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Dark solitons for a combined potential KdV and Schwarzian KdV equations with t-dependent coefficients and forcing term
Applied Mathematics and Computation 217 (2011) 8846–8851
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Dark solitons for a combined potential KdV and Schwarzian KdV equations with t-dependent coefficients and forcing term Houria Triki a, Abdul-Majid Wazwaz b,⇑ a b
Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria Department of Mathematics, Saint Xavier University, Chicago, IL 60655, United States
a r t i c l e
i n f o
Keywords: Potential KdV equation Schwarzian KdV equation Solitary wave ansatz
a b s t r a c t In this work we formally derive the dark soliton solutions for the combined potential KdV and Schwarzian KdV equations. The combined KdV and Schwarzian KdV equations with time-dependent coefficients and forcing term are then investigated to obtain dark soliton solutions. The solitary wave ansatz is used to carry out the analysis for both models. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction In past years, a growing interest has been given to the propagation of nonlinear waves in nonlinear dynamical systems. A nonlinear wave is one of the fundamental objects of nature [1]. These waves appear in a great array of contexts such as, hydrodynamics, nonlinear optics, plasmas, solid state physics, nuclear physics, and many other nonlinear phenomena. The systems are often described by nonlinear partial differential equations (NLPDEs) with constant or variable coefficients, supporting exact solutions with interesting properties which are commonly referred to as solitary waves. The existence of solitary wave solutions implies perfect balance between nonlinearity and dispersion effects which usually requires rather specific conditions and cannot be established in general [2]. They also exhibit remarkable stability and particle-like properties [3]. If they retain their identity after collisions, they are called solitons [4]. Solitons are ubiquitous in nature, appearing in diverse systems such as shallow water waves, DNA excitations, matter waves in Bose–Einstein condensates, and ultrashort pulses (or laser beams) in nonlinear optics [1–5]. Two different types of envelope solitons, bright and dark, can propagate in nonlinear dispersive media. Compared with the bright soliton which is a pulse on a zero-intensity background, the dark soliton appears as an intensity dip in an infinitely extended constant background [6,7]. From a mathematical point of view, there exists a certain class of NLPDEs that support soliton solutions in physical systems. Examples include the sine-Gordon (SG) equation, the nonlinear Schrödinger (NLS) equation, the Korteweg-de Vries (KdV) equation, the modified KdV equation, the Boussinesq equation, etc. In recent years, many powerful methods to construct exact solutions of NLPDEs have been established and developed, which lead to one of the most excited advances of nonlinear science and theoretical physics [8]. In fact, many kinds of exact soliton solutions have been obtained by using for example, the homogeneous balance principle and F-expansion method [9], the Jacobi elliptic functions method [10], the sine–cosine and tanh methods [11], the Hirota’s bilinear method, the Bäcklund transformation method, the solitary wave ansatz method [12–19], and other methods as well. Without these modern methods of integrability, many such equations would not have been solved, thus leaving many scientific questions unanswered [17].
⇑ Corresponding author. E-mail address: [email protected] (A.-M. Wazwaz). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.050
H. Triki, A.-M. Wazwaz / Applied Mathematics and Computation 217 (2011) 8846–8851
8847
The solitary wave ansatz method [12–19] is rather heuristic and possesses significant features that make it practical for the determination of single soliton solutions for a wide class of nonlinear evolution equations with constant and varying coefficients in a direct method. This method is not used to derive multiple soliton solutions for integrable equations. In this paper, the solitary wave ansatz method will be applied to carry out the integration of for the combined potential KdV and Schwarzian KdV equation with constant coefficients and with time-dependent coefficients and forcing term, respectively. The topological 1-soliton solution is obtained for each equation. 2. The combined potential KdV and Schwarzian KdV equations with constant coefficients The combined version of the potential KdV equation and the Schwarzian KdV equation is given by Li [20]:
9 3 ut þ ðux Þ2 þ uxxx þ 2ux SðuÞ ¼ 0; 2 2
