Labidi and Biswas, 2011; Pan and Zhang, 2012; Wang et al., 2011; Wazwaz, 2012; Zhang, 2007; Zuo, 2009; Zuo et al., 2010). Besides these, there are the coupled vector KdV equation, coupled vector Boussinesq equation, Gear–Grimshaw model or the Bona–Chen equation that all model two-layered shallow water waves. These kind of scenarios occur, for example, in an oil spill along an ocean shore like the Exxon Valdez spill at Prince William Sound in Alaska. This paper is, however, going to focus on the dispersive shallow water waves. Again, there are several models that describe these waves. A few of them are the Rosenau–Kawahara equation, sixth order Boussinesq equation, Rosenau–KdV (R–KdV) equation. Similar to R–KdV equation, a couple of authors studied
n
Corresponding author. Tel.: þ302 857 7913; fax: þ 302 857 7054. E-mail address: [email protected] (A. Biswas).
0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.01.014
the Rosenau-RLW equation that also models the dynamics of dispersive shallow water waves, but with the equal width factor taken into account (Pan and Zhang, 2012; Zuo et al., 2010). This paper is going to shine some light on the R–KdV equation with power law nonlinearity in the presence of several perturbation terms that arise due to numerous natural phenomena along ocean shores and beaches. The integrability aspect of this equation will be addressed along with the conservation laws and soliton perturbation theory. The integration phenomena will lead to several constraint conditions that will naturally fall out during the course of derivation of the soliton solution.
2. Governing equation The dimensionless form of the R–KdV equation that is going to be studied in this paper is given by Esfahani (2011) qt þaqx þ bqxxx þcqxxxxt þkðqn Þx ¼ 0
ð1Þ
Here in (1), the dependent variable qðx,tÞ represents the shallow water wave profile while the independent variables x and t represent the spatial and temporal variables, respectively. The coefficient of a represents the drifting term, the coefficient of b is the third order dispersion and the coefficient of c represents the higher order dispersion term. Finally, the last term with k is the nonlinear term where n is the power law nonlinearity parameter and it represents the strength of nonlinearity. However, the restriction is that n a 0,1
ð2Þ
From a historical standpoint, this equation was studied earlier by a couple of authors (Esfahani, 2011; Zuo, 2009). The solitary wave solutions were obtained earlier both with regular nonlinearity
2
P. Razborova et al. / Ocean Engineering 63 (2013) 1–7
and with power law nonlinearity. Unfortunately, in 2011, a complete picture of the solitary wave parameters was not presented (Esfahani, 2011). This paper first completes the job of giving a full scenario of the soliton parameters and subsequently proceeds to venture into further detailed analysis of this equation, in the presence of perturbation terms, that is important in the context of oceanography. The topological soliton solution or shock wave solutions of this equation was studied earlier (Ebadi et al., 2013). In this paper, therefore, two types of soliton solutions will be studied. The first one is the solitary wave. Subsequently, the singular solitary wave solution will be derived using the ansatz method. These two are detailed in the next two sections.
2.1.1. Conservation laws The two conserved quantities that (1) possess are given by Esfahani (2011) 1 2 Z 1 G G A 2 n1 M¼ ð13Þ q dx ¼ 1 2 B 1 þ G 2 n1 and E¼
Z
1
ðq2 þ cq2xx Þ dx
1
1 4 G A fðn1Þ ðn þ 7Þð3n þ5Þ þ 256ðn þ 2ÞcB g 2 n1 ¼ 1 4 Bðn1Þ2 ðn þ 7Þð3n þ 5Þ þ G 2 n1 2
2.1. Solitary wave solution The solitary wave solution or 1-soliton solution of the R–KdV equation is given by Esfahani (2011) and Zuo (2009) qðx,tÞ ¼ A sech
4=ðn1Þ
½BðxvtÞ
ð4Þ
2.2. Singular soliton
and the velocity (v) of the soliton is 2
aðn1Þ4 þ 16bB ðn1Þ2
ð6Þ
ðn1Þ þ 256cB4 4
or v¼
bðn1Þ2
ð7Þ
2
4cB ðn2 þ 2n þ5Þ
ð14Þ which represent the momentum and energy of the shallow water wave, respectively. The conserved quantities are computed by using the 1-soliton solution that is given by (3).
