1-Soliton solution of the generalized KdV equation with generalized evolution

Applied Mathematics and Computation 216 (2010) 1673–1679

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1-Soliton solution of the generalized KdV equation with generalized evolution M.S. Ismail a, Anjan Biswas b,* a

Department of Mathematics, College of Science, P.O. Box 80203, King Abdulaziz University, Jeddah-21589, Saudi Arabia Center for Research and Education in Optical Sciences and Applications, Applied Mathematics Research Center, Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA b

a r t i c l e Keywords: Solitary waves Integrals of motion

i n f o

a b s t r a c t This paper obtains the 1-soliton solution of three variants of the generalized KdV equation with generalized evolution. The solitary wave ansatz is used to carry out the integration of such equation. The parameter domain is also identified in the process. A couple of conserved quantities are also calculated for each of these variants. The numerical simulations are also carried out. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction The solitons play a very important role in various areas of Applied Mathematics and Theoretical Physics [1–10]. The solitary wave was first observed in the 19th century by a Scottish engineer named John Scott Russell which was then referred to as great primary wave [1]. Since then, the theory of solitary waves or solitons have come a long way through. In fact, nowadays solitons are studied in various areas of nonlinear science. They appear in Nonlinear Optics, Mathematical Biosciences, Fluid Dynamics, Plasma Physics, Nuclear Physics, just to name a few. Some of the important properties that are studied in the area of soliton theory are integrability issue, particle-like behavior, nonlinear super-position principle as well as the algebrogeometric aspects. The integrability aspects is one of the major issues that is very commonly addressed in modern times. There are various methods of integrability that are addressed in the past decade. They are the F-expansion method, G0 =G method, Lie symmetry method, Adomian decomposition method, He’s semi-inverse variational principle, He’s variational iteration method, Wadati trace method and many others. Once upon a time, there was the so-called method that is known as the Inverse Scattering Transform (IST) which one has to either master it or else had to suffer. Those days of suffering are now over. It needs to be noted that several equations are now integrable even though IST fails to integrate such equations. This area of nonlinear evolution equation is now blessed with these various modern methods of integrability. In this paper, one such modern method of integrability will be addressed to carry out the integration of three variants of the generalized Korteweg-de Vries (KdV) equation that appears with generalized evolution. This is the solitary wave ansatz method.

* Corresponding author. E-mail address: [email protected] (A. Biswas). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.02.045

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2. Mathematical analysis The three variants of the generalized KdV equation that will be studied in this paper are given by [5]

ðql Þt þ aðqnþ1 Þx þ b½qðqn Þxx x þ cqðqn Þxxx ¼ 0;

ð1Þ

ðql Þt þ aqn qx þ bðqn qxx Þx þ cqðqn Þxxx ¼ 0

ð2Þ

ðql Þt þ aqðqn Þx þ b½qðqn Þxx x þ cqðqn Þxxx ¼ 0;

ð3Þ

and

which will be respectively labeled as Variant-I, II and III. In (1)–(3), the first term represents the generalized evolution. The special case with l ¼ 1 is the regular evolution term. The coefficients of a are the nonlinear terms while the coefficients of b and c are the nonlinear dispersion terms. These equations with l ¼ 1 have been already studied in 2004 by Wazwaz [5], where, in addition to soliton solution, compactons and periodic solutions were also obtained. In this paper, where the evolution term is generalized, the focus is going to be on obtaining the 1-soliton solution only. The solitons are the result of a delicate balance between dispersion and nonlinearity. The hypothesis for solving these equations is

qðx; tÞ ¼

A p ; cosh s

ð4Þ

where

s ¼ Bðx  v tÞ:

ð5Þ

Here, in (4) and (5), A represents the amplitude of the soliton while B is the inverse width of the soliton and v is the velocity of the soliton. The exponent p is unknown at this point and will be evaluated during the course of the derivation of the solutions to (1)–(3). From (4), it is possible to obtain l

ðql Þt ¼

lpv A B tanh s

ð6Þ

; lp cosh s ðn þ 1ÞpAnþ1 B tanh s ðqnþ1 Þx ¼  ; ðnþ1Þp cosh s n2 p2 An B2 npðnp þ 1ÞAn B2  ; ðqn Þxx ¼ np npþ2 cosh s cosh s n3 p3 An B3 tanh s npðnp þ 1Þðnp þ 2ÞAn B3 tanh s ðqn Þxxx ¼   ; np npþ2 cosh s cosh s ½qðqn Þxx x ¼ 

n2 ðn þ 1Þp3 Anþ1 B2 tanh s cosh

ðnþ1Þp

s

þ

ð7Þ ð8Þ ð9Þ

npðnp þ 1Þðnp þ p þ 2ÞAnþ1 B2 tanh s cosh

ðnþ1Þpþ2

s

ð10Þ

:

These results will now be substituted in (1)–(3) to obtain the 1-soliton solution to the three variants of the generalized KdV equations. 2.1. Variant-I Eq. (1) by virtue of (6)–(10) reduces to l

lpv A B tanh s lp



aðn þ 1ÞpAnþ1 B tanh s ðnþ1Þp

2



bn ðn þ 1Þp3 Anþ1 B3 tanh s ðnþ1Þp

cosh s cosh s cosh s cn3 p3 Anþ1 B3 tanh s cnpðnp þ 1Þðnp þ 2ÞAnþ1 B3 tanh s  þ ¼ 0: ðnþ1Þp ðnþ1Þpþ2 cosh s cosh s

þ

bnpðnp þ 1Þðnp þ p þ 2ÞAnþ1 B3 tanh s ðnþ1Þpþ2

cosh

s ð11Þ

From (11), equating the exponents lp and np þ p þ 2 gives

np þ p þ 2 ¼ lp;

ð12Þ

that leads to

p¼

2 : ln1

ð13Þ npþpþj

Now from (11), the two linearly independent functions are 1=cosh gives

s for j ¼ 0; 2. Thus setting their coefficients to zero

M.S. Ismail, A. Biswas / Applied Mathematics and Computation 216 (2010) 1673–1679

v¼

aðn þ 1Þðl þ n  1Þfbl þ cðl  1Þg

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ð14Þ

2lnfbðn þ 1Þ þ cngAln1

and

ln1 B¼ 2n

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðn þ 1Þ  ; bðn þ 1Þ þ cn

ð15Þ

which leads to the constraint condition.

afbðn þ 1Þ þ cng < 0:

ð16Þ

Thus the 1-soliton solution of the variant-I of the generalized KdV equation with generalized evolution is given by

qðx; tÞ ¼

A 2 ln1

cosh

½Bðx  v tÞ

ð17Þ

:

This shows that the restriction on the nonlinear exponents of (1) must be

l > n þ 1;

ð18Þ

for the solitons to exist. The following Fig. 1(a) and (b) shows the profile of a 1-soliton solution. For Fig. 1(a), A ¼ 0:5; a ¼ 1; b ¼ 1; c ¼ 2; l ¼ 3 and n ¼ 1 while for Fig. 1(b), A ¼ 0:5; a ¼ 1; b ¼ 1; c ¼ 1; l ¼ 3 and n ¼ 1. 2.2. Variant-II In this section the 1-soliton solution to (2) will be obtained. So, substituting (6)–(10) in (2) yields l

lpv A B tanh s lp



apAnþ1 B tanh s ðnþ1Þp



bðn þ 1Þp3 Anþ1 B3 tanh s ðnþ1Þp

þ

bpðp þ 1Þðnp þ p þ 2ÞAnþ1 B3 tanh s

cosh s cosh s cosh s cn3 p3 Anþ1 B3 tanh s cnpðnp þ 1Þðnp þ 2ÞAnþ1 B3 tanh s  þ ¼ 0: ðnþ1Þp ðnþ1Þpþ2 cosh s cosh s

cosh

ðnþ1Þpþ2

s ð19Þ

Proceeding as in the previous sub-section, yields the same value of the unknown exponent p as in (13). Again similarly, as before,

v¼

afblðl  n þ 1Þ þ cnðl  1Þðl þ n  1Þg

ð20Þ

2lfbðn þ 1Þ þ cn3 gAln1

and

B¼

ln1 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ;  bðn þ 1Þ þ cn3

ð21Þ

which leads to the constraint condition

afbðn þ 1Þ þ cn3 g < 0:

ð22Þ

Thus finally the 1-soliton solution to (2) is still given by (17) where the velocity and the width of the soliton are given by (20) and (21) respectively. In this case the constraint (22) along with the same constraint given by (18) must be valid for the soliton to exist. The following Fig. 2(a) and (b) shows the profile of a 1-soliton solution. For Fig. 2(a), A ¼ 0:5; a ¼ 1; b ¼ 1; c ¼ 2; l ¼ 3 and n ¼ 1 while for Fig. 2(b), A ¼ 0:5; a ¼ 1; b ¼ 2; c ¼ 1; l ¼ 3 and n ¼ 1. 2.3. Variant-III In this sub-section the focus will be on solving Eq. (3). Thus, substituting (6)–(10) in (3) yield l

lpv A B tanh s lp



anpAnþ1 B tanh s ðnþ1Þp

2



bn ðn þ 1Þp3 Anþ1 B3 tanh s ðnþ1Þp

þ

cosh s cosh s cosh s cn3 p3 Anþ1 B3 tanh s cnpðnp þ 1Þðnp þ 2ÞAnþ1 B3 tanh s  þ ¼ 0: ðnþ1Þp ðnþ1Þpþ2 cosh s cosh s

bnpðnp þ 1Þðnp þ p þ 2ÞAnþ1 B3 tanh s cosh

ðnþ1Þpþ2

s ð23Þ

Proceeding as in the previous sub-section, yields the same value of the unknown exponent p as in (13). Again similarly, as before,