ð1Þ
where SðuÞ denotes the Schwarzian derivative of u, i.e.,
SðuÞ ¼
uxxx 3 u2xx : 2 u2x ux
ð2Þ
By using (2), one can rewrite (1) in the following form
9 7 ut ux þ ðux Þ3 þ ux uxxx 3u2xx ¼ 0: 2 2
ð3Þ
To obtain the topological 1-soliton solution of (3), we assume the solitary wave ansatz of the form [14,15] p
uðx; tÞ ¼ Atanh s;
ð4Þ
s ¼ Bðx v tÞ;
ð5Þ
where
where in (4) and (5), A and B are free parameters and v is the velocity of the wave. Also, the unknown exponent p will be determined during the course of the derivation of the soliton solution to (3). Therefore from (4), we get
ut ux ¼ v p2 A2 B2 ftanh
2pþ2
3pþ3
ðux Þ3 ¼ p3 A3 B3 ftanh
2p2
þ tanh
s tanh3p3 s 3tanh3pþ1 s þ 3tanh3p1 sg; 2p4
u2xx ¼ p2 A2 B4 fðp 1Þ2 tanh 2p2
4pðp 1Þtanh
s 2tanh2p sg;
ð6Þ ð7Þ
s þ ðp þ 1Þ2 tanh2pþ4 s þ ½2ðp 1Þðp þ 1Þ þ 4p2 tanh2p s 4pðp þ 1Þtanh2pþ2 s
sg;
ð8Þ 2pþ4
s þ ðp 1Þðp 2Þtanh2p4 s 2fp2 þ ðp 1Þðp 2Þgtanh2p2 s 2pþ2 2fp2 þ ðp þ 1Þðp þ 2Þgtanh s þ f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þgtanh2p sg:
ux uxxx ¼p2 A2 B4 fðp þ 1Þðp þ 2Þtanh
ð9Þ
Substituting (6)–(9) into (3), we have 2pþ2
v p2 A2 B2 ftanh
2p2
þ tanh
9 2
s 2tanh2p sg p3 A3 B3 ftanh3pþ3 s tanh3p3 s 3tanh3pþ1 s þ 3tanh3p1 sg
7 2pþ4 þ p2 A2 B4 fðp þ 1Þðp þ 2Þtanh s þ ðp 1Þðp 2Þtanh2p4 s 2fp2 þ ðp 1Þðp 2Þgtanh2p2 s 2 2pþ2 2fp2 þ ðp þ 1Þðp þ 2Þgtanh s þ f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þgtanh2p sg 3p2 A2 B4 fðp 1Þ2 tanh2p4 s 2pþ4
þ ðp þ 1Þ2 tanh
s þ 2½ðp 1Þðp þ 1Þ þ 2p2 tanh2p s 4pðp þ 1Þtanh2pþ2 s 4pðp 1Þtanh2p2 sg ¼ 0:
ð10Þ
From (10), equating the exponents 2p þ 4 and 3p þ 3 gives
2p þ 4 ¼ 3p þ 3;
ð11Þ
so that
p ¼ 1:
ð12Þ
It should be noted that the same value of p is yielded when the exponents pair 2p þ 2 and 3p þ 1; 2p and 3p 1; 2p 2 and 3p 3, are equated with each other, respectively. Again from (10), setting the coefficients of the linearly independent func2pþj tions tanh s, (with p ¼ 1) for j ¼ 4; 2; 0; 2; 4 to zero yields
7 2 2 4 9 p A B ðp þ 1Þðp þ 2Þ p3 A3 B3 3p2 A2 B4 ðp þ 1Þ2 ¼ 0; 2 2
ð13Þ
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v p2 A2 B2 þ
27 3 3 3 p A B 7p2 A2 B4 fp2 þ ðp þ 1Þðp þ 2Þg þ 12p3 A2 B4 ðp þ 1Þ ¼ 0; 2
9 v p2 A2 B2 þ p3 A3 B3 7p2 A2 B4 fp2 þ ðp 1Þðp 2Þg þ 12p3 ðp 1Þ ¼ 0; 2 2v p2 A2 B2
27 3 3 3 7 2 2 4 p A B þ p A B f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þg 6p2 A2 B4 ½ðp 1Þðp þ 1Þ þ 2p2 ¼ 0; 2 2
7 2 2 4 p A B ðp 1Þðp 2Þ 3p2 A2 B4 ðp 1Þ2 ¼ 0: 2
ð14Þ
ð15Þ
ð16Þ
ð17Þ
If we put p ¼ 1 in (13), one gets
A ¼ 2B:
ð18Þ
Substituting (18) into (14)–(16) gives
v¼
27 AB 49B2 þ 24B2 ; 2
ð19Þ
v¼
9 AB 7B2 2
ð20Þ
v¼
27 35 2 AB B þ 6B2 : 4 2
ð21Þ
and
Now, substituting (18) into (19), or (20), or (21) gives a unique value of 2
v ¼ 2B
v as follows: ð22Þ
;
which proves the consistency of the used method. Thus, finally, the topological 1-soliton solution to the combined KdV and Schwarzian KdV Eq. (3) is given by
uðx; tÞ ¼ A tanh½Bðx v tÞ;
ð23Þ
where the relation between the free parameters A and B is given by (18), and the velocity of the soliton is given by (22).
3. The combined potential KdV and Schwarzian KdV equations with time-dependent coefficients In this section we will study a family of the combined KdV and Schwarzian KdV equations with time-dependent coefficients and forcing term of the form
ut þ aðtÞðux Þ2 þ bðtÞuxxx þ cðtÞux SðuÞ ¼ aðtÞ;
ð24Þ
where aðtÞ; bðtÞ and cðtÞ are time-dependent coefficients, and aðtÞ is a forcing term. Here in (24), SðuÞ represents the Schwarzian derivative of u, given by (2) and aðtÞ; bðtÞ and cðtÞ are all real valued functions. If we set aðtÞ ¼ 9=2; bðtÞ ¼ 3=2; cðtÞ ¼ 2 and aðtÞ ¼ 0, (24) will give the standard combined KdV and Schwarzian KdV equations (3). As a matter of fact, the model (24) is much more general since nonlinear evolution equations with variable coefficients are more realistic in various physical situations than their constant-coefficients counterparts. In fact, due to the inhomogeneities of media and nonuniformities of boundaries, the variable-coefficient nonlinear evolution equations are needed to describe the propagation of pulses in these systems. It should be pointed out that the existence of the inhomogeneities in the media influences the accompanied physical effects giving rise to spatial or temporal dispersion and nonlinearity variations. Let us first set a generalized wave transformation formula [19]
qðx; tÞ ¼ uðx; tÞ bðtÞ;
ð25Þ
where
bðtÞ ¼
Z
aðtÞdt:
ð26Þ
Substituting (25) into (24) leads to the homogeneous combined KdV and Schwarzian KdV equations with time-dependent coefficients that reads
qt þ aðtÞðqx Þ2 þ bðtÞqxxx þ cðtÞqx SðqÞ ¼ 0:
ð27Þ
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Using (2), one can rewrite (27) in the following form:
3 qt qx þ aðtÞðqx Þ3 þ ðbðtÞ þ cðtÞÞqx qxxx cðtÞq2xx ¼ 0: 2
ð28Þ
To obtain the topological 1-soliton solution of (28), we assume the solitary wave ansatz of the form [15] p
qðx; tÞ ¼ AðtÞtanh fBðtÞðx v ðtÞtÞg;
ð29Þ
where AðtÞ; BðtÞ and v ðtÞ are unknown t-dependent parameters to be determined. Also, the unknown exponent p will be determined during the course of the derivation of the soliton solution to (28). From the ansatz (29), one obtains
qt ¼
dA dB dðtBv Þ p p1 pþ1 ftanh s tanh sg; tanh s þ Ap x dt dt dt p1
qx ¼ pABftanh
ð30Þ
s tanhpþ1 sg;
ð31Þ
dA dB dðtBv Þ 2p1 2p2 s tanh2pþ1 sg þ A2 p2 B x s þ tanh2pþ2 s 2tanh2p sg; qt qx ¼ pAB ftanh ftanh dt dt dt 3pþ3
ðqx Þ3 ¼ p3 A3 B3 ftanh
s tanh3p3 s 3tanh3pþ1 s þ 3tanh3p1 sg;
q2xx ¼ p2 A2 B4 fðp 1Þ2 tanh 4pðp 1Þtanh
2p2
2p4
ð32Þ ð33Þ
s þ ðp þ 1Þ2 tanh2pþ4 s þ ½2ðp 1Þðp þ 1Þ þ 4p2 tanh2p s 4pðp þ 1Þtanh2pþ2 s
sg;
ð34Þ 2pþ4
s þ ðp 1Þðp 2Þtanh2p4 s 2fp2 þ ðp 1Þðp 2Þgtanh2p2 s 2pþ2 2fp2 þ ðp þ 1Þðp þ 2Þgtanh s þ f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þgtanh2p sg;
qx qxxx ¼p2 A2 B4 fðp þ 1Þðp þ 2Þtanh
ð35Þ
where
s ¼ BðtÞðx v ðtÞtÞ:
ð36Þ
Substituting (32)–(35) into (28) gives
pAB
dA dB dðtBv Þ 2p1 2p2 ftanh ftanh s tanh2pþ1 sg þ A2 p2 B x s þ tanh2pþ2 s 2tanh2p sg dt dt dt 3pþ3
aðtÞp3 A3 B3 ftanh
þ ðp 1Þðp 2Þtanh
s tanh3p3 s 3tanh3pþ1 s þ 3tanh3p1 sg þ ðbðtÞ þ cðtÞÞp2 A2 B4 fðp þ 1Þðp þ 2Þtanh2pþ4 s
2p4
s 2fp2 þ ðp 1Þðp 2Þgtanh2p2 s 2fp2 þ ðp þ 1Þðp þ 2Þgtanh2pþ2 s 2p
þ f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þgtanh 2p
þ ½2ðp 1Þðp þ 1Þ þ 4p2 tanh
3 2
sg cðtÞp2 A2 B4 fðp 1Þ2 tanh2p4 s þ ðp þ 1Þ2 tanh2pþ4 s
s 4pðp þ 1Þtanh2pþ2 s 4pðp 1Þtanh2p2 sg ¼ 0: 2pþ4
Now, from (37), matching the exponents of tanh
3pþ3
s and tanh
ð37Þ
s, one gets
2p þ 4 ¼ 3p þ 3;
ð38Þ
so that
p ¼ 1:
ð39Þ
It needs to be noted that the same value of p is yielded when the exponents 2p þ 2 and 3p þ 1; 2p and 3p 1; 2p 2 and 3p 3 are equated with each other. Again, from (37), setting the coefficients of the stand-alone linearly independent func2p1 tions tanh s and tanh2pþ1 s to zero yields
dA ¼ 0; dt
ð40Þ
which gives after integration
AðtÞ ¼ A0 ;
ð41Þ
where A0 is a constant. We can determine the other soliton parameters BðtÞ and v ðtÞ by setting the coefficients of the linearly 2pþj independent functions tanh s, for j ¼ 4; 2; 0; 2; 4, to zero such that we get the following parametric equations:
dB dðtBv Þ 3 þ aðtÞp3 A3 B3 2ðbðtÞ þ cðtÞÞp2 A2 B4 fp2 þ ðp 1Þðp 2Þg þ cðtÞp2 A2 B4 4pðp 1Þ ¼ 0; A2 p2 B x dt dt 2
ð42Þ
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dB dðtBv Þ 3 þ 3aðtÞp3 A3 B3 2ðbðtÞ þ cðtÞÞp2 A2 B4 fp2 þ ðp þ 1Þðp þ 2Þg þ cðtÞp2 A2 B4 4pðp þ 1Þ ¼ 0; A2 p2 B x dt dt 2 dB dðtBv Þ 2A2 p2 B x 3aðtÞp3 A3 B3 þ ðbðtÞ þ cðtÞÞp2 A2 B4 f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þg dt dt 3 cðtÞp2 A2 B4 ½2ðp 1Þðp þ 1Þ þ 4p2 ¼ 0; 2
ð43Þ
ð44Þ
3 aðtÞp3 A3 B3 þ ðbðtÞ þ cðtÞÞp2 A2 B4 ðp þ 1Þðp þ 2Þ cðtÞp2 A2 B4 ðp þ 1Þ2 ¼ 0; 2
ð45Þ
3 ðbðtÞ þ cðtÞÞp2 A2 B4 ðp 1Þðp 2Þ cðtÞp2 A2 B4 ðp 1Þ2 ¼ 0: 2
ð46Þ
If we put p ¼ 1 in (42)–(46), the above system reduces to
dB dðtBv Þ þ aðtÞA3 B3 2ðbðtÞ þ cðtÞÞA2 B4 ¼ 0; A2 B x dt dt
ð47Þ
dB dðtBv Þ A2 B x þ 3aðtÞA3 B3 14ðbðtÞ þ cðtÞÞA2 B4 þ 12cðtÞA2 B4 ¼ 0; dt dt
ð48Þ
dB dðtBv Þ 3aðtÞA3 B3 þ 10ðbðtÞ þ cðtÞÞA2 B4 6cðtÞA2 B4 ¼ 0; 2A2 B x dt dt
ð49Þ
aðtÞA3 B3 þ 6ðbðtÞ þ cðtÞÞA2 B4 6cðtÞA2 B4 ¼ 0:
ð50Þ
From (50), one gets
A¼
6bðtÞ B: aðtÞ
ð51Þ
Now, substituting (51) into (47), or (48), or (49) gives a unique expression of the form:
dB dðtBv Þ þ 4bðtÞB3 2cðtÞB3 ¼ 0: x dt dt
ð52Þ
Taking into account the fact that the soliton parameter v we want to determine from (52) is a function of time, one can split (52) into two equations as follows:
dB ¼ 0; dt dðtBv Þ þ 4bðtÞB3 2cðtÞB3 ¼ 0: dt
ð53Þ ð54Þ
By integrating the above equations with respect to the time variable t, one obtains
BðtÞ ¼ B0 ; 1 v ðtÞ ¼ tBðtÞ
ð55Þ
Z
t
0
f4bðt 0 Þ 2cðt0 ÞgB3 ðt0 Þdt ;
ð56Þ
0
where B0 is a constant. From (55), it is apparent that the free parameter B remains constant when the pulse propagates in the physical system. We remark also from (56) that the pulse velocity is affected by the time-dependent coefficients bðtÞ and cðtÞ. It is interesting to note also from (51) that, since AðtÞ ¼ A0 and BðtÞ ¼ B0 , then it is necessary that
bðtÞ ¼ k; aðtÞ
ð57Þ
where k is a constant. Thus, finally, the topological 1-soliton solution to the family of the combined KdV and Schwarzian KdV equations with time-dependent coefficients and forcing term (24) is given by
uðx; tÞ ¼ AðtÞ tanhfBðtÞðx v ðtÞtÞg þ bðtÞ;
ð58Þ
where the free parameters AðtÞ and BðtÞ are constants, while the velocity v ðtÞ of the soliton is given by (56). Note that the constraint relation between the varying parameters aðtÞ and bðtÞ is displayed in (57).