The relation between the soliton amplitude (A) and the inverse width (B) is given by " # 2 1=ðn1Þ 2ðn þ 1Þðn þ 3Þð3n þ 1ÞbB ð5Þ A¼ kðn1Þ2 ðn2 þ 2n þ 5Þ
v¼
The singular solitary wave solution of the R–KdV equation will now be the focus in this section. It is the same method that was employed to derive the solitary wave solution earlier in 2011 (Chen et al., 2007). Singular solitons are another kind of solitary waves that appear with a singularity, typically infinite discontinuity (Wazwaz, 2012). Singular solitons can be connected to solitary waves when the center position of the solitary wave is imaginary (Drazin and Johnson, 1989). Therefore it is not out of place to address the issue of singular solitons in this paper. This paper will, however, obtain the singular soliton solution to R–KdV equation by the ansatz method (Esfahani, 2011). The starting hypothesis is therefore given by p
qðx,tÞ ¼ A csch t
Upon equating the two expressions of the velocity of the soliton leads to the constraint condition 2
bðn1Þ4 ¼ 4cB2 f16ðn þ1Þ2 bB þ aðn1Þ2 ðn2 þ2n þ5Þg
ð8Þ
and bk 40
ð15Þ
where
t ¼ BðxvtÞ
ð16Þ
and p 40
ð9Þ
for odd values of n. Now, from (8), the soliton width (B) can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31=2 2 2 2 2 2 n1 4acðn þ 2n þ 5Þ þ a2 c2 ðn2 þ 2n þ 5Þ þ16b cðn þ1Þ 5 B¼ n þ1 32bc
ð17Þ
for solitons to exist. Here, in (15) and (16), A and B are free parameters, while v is the velocity of the wave. The unknown exponent p will be determined in terms of n during the process of the derivation of the soliton solution to (1). Substituting (15) into (1), leads to 2
p
ðvabp B2 þ cvp4 B4 Þcsch t 2
þ B ðp þ1Þðp þ 2Þf2cvB2 ðp2 þ 2p þ 2Þbgcsch
ð10Þ
4
Substituting (10) into (5) leads to
pþ4
þ cvB ðp þ1Þðp þ 2Þðp þ3Þðp þ 4Þcsch
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 31=ðn1Þ 2 acðn2 þ 2nþ 5Þ þ a2 c2 ðn2 þ 2n þ 5Þ2 þ 16b cðnþ 1Þ2 ðnþ 3Þð3n þ 1Þ 6 7 7 A¼6 4 5 16ckðn þ 1Þðn2 þ 2nþ 5Þ
ð11Þ which is the amplitude of the soliton. It is easily observed from (10) and (11) that the soliton amplitude and the width are guaranteed to exist provided c40
G
ð3Þ
where n41
4
2
ð12Þ
It needs to be noted that these exact values of the soliton width and amplitude, as given by (10) and (11), respectively, were not reported in 2011 and is thus being reported for the first time, in this paper (Esfahani, 2011).
