The compact and noncompact structures for two types of generalized Camassa–Holm–KP equations

Mathematical and Computer Modelling 47 (2008) 1089–1098 www.elsevier.com/locate/mcm

The compact and noncompact structures for two types of generalized Camassa–Holm–KP equationsI Shaoyong Lai a,∗ , Yang Xu b a Department of Economic Mathematics, Southwestern University of Finance and Economics, 610074, Chengdu, China b Department of Applied Mathematics, Southwest JiaoTong University, Chengdu, 610031, China

Received 31 January 2007; received in revised form 3 June 2007; accepted 22 June 2007

Abstract The objective of this paper is to investigate two types of generalized nonlinear Camassa–Holm–KP equations in (2 + 1) dimensional space. Compactons, solitons, solitary patterns, periodic solutions and algebraic travelling wave solutions are expressed analytically under various circumstances. The conditions that cause the qualitative change in the physical structures of the solutions are emphasized. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Generalized Camassa–Holm–KP equations; Compactons; Solitons; Variants; Periodic solutions

1. Introduction Nonlinear partial differential equations which describe the nonlinear waves existing in the natural world have been investigated by many mathematical approaches, such as the Darboux transformation, the inverse scattering method, the Backlund ¨ transformation, the painleve´ analysis, the tri-Hamiltonian operators, the finite difference method, the Adomian decomposition method, the tanh method, the sine–cosine method, and so on [1–4,9–14]. Wazwaz [7–12] made some important discoveries that stated the causes of the qualitative change of the physical structures of solitary wave solutions for several nonlinear dispersive equations. Fan and Zhang [1,2] derived the extended tanh method and homogeneous balance method and used them to study the generalized mKdV equation and the generalized Z K equation. The method presented in [1,2] was shown as a powerful tool to seek exact solutions of nonlinear equations. Many researchers have used the extended tanh method to find travelling wave solutions for various forms of partial differential equations. Boyd [5] pointed out that the Camassa–Holm equation is a model for small amplitude shallow water waves and it gives peaked periodic solutions which have a discontinuous first derivative at each peak. New peaked solitary wave

I This work is supported in part by the Scientific Research Fund of Southwestern University of Finance and Economics (Grant No. 07YB53) and in part by the National Science Foundation of China (Grant No. 60474022). ∗ Corresponding author. E-mail address: [email protected] (S. Lai).

c 2007 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2007.06.020

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solutions for a modified Camassa–Holm equation were obtained by Tian and Song [6] in which many features of the solutions were discussed. Using the sine–cosine method and the tanh technique, Wazwaz [7] studied the following Camassa–Holm–KP equations given by (u t + 2ku x − u x xt − au n u x )x + u yy = 0

(1)

(u t + 2ku x − u x xt + au n (u n )x )x + u yy = 0,

(2)

and

where a, k and n are constants. The compactons, solitary patterns, solitons and periodic solutions for Eqs. (1) and (2) were obtained and expressed analytically. Also, Wazwaz [7] analyzed what causes the qualitative change in the physical structures of the solutions. In the present paper, we will study the generalized forms of Eqs. (1) and (2), which are written by   u t + 2ku x − (u m )x xt − au n u x x + u yy = 0 (3) and 

u t + 2ku x − (u m )x xt + au n (u n )x

 x

+ u yy = 0,

(4)

where nonzero constants a, k, m and n satisfy some assumptions mentioned later. The objective of this paper is to develop a mathematical method, which is different from those in previous works [1–8], to study Eqs. (3) and (4). More precisely, we will discuss Eq. (3) in four cases, where m = 2n + 1, m = n + 1, m + n = 1 and m = 1, and investigate Eq. (4) in the cases, where m = 4n − 1, m = 2n, m + 2n = 2 and m = 1. Compactons, solitons, solitary patterns, periodic solutions and algebraic travelling wave solutions of the two equations are acquired. For both equations when m = 1, we examine that the results in this paper are in full agreement with those presented by Wazwaz [7]. 2. The generalized Camassa–Holm–KP equation (3) We seek the travelling wave solutions for Eq. (3) in the form u = u(ξ ) with wave variable ξ = µ(x + y −ct), where µ and c are two nonzero constants. The wave variable ξ transforms Eq. (3) into the following ordinary differential equation (ODE) µ[−µcu ξ + 2kµu ξ + µ3 c(u m )ξ ξ ξ − au n µu ξ ]ξ + µ2 u ξ ξ = 0,