H. Triki, A.-M. Wazwaz / Applied Mathematics and Computation 217 (2011) 8846–8851
8851
It is interesting to note that for aðtÞ ¼ 9=2; bðtÞ ¼ 3=2; cðtÞ ¼ 2 and aðtÞ ¼ 0, (51) and (56) will be reduced, respectively, to A ¼ 2B and v ¼ 2B2 which are the same values given in (18) and (22) and hence the soliton solution (58) will be reduced to the form of (23). 4. Conclusion In this work, we have derived the dark soliton solutions for the combined potential KdV and Schwarzian KdV equations with constant coefficients and time-dependent coefficients by using the solitary wave ansatz method. For the family of timedependent equations, all the physical parameters in the solitary wave solutions are obtained as functions of the time varying model coefficients. We have found that the velocity of the soliton is affected by certain t-dependent model coefficients, however the other soliton parameters remain constant during the soliton propagation. In view of the analysis, we clearly see the consistency of the solitary wave ansatz method which has recently been applied successfully to wide range of NLPDEs with constant and time-dependent coefficients. References [1] A.I. Maimistov, Completely integrable models of nonlinear optics, Pramana J. Phys. 57 (5–6) (2001) 953–968. [2] M. Gedalin, T.C. Scott, Y.B. Band, Optical solitary waves in the higher order nonlinear Schrödinger equation, Phys. Rev. Lett. 78 (3) (1997) 448–451. [3] E. Yomba, Jacobi elliptic function solutions of the generalized Zakharov–Kuznetsov equation with nonlinear dispersion and t-dependent coefficients, Phys. Lett. A 374 (2010) 1611–1615. [4] J.F. Currie, J.A. Krumhansl, A.R. Bishop, S.E. Trullinger, Phys. Rev. B. 22 (1980) 477. [5] Y.S. Kivshar, G.P. Agarwal, Optical Solitons: From Fibers to Photonic Crystal, Academic, San Diego, 2003. [6] M.M. Scott, M.P. Kostylev, B.A. Kalinikos, C.E. Patton, Excitation of bright and dark envelope solitons for magnetostatic waves with attractive nonlinearity, Phys. Rev. B 71 (174440) (2005) 1–4. [7] P. Emplit, J.P. Hamaide, F. Reinaud, C. Froehly, A. Bartelemy, Opt. Commun. 62 (1987) 374. [8] X. Li, M. Wang, A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms, Phys. Lett. A. 361 (2007) 115–118. [9] L.-Ping Xu, J.-Liang Zhang, Exact solutions to two higher order nonlinear Schrödinger equations, Chaos, Solitons Fractals 31 (2007) 937–942. [10] E.M.E. Zayed, H.A. Zedan, K.A. Gepreel, On the solitary wave solutions for nonlinear Hirota-satsuma coupled KdV of equations, Chaos, Solitons Fractals 22 (2004) 285–303. [11] A.-M. Wazwaz, Analytic study for fifth-order KdV-type equations with arbitrary power nonlinearities, Commun. Nonlinear Sci. Numer. Simul. 12 (2007) 904–909. [12] A. Biswas, 1-soliton solution of the Bðm; nÞ equation with generalized evolution, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3226–3229. [13] A. Biswas, D. Milovic, Bright and dark solitons of the generalized nonlinear Schrödinger’s equation, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1473–1484. [14] M. Saha, A.K. Sarma, A. Biswas, Dark optical solitons in power law media with time-dependent coefficients, Phys. Lett. A 373 (2009) 4438–4441. [15] H. Triki, A.M. Wazwaz, Bright and dark soliton solutions for a Kðm; nÞ equation with t-dependent coefficients, Phys. Lett. A 373 (2009) 2162–2165. [16] A. Biswas, 1-soliton solution of (1 + 2) dimensional nonlinear Schrödinger’s equation in dual-power law media, Phys. Lett. A 372 (2008) 5941–5943. [17] A. Biswas, 1-soliton solution of the K(m, n) equation with generalized evolution, Phys. Lett. A 372 (2008) 4601–4602. [18] A. Biswas, Solitary wave solution for the generalized Kawahara equation, Appl. Math. Lett. 22 (2009) 208–210. [19] A.M. Wazwaz, A study on KdV and Gardner equations with time-dependent coefficients and forcing terms, Appl. Math. Comput. 217 (2010) 2277– 2281. [20] Z. Li, New exact kink solutions solitons and periodic form solutions for a combined KdV and Schwarzian KdV equation, Appl. Math. Comput. 215 (2009) 2886–2890.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Dark solitons for a combined potential KdV and Schwarzian KdV equations with t-dependent coefficients and forcing term Houria Triki a, Abdul-Majid Wazwaz b,⇑ a b
Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria Department of Mathematics, Saint Xavier University, Chicago, IL 60655, United States
a r t i c l e
i n f o
Keywords: Potential KdV equation Schwarzian KdV equation Solitary wave ansatz