pþ2
tkA
t
n1
np
n csch t ¼ 0: ð18Þ
Now, from (18), by the aid of balancing principle, equating the exponents pn and p þ4 implies pn ¼ p þ4
ð19Þ
so that p¼
4 n1
ð20Þ
which shows that (4) must hold in order for solitons to exist. Again from (18) the linearly independent functions are pþj csch t for j ¼0,2,4. Hence, setting their respective coefficients to zero gives the same value of the soliton velocity as in (6) and (7). The relation between the soliton parameters A and B is the
P. Razborova et al. / Ocean Engineering 63 (2013) 1–7
same as in (8). Then the free parameters are the same as given by (10) and (11) with their respective constraint conditions. Thus, finally, the singular 1-soliton solution for (1) is given by qðx,tÞ ¼ A csch
4=ðn1Þ
½BðxvtÞ
ð21Þ
3
and Girgis and Biswas (2010) 1 2 G G dM EaA 2 n1 ¼ EaM ¼ 1 2 dt B þ G 2 n1
ð29Þ
which shows that MðtÞ ¼ M0 eEat
3. Soliton perturbation theory In the presence of perturbation terms, the R–KdV equation is given by Antonova and Biswas (2009) and Girgis and Biswas (2010) qt þ aqx þbqxxx þ cqxxxxt þ kðqn Þx ¼ ER
ð22Þ
where E represents the perturbation parameter and R represents the perturbation terms. When these perturbation terms are turned on, the modified conservation laws are given by Z 1 dM ¼E R dx ð23Þ dt 1 dE ¼ 2E dt
Z
1
qR dx
ð24Þ
1
which respectively represent the adiabatic variation of the soliton momentum and the soliton energy. The slow change in the soliton velocity is given by Z 2 aðn1Þ4 þ 16bB ðn1Þ2 E 1 þ xR dx ð25Þ v¼ M 1 ðn1Þ4 þ 256cB4
ð30Þ
where M0 is the initial momentum of the soliton. Hence, in the limiting case lim MðtÞ ¼ 0
ð31Þ
t-1
for a o0. This shows that the momentum adiabatically dissipates with time, in the presence of shoaling since this is a dissipative perturbation term. The adiabatic change of the energy is given by Antonova and Biswas (2009) and Girgis and Biswas (2010) ( )G 1 G 4 3 dE a 16bB 256rB ðn þ 2Þ 2 n1 ¼ 2EA2 þ 1 4 dt B ðn1Þðn þ 7Þ ðn1Þ2 ðn þ 7Þð3n þ 5Þ G þ 2 n1 ð32Þ This means that when dE=dt ¼ 0, the solitons will travel with constant energy for a stable fixed value of the width given by " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#1=2 bðn1Þð3n þ 5Þ 7 ðn1Þ ð3n þ 5ÞD B¼ ð33Þ 32rðn þ 2Þ where the discriminant D is given by 2
D ¼ b ð3n þ 5Þ4arðn þ 2Þðn þ7Þ
or v¼
bðn1Þ2 4cB
2
ðn2 þ 2n þ5Þ
þ
E M
Z
ð34Þ
whenever
1
xR dx
ð26Þ
1
b2 ð3n þ 5Þ 4 4arðn þ 2Þðn þ 7Þ
ð35Þ
and by 2
D ¼ 4arðn þ 2Þðn þ 7Þb ð3n þ5Þ 3.1. Perturbation terms In this paper, the specific perturbation terms that will be taken into consideration are given by Antonova and Biswas (2009) and Girgis and Biswas (2010) R ¼ aq þ bqxx þ gqx qxx þ dqm qx þ lqqxxx þ nqqx qxx þ sq3x þ xqx qxxxx þ Zqxx qxxx þ rqxxxx þ cqxxxxx þ kqqxxxxx