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Fig. 1. Profile for solitons in Variant-I.

v¼

anðl þ n  1Þfbl þ cðl  1Þg 2lnfbðn þ 1Þ þ cngAln1

ð24Þ

and

B¼

ln1 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a  ; nfðn þ 1Þb þ cng

ð25Þ

which leads to the constraint condition

afbðn þ 1Þ þ cng < 0:

ð26Þ

Thus finally the 1-soliton solution to (3) is again given by (17) where the velocity and the width of the soliton are given by (24) and (25) respectively. In this case the constraint (26) along with the same constraint given by (18) must be valid for the

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Fig. 2. Profile for solitons in Variant-II.

soliton to exist. The following Fig. 3(a) and (b) shows the profile of a 1-soliton solution. For Fig. 3(a), A ¼ 0:5; a ¼ 1; b ¼ 2; c ¼ 2; l ¼ 4 and n ¼ 2 while for Fig. 3(b), A ¼ 0:5; a ¼ 1; b ¼ 3; c ¼ 1; l ¼ 3 and n ¼ 1. 3. Integrals of motion The three variants of the generalized KdV equation that are studied in this paper given by (1)–(3) permit at least two integrals of motion or conserved quantities. They are the momentum (M) and the energy (E) that are respectively given by

  l  A C 12 C ln1  ; l B C 12 þ ln1 1     Z 1 lþ1 Alþ1 C 12 C ln1 1  qlþ1 dx ¼ E¼ : lþ1 B C 2 þ ln1 1

M¼

Z

1

ql dx ¼

ð27Þ ð28Þ

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M.S. Ismail, A. Biswas / Applied Mathematics and Computation 216 (2010) 1673–1679

Fig. 3. Profile for solitons in Variant-III.

In each of the above cases the integrals are evaluated by using the 1-soliton solution that was obtained in the previous section. It needs to be noted that the center of the soliton in each of the three variants can be written as

R1 xql dx xðtÞ ¼ R1 ; 1 ql dx 1

ð29Þ

so that the velocity of the soliton is

v¼

dx ¼ dt

R1 R1 xðql Þ dx xðql Þt dx 1 R 1 l t ¼ 1 M q dx 1

ð30Þ

This will lead to the same results as obtained before in each of the three variants that are studied in this paper. This shows consistency of the method.

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4. Conclusions This paper obtains the 1-soliton solution to the three variants of the generalized KdV equation with generalized dispersion. The two conserved quantities are also calculated for each of these equations. The numerical simulations are also given for each of these three variants of the KdV equation. In future, these results will be used to study the perturbed generalized KdV equation. The soliton perturbation theory, stochastic perturbation integrability aspects will be touched upon in those cases. Those results will be reported in future. Acknowledgment The research work of the second author (A. B.) was fully supported by NSF-CREST Grant No: HRD-0630388 and this support is genuinely and sincerely appreciated. References [1] M. Antonova, A. Biswas, Adiabatic parameter dynamics of perturbed solitary waves, Communications in Nonlinear Science and Numerical Simulation 14 (3) (2009) 734–748. [2] M.S. Ismail, Numerical solution of complex modified Korteweg-de Vries equation by collocation method, Communications in Nonlinear Science and Numerical Simulation 14 (3) (2009) 749–759. [3] D. Lu, B. Hong, L. Tian, New solitary wave and periodic wave solutions for general types of KdV and KdV-Burgers equations, Communications in Nonlinear Science and Numerical Simulation 14 (1) (2009) 77–84. [4] H. Triki, M.S. Ismail, Solitary wave solutions for a coupled pair of mKdV equations, in press. [5] A.M. Wazwaz, Variants of the generalized KdV equation with compact and noncompact structures, Computers and Mathematics with Applications 47 (583–591) (2004). [6] A.M. Wazwaz, Compactons solitons and periodic solutions for variants of the KdV and the KP equations, Applied Mathematics and Computation 161 (2) (2005) 561–575. [7] A.M. Wazwaz, The extended tanh method for new solitons for many forms of the fifth-order KdV equations, Applied Mathematics and Computation 184 (2) (2007) 1002–1014. [8] L. Wazzan, A modified tanh–coth method for solving KdV and the KdV-Burger’s equation, Communications in Nonlinear Science and Numerical Simulation 14 (2) (2009) 443–450. [9] H. Zhang, New exact solutions for two generalized Hirota–Satsuma coupled KdV systems, Communications in Nonlinear Science and Numerical Simulation 12 (7) (2009) 1120–1127. [10] P.E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger’s Equations: Qualitative Theory, Springer Verlag, New York, NY, 2001.