a b s t r a c t In this work we formally derive the dark soliton solutions for the combined potential KdV and Schwarzian KdV equations. The combined KdV and Schwarzian KdV equations with time-dependent coefficients and forcing term are then investigated to obtain dark soliton solutions. The solitary wave ansatz is used to carry out the analysis for both models. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction In past years, a growing interest has been given to the propagation of nonlinear waves in nonlinear dynamical systems. A nonlinear wave is one of the fundamental objects of nature [1]. These waves appear in a great array of contexts such as, hydrodynamics, nonlinear optics, plasmas, solid state physics, nuclear physics, and many other nonlinear phenomena. The systems are often described by nonlinear partial differential equations (NLPDEs) with constant or variable coefficients, supporting exact solutions with interesting properties which are commonly referred to as solitary waves. The existence of solitary wave solutions implies perfect balance between nonlinearity and dispersion effects which usually requires rather specific conditions and cannot be established in general [2]. They also exhibit remarkable stability and particle-like properties [3]. If they retain their identity after collisions, they are called solitons [4]. Solitons are ubiquitous in nature, appearing in diverse systems such as shallow water waves, DNA excitations, matter waves in Bose–Einstein condensates, and ultrashort pulses (or laser beams) in nonlinear optics [1–5]. Two different types of envelope solitons, bright and dark, can propagate in nonlinear dispersive media. Compared with the bright soliton which is a pulse on a zero-intensity background, the dark soliton appears as an intensity dip in an infinitely extended constant background [6,7]. From a mathematical point of view, there exists a certain class of NLPDEs that support soliton solutions in physical systems. Examples include the sine-Gordon (SG) equation, the nonlinear Schrödinger (NLS) equation, the Korteweg-de Vries (KdV) equation, the modified KdV equation, the Boussinesq equation, etc. In recent years, many powerful methods to construct exact solutions of NLPDEs have been established and developed, which lead to one of the most excited advances of nonlinear science and theoretical physics [8]. In fact, many kinds of exact soliton solutions have been obtained by using for example, the homogeneous balance principle and F-expansion method [9], the Jacobi elliptic functions method [10], the sine–cosine and tanh methods [11], the Hirota’s bilinear method, the Bäcklund transformation method, the solitary wave ansatz method [12–19], and other methods as well. Without these modern methods of integrability, many such equations would not have been solved, thus leaving many scientific questions unanswered [17].
⇑ Corresponding author. E-mail address: [email protected] (A.-M. Wazwaz). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.050
H. Triki, A.-M. Wazwaz / Applied Mathematics and Computation 217 (2011) 8846–8851
8847
The solitary wave ansatz method [12–19] is rather heuristic and possesses significant features that make it practical for the determination of single soliton solutions for a wide class of nonlinear evolution equations with constant and varying coefficients in a direct method. This method is not used to derive multiple soliton solutions for integrable equations. In this paper, the solitary wave ansatz method will be applied to carry out the integration of for the combined potential KdV and Schwarzian KdV equation with constant coefficients and with time-dependent coefficients and forcing term, respectively. The topological 1-soliton solution is obtained for each equation. 2. The combined potential KdV and Schwarzian KdV equations with constant coefficients The combined version of the potential KdV equation and the Schwarzian KdV equation is given by Li [20]:
9 3 ut þ ðux Þ2 þ uxxx þ 2ux SðuÞ ¼ 0; 2 2
ð1Þ
where SðuÞ denotes the Schwarzian derivative of u, i.e.,
SðuÞ ¼
uxxx 3 u2xx : 2 u2x ux
ð2Þ
By using (2), one can rewrite (1) in the following form
9 7 ut ux þ ðux Þ3 þ ux uxxx 3u2xx ¼ 0: 2 2
ð3Þ
To obtain the topological 1-soliton solution of (3), we assume the solitary wave ansatz of the form [14,15] p
uðx; tÞ ¼ Atanh s;
ð4Þ
s ¼ Bðx v tÞ;
ð5Þ
where