ð27Þ
So, the perturbed R–KdV equation that is going to be considered in this paper is qt þ aqx þbqxxx þ cqxxxxt þ kðqn Þx
b2 ð3n þ 5Þ o 4arðn þ 2Þðn þ 7Þ
ð28Þ
Amongst these perturbations, a represents the shoaling coefficient (Chen et al., 2007) and b is coefficient of dissipation (Chen et al., 2007). The coefficient of d is the higher order nonlinear dispersion with 1 rm r 4 (Antonova and Biswas, 2009; Girgis and Biswas, 2010) while the coefficient of c represents the fifth order spatial dispersion (Antonova and Biswas, 2009; Girgis and Biswas, 2010). The coefficient of r is due to higher order stabilizing term (Antonova and Biswas, 2009; Biswas et al., 2011; Girgis and Biswas, 2010). The remaining coefficients appear in the context of Whitham hierarchy of shallow water wave equation (Antonova and Biswas, 2009; Drazin and Johnson, 1989). In the presence of these perturbation terms, the adiabatic change of the momentum is given by Antonova and Biswas (2009)
ð37Þ
If, however, D ¼0, namely
b2 ð3n þ 5Þ ¼ 4arðn þ 2Þðn þ7Þ the fixed value of the soliton width is given by bðn1Þð3n þ5Þ 1=2 B¼ 32rðn þ 2Þ
ð38Þ
ð39Þ
in which case the constraint condition
br 4 0
¼ Eðaqþ bqxx þ gqx qxx þ dqm qx þ lqqxxx þ nqqx qxx þ sq3x þ xqx qxxxx þ Zqxx qxxx þ rqxxxx þ cqxxxxx þ kqqxxxxx Þ
ð36Þ
for
ð40Þ
is valid. Also, the corresponding fixed value of the amplitude, in this case by virtue of (5), is n 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio31=ðn1Þ ðn þ 1Þðn þ 3Þð3n þ 1Þb bð3n þ 5Þ þ ð3n þ 5ÞD 5 A ¼4 ð41Þ 16krðn2 þ 2n þ 5Þðn1Þðn þ2Þ where the same analysis as in (34)–(37) holds. If, however, the discriminant vanishes by virtue of (38), the fixed value of the amplitude is then given by 1=ðn1Þ ðn þ 1Þðn þ3Þð3n þ 1Þð3n þ 5Þbb A¼ ð42Þ 16krðn2 þ 2n þ5Þðn1Þðn þ 2Þ in which case the constraint condition reduces to bkrb 4 0 for odd values of n.
ð43Þ
4
P. Razborova et al. / Ocean Engineering 63 (2013) 1–7
For these specific perturbation terms, given by (27), the slow change in the velocity of the soliton is
For solitary wave solution to (46), the starting hypothesis is given by
2
v¼
aðn1Þ4 þ 16bB ðn1Þ2 ðn1Þ4 þ 256cB4 2 2m þ 2 G m 6 dA 16A2 B2 fnðn1Þ2 þ sðn þ 17Þg n1 þ EB6 4m þ 1 2m þ2 3ðn1Þðn þ5Þðn þ 11Þ þ 12 G n1 6 G n1 6 1 þ G n1 2 þ
4.1. Solitary waves
p
qðx,tÞ ¼ A sech ½BðxvtÞ
where p is the unknown exponent that will be determined by the aid of the balancing principle. Then, by the ansatz method, the amplitude (A), width (B) and velocity (v) will be determined. Substituting (47) into (46) leads to 2
ðn1Þ2 ðn þ7Þð3n þ 5Þ 4 2 1 þ G G 7 n1 n1 2 7 4 1 5 2 þ G G n1 2 n1