where in (4) and (5), A and B are free parameters and v is the velocity of the wave. Also, the unknown exponent p will be determined during the course of the derivation of the soliton solution to (3). Therefore from (4), we get
ut ux ¼ v p2 A2 B2 ftanh
2pþ2
3pþ3
ðux Þ3 ¼ p3 A3 B3 ftanh
2p2
þ tanh
s tanh3p3 s 3tanh3pþ1 s þ 3tanh3p1 sg; 2p4
u2xx ¼ p2 A2 B4 fðp 1Þ2 tanh 2p2
4pðp 1Þtanh
s 2tanh2p sg;
ð6Þ ð7Þ
s þ ðp þ 1Þ2 tanh2pþ4 s þ ½2ðp 1Þðp þ 1Þ þ 4p2 tanh2p s 4pðp þ 1Þtanh2pþ2 s
sg;
ð8Þ 2pþ4
s þ ðp 1Þðp 2Þtanh2p4 s 2fp2 þ ðp 1Þðp 2Þgtanh2p2 s 2pþ2 2fp2 þ ðp þ 1Þðp þ 2Þgtanh s þ f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þgtanh2p sg:
ux uxxx ¼p2 A2 B4 fðp þ 1Þðp þ 2Þtanh
ð9Þ
Substituting (6)–(9) into (3), we have 2pþ2
v p2 A2 B2 ftanh
2p2
þ tanh
9 2
s 2tanh2p sg p3 A3 B3 ftanh3pþ3 s tanh3p3 s 3tanh3pþ1 s þ 3tanh3p1 sg
7 2pþ4 þ p2 A2 B4 fðp þ 1Þðp þ 2Þtanh s þ ðp 1Þðp 2Þtanh2p4 s 2fp2 þ ðp 1Þðp 2Þgtanh2p2 s 2 2pþ2 2fp2 þ ðp þ 1Þðp þ 2Þgtanh s þ f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þgtanh2p sg 3p2 A2 B4 fðp 1Þ2 tanh2p4 s 2pþ4
þ ðp þ 1Þ2 tanh
s þ 2½ðp 1Þðp þ 1Þ þ 2p2 tanh2p s 4pðp þ 1Þtanh2pþ2 s 4pðp 1Þtanh2p2 sg ¼ 0:
ð10Þ
From (10), equating the exponents 2p þ 4 and 3p þ 3 gives
2p þ 4 ¼ 3p þ 3;
ð11Þ
so that
p ¼ 1:
ð12Þ
It should be noted that the same value of p is yielded when the exponents pair 2p þ 2 and 3p þ 1; 2p and 3p 1; 2p 2 and 3p 3, are equated with each other, respectively. Again from (10), setting the coefficients of the linearly independent func2pþj tions tanh s, (with p ¼ 1) for j ¼ 4; 2; 0; 2; 4 to zero yields
7 2 2 4 9 p A B ðp þ 1Þðp þ 2Þ p3 A3 B3 3p2 A2 B4 ðp þ 1Þ2 ¼ 0; 2 2
ð13Þ
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v p2 A2 B2 þ
27 3 3 3 p A B 7p2 A2 B4 fp2 þ ðp þ 1Þðp þ 2Þg þ 12p3 A2 B4 ðp þ 1Þ ¼ 0; 2
9 v p2 A2 B2 þ p3 A3 B3 7p2 A2 B4 fp2 þ ðp 1Þðp 2Þg þ 12p3 ðp 1Þ ¼ 0; 2 2v p2 A2 B2
27 3 3 3 7 2 2 4 p A B þ p A B f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þg 6p2 A2 B4 ½ðp 1Þðp þ 1Þ þ 2p2 ¼ 0; 2 2
7 2 2 4 p A B ðp 1Þðp 2Þ 3p2 A2 B4 ðp 1Þ2 ¼ 0: 2
ð14Þ
ð15Þ
ð16Þ
ð17Þ
If we put p ¼ 1 in (13), one gets
A ¼ 2B:
ð18Þ
Substituting (18) into (14)–(16) gives
v¼
27 AB 49B2 þ 24B2 ; 2
ð19Þ
v¼
9 AB 7B2 2
ð20Þ
v¼
27 35 2 AB B þ 6B2 : 4 2
ð21Þ
and
Now, substituting (18) into (19), or (20), or (21) gives a unique value of 2
v ¼ 2B
v as follows: ð22Þ
;
which proves the consistency of the used method. Thus, finally, the topological 1-soliton solution to the combined KdV and Schwarzian KdV Eq. (3) is given by
uðx; tÞ ¼ A tanh½Bðx v tÞ;
ð23Þ
where the relation between the free parameters A and B is given by (18), and the velocity of the soliton is given by (22).
3. The combined potential KdV and Schwarzian KdV equations with time-dependent coefficients In this section we will study a family of the combined KdV and Schwarzian KdV equations with time-dependent coefficients and forcing term of the form
ut þ aðtÞðux Þ2 þ bðtÞuxxx þ cðtÞux SðuÞ ¼ aðtÞ;
ð24Þ
where aðtÞ; bðtÞ and cðtÞ are time-dependent coefficients, and aðtÞ is a forcing term. Here in (24), SðuÞ represents the Schwarzian derivative of u, given by (2) and aðtÞ; bðtÞ and cðtÞ are all real valued functions. If we set aðtÞ ¼ 9=2; bðtÞ ¼ 3=2; cðtÞ ¼ 2 and aðtÞ ¼ 0, (24) will give the standard combined KdV and Schwarzian KdV equations (3). As a matter of fact, the model (24) is much more general since nonlinear evolution equations with variable coefficients are more realistic in various physical situations than their constant-coefficients counterparts. In fact, due to the inhomogeneities of media and nonuniformities of boundaries, the variable-coefficient nonlinear evolution equations are needed to describe the propagation of pulses in these systems. It should be pointed out that the existence of the inhomogeneities in the media influences the accompanied physical effects giving rise to spatial or temporal dispersion and nonlinearity variations. Let us first set a generalized wave transformation formula [19]
qðx; tÞ ¼ uðx; tÞ bðtÞ;
ð25Þ
where
bðtÞ ¼
Z
aðtÞdt:
ð26Þ
Substituting (25) into (24) leads to the homogeneous combined KdV and Schwarzian KdV equations with time-dependent coefficients that reads
qt þ aðtÞðqx Þ2 þ bðtÞqxxx þ cðtÞqx SðqÞ ¼ 0:
ð27Þ
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Using (2), one can rewrite (27) in the following form:
3 qt qx þ aðtÞðqx Þ3 þ ðbðtÞ þ cðtÞÞqx qxxx cðtÞq2xx ¼ 0: 2
ð28Þ
To obtain the topological 1-soliton solution of (28), we assume the solitary wave ansatz of the form [15] p
qðx; tÞ ¼ AðtÞtanh fBðtÞðx v ðtÞtÞg;
ð29Þ
where AðtÞ; BðtÞ and v ðtÞ are unknown t-dependent parameters to be determined. Also, the unknown exponent p will be determined during the course of the derivation of the soliton solution to (28). From the ansatz (29), one obtains
qt ¼
dA dB dðtBv Þ p p1 pþ1 ftanh s tanh sg; tanh s þ Ap x dt dt dt p1
qx ¼ pABftanh
ð30Þ
s tanhpþ1 sg;
ð31Þ
dA dB dðtBv Þ 2p1 2p2 s tanh2pþ1 sg þ A2 p2 B x s þ tanh2pþ2 s 2tanh2p sg; qt qx ¼ pAB ftanh ftanh dt dt dt 3pþ3
ðqx Þ3 ¼ p3 A3 B3 ftanh
s tanh3p3 s 3tanh3pþ1 s þ 3tanh3p1 sg;
q2xx ¼ p2 A2 B4 fðp 1Þ2 tanh 4pðp 1Þtanh
2p2
2p4
ð32Þ ð33Þ
s þ ðp þ 1Þ2 tanh2pþ4 s þ ½2ðp 1Þðp þ 1Þ þ 4p2 tanh2p s 4pðp þ 1Þtanh2pþ2 s
sg;
ð34Þ 2pþ4
s þ ðp 1Þðp 2Þtanh2p4 s 2fp2 þ ðp 1Þðp 2Þgtanh2p2 s 2pþ2 2fp2 þ ðp þ 1Þðp þ 2Þgtanh s þ f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þgtanh2p sg;
qx qxxx ¼p2 A2 B4 fðp þ 1Þðp þ 2Þtanh
ð35Þ
where
s ¼ BðtÞðx v ðtÞtÞ:
ð36Þ
Substituting (32)–(35) into (28) gives
pAB
dA dB dðtBv Þ 2p1 2p2 ftanh ftanh s tanh2pþ1 sg þ A2 p2 B x s þ tanh2pþ2 s 2tanh2p sg dt dt dt 3pþ3
aðtÞp3 A3 B3 ftanh
þ ðp 1Þðp 2Þtanh
s tanh3p3 s 3tanh3pþ1 s þ 3tanh3p1 sg þ ðbðtÞ þ cðtÞÞp2 A2 B4 fðp þ 1Þðp þ 2Þtanh2pþ4 s
2p4
s 2fp2 þ ðp 1Þðp 2Þgtanh2p2 s 2fp2 þ ðp þ 1Þðp þ 2Þgtanh2pþ2 s 2p
þ f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þgtanh 2p
þ ½2ðp 1Þðp þ 1Þ þ 4p2 tanh
3 2
sg cðtÞp2 A2 B4 fðp 1Þ2 tanh2p4 s þ ðp þ 1Þ2 tanh2pþ4 s
s 4pðp þ 1Þtanh2pþ2 s 4pðp 1Þtanh2p2 sg ¼ 0: 2pþ4
Now, from (37), matching the exponents of tanh
3pþ3
s and tanh
ð37Þ
s, one gets
2p þ 4 ¼ 3p þ 3;
ð38Þ
so that
p ¼ 1:
ð39Þ
It needs to be noted that the same value of p is yielded when the exponents 2p þ 2 and 3p þ 1; 2p and 3p 1; 2p 2 and 3p 3 are equated with each other. Again, from (37), setting the coefficients of the stand-alone linearly independent func2p1 tions tanh s and tanh2pþ1 s to zero yields
dA ¼ 0; dt
ð40Þ
which gives after integration
AðtÞ ¼ A0 ;
ð41Þ
where A0 is a constant. We can determine the other soliton parameters BðtÞ and v ðtÞ by setting the coefficients of the linearly 2pþj independent functions tanh s, for j ¼ 4; 2; 0; 2; 4, to zero such that we get the following parametric equations:
dB dðtBv Þ 3 þ aðtÞp3 A3 B3 2ðbðtÞ þ cðtÞÞp2 A2 B4 fp2 þ ðp 1Þðp 2Þg þ cðtÞp2 A2 B4 4pðp 1Þ ¼ 0; A2 p2 B x dt dt 2
ð42Þ
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dB dðtBv Þ 3 þ 3aðtÞp3 A3 B3 2ðbðtÞ þ cðtÞÞp2 A2 B4 fp2 þ ðp þ 1Þðp þ 2Þg þ cðtÞp2 A2 B4 4pðp þ 1Þ ¼ 0; A2 p2 B x dt dt 2 dB dðtBv Þ 2A2 p2 B x 3aðtÞp3 A3 B3 þ ðbðtÞ þ cðtÞÞp2 A2 B4 f4p2 þ ðp 1Þðp 2Þ þ ðp þ 1Þðp þ 2Þg dt dt 3 cðtÞp2 A2 B4 ½2ðp 1Þðp þ 1Þ þ 4p2 ¼ 0; 2
ð43Þ
ð44Þ
3 aðtÞp3 A3 B3 þ ðbðtÞ þ cðtÞÞp2 A2 B4 ðp þ 1Þðp þ 2Þ cðtÞp2 A2 B4 ðp þ 1Þ2 ¼ 0; 2
ð45Þ
3 ðbðtÞ þ cðtÞÞp2 A2 B4 ðp 1Þðp 2Þ cðtÞp2 A2 B4 ðp 1Þ2 ¼ 0: 2
ð46Þ
If we put p ¼ 1 in (42)–(46), the above system reduces to
dB dðtBv Þ þ aðtÞA3 B3 2ðbðtÞ þ cðtÞÞA2 B4 ¼ 0; A2 B x dt dt
ð47Þ
dB dðtBv Þ A2 B x þ 3aðtÞA3 B3 14ðbðtÞ þ cðtÞÞA2 B4 þ 12cðtÞA2 B4 ¼ 0; dt dt
ð48Þ
dB dðtBv Þ 3aðtÞA3 B3 þ 10ðbðtÞ þ cðtÞÞA2 B4 6cðtÞA2 B4 ¼ 0; 2A2 B x dt dt
ð49Þ
aðtÞA3 B3 þ 6ðbðtÞ þ cðtÞÞA2 B4 6cðtÞA2 B4 ¼ 0:
ð50Þ
From (50), one gets
A¼
6bðtÞ B: aðtÞ
ð51Þ
Now, substituting (51) into (47), or (48), or (49) gives a unique expression of the form:
dB dðtBv Þ þ 4bðtÞB3 2cðtÞB3 ¼ 0: x dt dt
ð52Þ
Taking into account the fact that the soliton parameter v we want to determine from (52) is a function of time, one can split (52) into two equations as follows:
dB ¼ 0; dt dðtBv Þ þ 4bðtÞB3 2cðtÞB3 ¼ 0: dt
ð53Þ ð54Þ
By integrating the above equations with respect to the time variable t, one obtains
BðtÞ ¼ B0 ; 1 v ðtÞ ¼ tBðtÞ
ð55Þ
Z
t
0
f4bðt 0 Þ 2cðt0 ÞgB3 ðt0 Þdt ;
ð56Þ
0
where B0 is a constant. From (55), it is apparent that the free parameter B remains constant when the pulse propagates in the physical system. We remark also from (56) that the pulse velocity is affected by the time-dependent coefficients bðtÞ and cðtÞ. It is interesting to note also from (51) that, since AðtÞ ¼ A0 and BðtÞ ¼ B0 , then it is necessary that
bðtÞ ¼ k; aðtÞ
ð57Þ
where k is a constant. Thus, finally, the topological 1-soliton solution to the family of the combined KdV and Schwarzian KdV equations with time-dependent coefficients and forcing term (24) is given by
uðx; tÞ ¼ AðtÞ tanhfBðtÞðx v ðtÞtÞg þ bðtÞ;
ð58Þ
where the free parameters AðtÞ and BðtÞ are constants, while the velocity v ðtÞ of the soliton is given by (56). Note that the constraint relation between the varying parameters aðtÞ and bðtÞ is displayed in (57).