4
n1
2 2
ð44Þ
4cB2 ðn2 þ 2n þ5Þ 2 2mþ 2 G 6 dAm 16A2 B2 fnðn1Þ2 þ sðn þ17Þg n1 EB6 4m þ 1 2m þ2 1 þ 3ðn1Þðn þ 5Þðn þ 11Þ þ G n1 2 6 G n1 6 1 þ G n1 2
t
t
np
t þðg þ l þ xp2 B2 þ Zp2 B2 2p þ kp B Þp AB sech t 2 ðp þ 1ÞAB ½gp þ ðp þ 2Þl þ 2p2 ðp þ 1ÞZB2 2p þ 2 þ 2ðp2 þ 2p þ 2Þfxp þ ðp þ2ÞkgB2 sech t 4 þ ðp þ1Þðp þ 2ÞAB fpðp þ 3Þx þpðp þ1ÞZ 2p þ 4 þ ðp þ3Þðp þ 4Þkg sech t 3p 3p þ 2 2 2 2 þ p A B ðs þ nÞ sech tpA2 B2 fps þ ðp þ1Þng sech t¼0 knA
3
sech 2
2
ð48Þ Then, from (48) equating the exponents np and p þ 4, by the aid of balancing principle, leads to the same situation as in (19) and (20). Again, as before, from the linearly independent functions, the amplitude–width relation is still given by (5) while the velocity of the soliton is given by 2
v¼
8AB2 fðn1Þð3n þ 5Þðg3lÞ16ðn þ 2Þð3xZ5kÞB2 g
ðn1Þ2 ðn þ7Þð3n þ 5Þ 3 4 2 1 þ G G 7 n1 n1 2 7 4 1 5 2 þ G G n1 2 n1
pþ4
þ ðp þ1Þðp þ 2Þðp þ3Þðp þ 4Þðcv þ cÞB sech
bðn1Þ2
þ
pþ2
þ ðp þ1Þðp þ 2Þ½b2ðp þ2p þ 2Þðcv þ cÞB2 B2 sech 2
or v¼
p
ðvabp B2 þ cvp4 B4 þ cp4 B4 Þsech t
8AB2 fðn1Þð3n þ 5Þðg3lÞ16ðn þ 2Þð3xZ5kÞB2 g
ð47Þ
aðn1Þ4 þ 16bB ðn1Þ2 256cB4
ð49Þ
ðn1Þ þ 256cB4 4
or v¼ ð45Þ
bðn1Þ2 4cðn2 þ 2n þ5ÞB2
ð50Þ
4cB2 ðn2 þ2n þ 5Þ
Then equating these two expression for velocity leads to the relation 2
2
by virtue of (25) and (26).
bðn1Þ4 ¼ 4B2 ½ðn2 þ 2n þ 5Þfðn1Þ2 ðac þ cÞ þ 16bcB g64bcB
4. Exact soliton solution of the perturbed R–KdV equation
and solving this bi-quadratic equation for the width B leads to
ð51Þ
The exact 1-soliton solution to the perturbed R–KdV equation for strong perturbation terms will be now obtained by the aid of the ansatz method. The perturbed R–KdV equation that is going to be studied in this section is given by Antonova and Biswas (2009) and Girgis and Biswas (2010)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31=2 2 2 2 2 2 2 n1 4ðac þ cÞðn þ 2n þ 5Þ þ ðac þ cÞ ðn2 þ 2n þ 5Þ þ 16b cðn þ 1Þ 5 B¼ 32bc nþ1
ð52Þ Substituting (52) into (5) leads to
qt þaqx þbqxxx þ cqxxxxt þ kðqn Þx 2
2 6 ðac þ cÞðn þ 2n þ 5Þ þ A¼6 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=ðn1Þ 2 ðac þ cÞ2 ðn2 þ 2n þ 5Þ2 þ 16b cðn þ 1Þ2 ðn þ 3Þð3n þ1Þ7 7 5 16ckðn þ1Þðn2 þ 2n þ 5Þ
¼ gqx qxx þ lqqxxx þ nqqx qxx þ sq3x þ xqx qxxxx þ Zqxx qxxx þ cqxxxxx þ kqqxxxxx
ð46Þ
where certain perturbation terms due to shoaling, stabilization and second order dispersion are removed. These removed terms are dissipative, as seen earlier, and do not permit integrability of the perturbed R–KdV equation. Eq. (46) will now be integrated by the ansatz method. The study is going to be split into the following two sections for solitary waves and singular solitons.
ð53Þ
It needs to be noted that, when c ¼ 0, the relations (51)–(53) collapse to (8), (10) and (11), respectively. The additional linearly independent functions from (48) leads to a set of constraint relations given by
sþn ¼ 0
ð54Þ
4s þðn þ 3Þn ¼ 0
ð55Þ
P. Razborova et al. / Ocean Engineering 63 (2013) 1–7
This linear system of equations given by (54) and (55) implies the unique solution s 0 ¼ ð56Þ n 0 since the determinant of the coefficient matrix is given by D ¼ n1
ð57Þ
and consequently D a0 since n a 1 as indicated in (2). Furthermore, there is another set of constraint relations that must remain valid for the integrability of (46) by the ansatz method. These are given by ðn1Þ2 f2g þðn þ 1Þlg þ2½8ðn þ 3ÞZ þ ðn2 þ 2n þ 5Þfðn þ3Þx þ 2ðn þ 1ÞkgB2 ¼ 0 ð3n þ 1Þx þðn þ 3ÞZ þ nð3n þ 1Þk ¼ 0
ð59Þ
ðn1Þ ðg þ lÞ þ16ðx þ Z þ kÞB ¼ 0
ð60Þ
This set of linear equations, given by (58)–(60) gives the solution 2 3 2 3 2 2 2 2 ðn 4n9Þ ðn þ 8n þ 3Þ 2 3 8B2ðn1Þ 8B2 ðn1Þ 3 þ 17n2 þ 29n1Þ 3 þ 17n2 þ 29n1Þ ð3n ð3n x 6 7 6 7 7 6 ðn1Þðn þ 2Þð3n þ 1Þðn2 þ 1Þ 7 6Z7 6 þ 1Þðn3 þ 4n2 þ 9n6Þ 7 6 7 ðn1Þð3n 6 7 6 2 2 3 þ 17n2 þ 29n1Þ 3 þ 17n2 þ 29n1Þ 7 6 7 6 16B 16B ð3n ð3n 6 7 6 6 7 6 k 7 ¼ 6 ðn1Þðn3 þ 9n2 þ 11n þ 11Þ 7 3 2 g þ 7 6 7l ðn1Þðn þ 9n þ 27n5Þ 6 7 6 6 16B 7 2 3 2 6 7 6 16B2 ð3n3 þ 17n2 þ 29n1Þ 7 ð3n þ 17n þ 29n1Þ 7 7 6 4g5 6 7 6 7 1 0 4 5 4 5 l 0 1 ð61Þ Thus, the system of linear equations (58)–(60) for the perturbation coefficients lead to the conclusion that the five perturbation coefficients, x, Z, k, g and l can be solved in terms of two linearly independent parameters, namely g and l only. This is because the matrix coefficient of the system of linear equations as indicated by (58)–(60) has rank 2. Hence, their solution matrix given by (61) has dimension 2. Therefore, in principle, Eq. (46) can be solved by the ansatz method provided the perturbation coefficients satisfy relations (56) and (61). 4.2. Singular solitons For singular solitons of the perturbed R–KdV equation given by (46), the starting hypothesis is still taken to be (15). Then substituting (15) into (46) gives 2
p
ðvabp B2 þcvp4 B4 þ cp4 B4 Þcsch t
4
þðp þ 1Þðp þ2Þðp þ 3Þðp þ 4Þðcvþ cÞB csch n1
pþ4
t
t
np
2p
þ kp2 B2 Þp2 AB2 csch t þðp þ 1ÞAB2 ½gp þ ðp þ2Þl þ 2p2 ðp þ1ÞZB2 þ2ðp2 þ2p þ2Þfxp þðp þ 2ÞkgB2 csch
2p þ 2
t
4
þðp þ 1Þðp þ2ÞAB fpðp þ3Þx þ pðp þ 1ÞZ 2p þ 4
2 2 2
3p
ð63Þ
As seen from (63), it is only the nonlinear dispersion and the fifth order dispersion terms that permit integrability of the perturbed R–KdV equation. The first step is to consider the traveling wave hypothesis that is taken to be qðx,tÞ ¼ gðxvtÞ ¼ gðsÞ
ð64Þ
where s ¼ xvt
ð65Þ
Substituting this traveling wave assumption into (63) and integrating once while taking the integration constant to be zero gives
d
n
00
ðvaÞgbg þ ðcvþ cÞg 0000 kg þ 2
mþ1
gm þ 1 ¼ 0
2
4
ð66Þ 4
where the notations g 00 ¼ d g=ds and g 0000 ¼ d g=ds are adopted. Now, multiplying both sides of (66) by g 0 ¼ dg=ds and integrating leads to ðvaÞg 2 bðg 0 Þ2 þðcv þ cÞf2g 0 g 000 ðg 00 Þ2 g 2k