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8851
It is interesting to note that for aðtÞ ¼ 9=2; bðtÞ ¼ 3=2; cðtÞ ¼ 2 and aðtÞ ¼ 0, (51) and (56) will be reduced, respectively, to A ¼ 2B and v ¼ 2B2 which are the same values given in (18) and (22) and hence the soliton solution (58) will be reduced to the form of (23). 4. Conclusion In this work, we have derived the dark soliton solutions for the combined potential KdV and Schwarzian KdV equations with constant coefficients and time-dependent coefficients by using the solitary wave ansatz method. For the family of timedependent equations, all the physical parameters in the solitary wave solutions are obtained as functions of the time varying model coefficients. We have found that the velocity of the soliton is affected by certain t-dependent model coefficients, however the other soliton parameters remain constant during the soliton propagation. In view of the analysis, we clearly see the consistency of the solitary wave ansatz method which has recently been applied successfully to wide range of NLPDEs with constant and time-dependent coefficients. References [1] A.I. Maimistov, Completely integrable models of nonlinear optics, Pramana J. Phys. 57 (5–6) (2001) 953–968. [2] M. Gedalin, T.C. Scott, Y.B. Band, Optical solitary waves in the higher order nonlinear Schrödinger equation, Phys. Rev. Lett. 78 (3) (1997) 448–451. [3] E. Yomba, Jacobi elliptic function solutions of the generalized Zakharov–Kuznetsov equation with nonlinear dispersion and t-dependent coefficients, Phys. Lett. A 374 (2010) 1611–1615. [4] J.F. Currie, J.A. Krumhansl, A.R. Bishop, S.E. Trullinger, Phys. Rev. B. 22 (1980) 477. [5] Y.S. Kivshar, G.P. Agarwal, Optical Solitons: From Fibers to Photonic Crystal, Academic, San Diego, 2003. [6] M.M. Scott, M.P. Kostylev, B.A. Kalinikos, C.E. Patton, Excitation of bright and dark envelope solitons for magnetostatic waves with attractive nonlinearity, Phys. Rev. B 71 (174440) (2005) 1–4. [7] P. Emplit, J.P. Hamaide, F. Reinaud, C. Froehly, A. Bartelemy, Opt. Commun. 62 (1987) 374. [8] X. Li, M. Wang, A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms, Phys. Lett. A. 361 (2007) 115–118. [9] L.-Ping Xu, J.-Liang Zhang, Exact solutions to two higher order nonlinear Schrödinger equations, Chaos, Solitons Fractals 31 (2007) 937–942. [10] E.M.E. Zayed, H.A. Zedan, K.A. Gepreel, On the solitary wave solutions for nonlinear Hirota-satsuma coupled KdV of equations, Chaos, Solitons Fractals 22 (2004) 285–303. [11] A.-M. Wazwaz, Analytic study for fifth-order KdV-type equations with arbitrary power nonlinearities, Commun. Nonlinear Sci. Numer. Simul. 12 (2007) 904–909. [12] A. Biswas, 1-soliton solution of the Bðm; nÞ equation with generalized evolution, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3226–3229. [13] A. Biswas, D. Milovic, Bright and dark solitons of the generalized nonlinear Schrödinger’s equation, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1473–1484. [14] M. Saha, A.K. Sarma, A. Biswas, Dark optical solitons in power law media with time-dependent coefficients, Phys. Lett. A 373 (2009) 4438–4441. [15] H. Triki, A.M. Wazwaz, Bright and dark soliton solutions for a Kðm; nÞ equation with t-dependent coefficients, Phys. Lett. A 373 (2009) 2162–2165. [16] A. Biswas, 1-soliton solution of (1 + 2) dimensional nonlinear Schrödinger’s equation in dual-power law media, Phys. Lett. A 372 (2008) 5941–5943. [17] A. Biswas, 1-soliton solution of the K(m, n) equation with generalized evolution, Phys. Lett. A 372 (2008) 4601–4602. [18] A. Biswas, Solitary wave solution for the generalized Kawahara equation, Appl. Math. Lett. 22 (2009) 208–210. [19] A.M. Wazwaz, A study on KdV and Gardner equations with time-dependent coefficients and forcing terms, Appl. Math. Comput. 217 (2010) 2277– 2281. [20] Z. Li, New exact kink solutions solitons and periodic form solutions for a combined KdV and Schwarzian KdV equation, Appl. Math. Comput. 215 (2009) 2886–2890.