gn þ 1 gm þ 2 þ 2d ¼K n þ1 ðm þ 1Þðm þ 2Þ
ð67Þ
where K is the integration constant. The stationary integral is then defined as Z 1 Z 1 J¼ K ds ¼ ½ðvaÞg 2 bðg 0 Þ2 þ ðcv þ cÞf2g 0 g 000 ðg 00 Þ2 g 1 1 gn þ 1 gm þ 2 þ2d ds ð68Þ 2k n þ1 ðm þ1Þðm þ2Þ
gðsÞ ¼ A sech
csch t þðg þ l þ xp2 B2 þ Zp2 B2
þðp þ 3Þðp þ4Þkgcsch
The previous section carried out the integration of the perturbed R–KdV equation by the ansatz method. In this section, there is another analytical approach that will be adopted to integrate the perturbed R–KdV equation. This is the semiinverse variational principle (SVP) (El-Ganaini, 2012a,b; El-Sabbagh and El-Ganaini, 2012; Girgis et al., 2010; Girgis and Biswas, 2011; Labidi and Biswas, 2011; Zhang, 2007). This technique, however, does not give an exact solution, unlike the ansatz method as discussed in the previous section, although this is a purely analytical approach to solve the equation. Therefore, there are fewer perturbation terms that can be incorporated in order to be able to apply the SVP. Thus, the perturbed R–KdV equation that will be considered in this section is given by Girgis et al. (2010), Girgis and Biswas (2011) and Labidi and Biswas (2011)
Now, choosing pþ2
ðp þ 1Þðp þ 2Þ½b2ðp2 þ 2p þ 2Þðcv þ cÞB2 B2 csch
knA
5. Semi-inverse variational principle
qt þaqx þ bqxxx þcqxxxxt þkðqn Þx ¼ dqm qx þ cqxxxxx
2
2
ð58Þ
5
t 3p þ 2
þp A B ðs þ nÞcsch t þ pA2 B2 fps þ ðp þ 1Þngcsch
t¼0 ð62Þ
Thus, following the same procedure as in the previous section for solitary waves, this analysis leads to exactly the same results as given by (54)–(61). The free parameters A and B are still given by (53) and (52), respectively.
4=ðn1Þ
ðBsÞ
ð69Þ
as a solution hypothesis for (63), where A and B are still the amplitude and inverse width of the soliton, respectively, the stationary integral J reduces to 2m þ 4 1 G G 2m þ 2 2dA n1 2 J¼ 2m þ4 1 ðm þ1Þðm þ2ÞB þ G n1 2 " 2 ðvaÞA2 16bA B 768ðn þ 2Þðcv þ cÞA2 B3 þ ðn1Þðn þ7Þ B ðn1Þ2 ðn þ 7Þð3n þ 5Þ #G 4 G 1 nþ1 32ðn þ 3ÞKA n1 2 ð70Þ 4 1 ðn þ 1Þðn þ 7Þð3n þ 5ÞB þ G n1 2 Then, the SVP states that the amplitude (A) and the width (B) of the perturbed R–KdV equation (63) is given by the solution of the
6
P. Razborova et al. / Ocean Engineering 63 (2013) 1–7
coupled system of equations (Girgis et al., 2010; Girgis and Biswas, 2011; Labidi and Biswas, 2011) @J ¼0 @A
ð71Þ
and @J ¼0 @B
ð72Þ
From (70), Eqs. (71) and (72) are, respectively, given by 2
16bB 768ðn þ 2Þðcv þ cÞB4 ðn1Þðn þ7Þ ðn1Þ2 ðn þ7Þð3n þ 5Þ 2mþ 4 4 1 þ G m G dA 16kðn þ 3ÞAn1 n1 n1 2 ¼0 þ 2mþ 4 1 4 m þ1 ðn þ 7Þð3n þ 5Þ þ G G n1 2 n1
va
ð73Þ
and 2
16bB 2304ðn þ2Þðcvþ cÞB4 þ ðn1Þðn þ 7Þ ðn1Þ2 ðn þ 7Þð3n þ 5Þ 2m þ4 4 1 þ G G m 2dA n1 n1 2 þ 2m þ4 1 4 ðm þ 1Þðm þ 2Þ þ G G n1 2 n1
va þ
32kðn þ 3ÞAn1 ¼0 ðn þ 1Þðn þ 7Þð3n þ5Þ
ð74Þ
after simplification. Subtracting (73) from (74) leads to the biquadratic equation for the width of the soliton as 2
256NB4 þ 32ðn1Þðn þ 7ÞbB Mðn1Þ2 ðn þ 7Þ2 ¼ 0
ð75Þ
where
2m þ4 4 1 þ G mdA 16kðn1Þðn þ3ÞAn1 n1 n1 2 M¼ 2m þ4 1 4 ðm þ 1Þðm þ 2Þ ðn þ 1Þðn þ 7Þð3n þ 5Þ þ G G n1 2 n1 ð76Þ m
G
and N¼
12ðn þ 2Þðn þ 7Þðcv þ cÞ 3n þ 5
The solution of (75) is then n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio31=2 2 2 1 4ðn1Þðn þ 7Þ b þ b þ NM 5 B¼ N 4
ð77Þ
ð78Þ
for MN 4 0
ð79Þ
and Nðb þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b þNM Þ 4 0
ð80Þ
Finally, the amplitude can be obtained from the polynomial equation (73) or (74). Also, the velocity of the soliton can also be obtained directly from either (73) or (74). Once these parameters are obtained, the soliton solution of the perturbed R–KdV equation can be written as (69), which is the analytical soliton solution of the perturbed R–KdV equation (63).
6. Conclusions This paper addresses the dynamics of dispersive solitary waves that is modeled by the R–KdV equation with power law
nonlinearity in the presence of perturbation terms. The 1soliton solution is obtained by the ansatz method for solitary waves and singular solitons. Then, the soliton perturbation theory is implemented in order to describe the adiabatic dynamics of the perturbed soliton. It was observed that the momentum dissipates in the presence of shoaling, while the solitons travel with a fixed value of the energy when the amplitude and the width of the soliton get locked to a definitive value. Finally, the integration of the perturbed R–KdV equation is carried out by the aid of two integration tools. They are the ansatz method as well as the SVP. These results lead to a set of constraint conditions that must hold in order for the soliton dynamics to hold. In this context, the future of this research holds pretty strong. These results will be extended further. The soliton solutions will be obtained with stochastic perturbation terms after integrating the corresponding Langevin equation. This will lead to a mean free value of the soliton parameters in the presence of such perturbation terms. The quasi-stationary soliton solution will also be obtained by the aid of multiple-scale perturbation theory, and finally the quasi-particle theory of soliton–soliton interaction will be developed. These just form the tip of the iceberg. From a different perspective, the R–KdV equation with power law nonlinearity can be explored further along. The cnoidal waves and other doubly periodic function solutions will be obtained by the aid of mapping method and the ansatz method. Additionally, the F-expansion method will be applied in order to reveal further spectrum of solutions. Moreover the Lie symmetry analysis will be applied to give an additional display of solutions. Finally, the exp-function method as well as the G0 =G-expansion method will imply another set of solutions. These solutions will then be connected to each other. From the numerical standpoint, this equation will be addressed by the variational iteration method and those results will be compared with the analytical solutions that are obtained by the above methods. These results will be reported in future publications.
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