(5)

where m 6= 0, −1, n 6= 0, −1 and m + n + 1 6= 0. Integrating Eq. (5) twice and ignoring the additive constants yield the second order ODE (−c + 2k + 1)u − Using transformation µ2 cZ

au n+1 + µ2 c(u m )ξ ξ = 0. n+1

du m dξ

dZ mu m−1 du

(6)

dZ = Z , we have (u m )ξ ξ = Z du m , which changes Eq. (6) into the first order ODE

= (c − 2k − 1)u +

au n+1 . n+1

(7)

Integrating Eq. (7) with respect to u and ignoring the constant of integration, we have (c − 2k − 1)u m+1 au n+m+1 µ2 c 2 Z = + , 2m m+1 (n + 1)(m + n + 1)

(8)

from which we obtain µ2 c 2m

mu

m−3 2

dξ

du

!2 =

(c − 2k − 1) au n + . m+1 (n + 1)(m + n + 1)

(9)

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Letting u n = V 2 , we have 2

u = V n,

du =

2 2 −1 V n dV, n

(10)

which makes Eq. (9) become 2µ2 mc n2

V

m−1 n −1

dξ

dV

!2 =

(c − 2k − 1) aV 2 + . m+1 (n + 1)(m + n + 1)

(11)

2.1. m = 2n + 1 Assuming m = 2n + 1 lets Eq. (11) turn into the following   aV 2 2µ2 (2n + 1)c V dV 2 (c − 2k − 1) + , = dξ 2(n + 1) (n + 1)(3n + 2) n2 where n 6= − 12 , − 32 , −1 and c − 2k − 1 6= 0. Solving the above equation directly, we acquire the algebraic travelling wave solution in the form  u=

(2k + 1 − c)(3n + 2) an 2 (x + y − ct)2 + 2a 2c(2n + 1)(n + 1)(3n + 2)

 n1

.

2.2. m = n + 1 If m = n + 1, we know that Eq. (11) becomes 

dV dξ

2 =

  c − 2k − 1 n2 aV 2 , + n+2 2µ2 c(n + 1) 2(n + 1)2

in which we assume n 6= −1, −2 and c − 2k − 1 6= 0. a 2.2.1. c(n+1) < 0, n > 0 Solving Eq. (12), we obtain the compacton solutions given in the forms     n1 r 2  2(2k + 1 − c)(n + 1) n a  2 u = cos (x + y − ct) − ,    a(n + 2) 2(n + 1) c(n + 1)  r π n a  < ,  (x + y − ct) −    2(n + 1) c(n + 1) 2   u = 0, otherwise

and     n1 r  2(2k + 1 − c)(n + 1)2 n a  2   u= sin − (x + y − ct) ,   a(n + 2) 2(n + 1) c(n + 1)   r  n a  < π,  − (x + y − ct)   2(n + 1) c(n + 1)    u = 0, otherwise.

(12)

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a 2.2.2. c(n+1) > 0, n > 0 The solitary pattern solutions are given by

 u=

  n1 r 2(2k + 1 − c)(n + 1)2 n a 2 cosh (x + y − ct) a(n + 2) 2(n + 1) c(n + 1)

and   n1 r a 2(2k + 1 − c)(n + 1)2 n 2 u= − . sinh (x + y − ct) a(n + 2) 2(n + 1) c(n + 1) 

a 2.2.3. c(n+1) < 0, n < 0 The periodic solutions take in the forms



 − n1 r n a(n + 2) a 2 sec − (x + y − ct) 2(n + 1) c(n + 1) 2(2k + 1 − c)(n + 1)2



 − n1 r n a(n + 2) a 2 csc (x + y − ct) . − 2(n + 1) c(n + 1) 2(2k + 1 − c)(n + 1)2

u= and u=

a 2.2.4. c(n+1) > 0, n < 0 This case admits Eq. (12) to have solitons



 − n1 r n a(n + 2) a 2 sech (x + y − ct) 2(n + 1) c(n + 1) 2(2k + 1 − c)(n + 1)2



 − n1 r −a(n + 2) n a 2 (x + y − ct) csch . 2(n + 1) c(n + 1) 2(2k + 1 − c)(n + 1)2

u= and u=

2.3. m + n = 1 It follows from Eq. (11) and assumption m + n = 1 that     dV 2 c − 2k − 1 aV 2 n2 + , = 2−n 2(n + 1) V 2 dξ 2cµ2 (1 − n)

(13)

where n 6= −1, 1, 2 and c − 2k − 1 6= 0. From (13), we obtain the algebraic travelling wave solution in the form  u=

2c(1 − n 2 )(2 − n)(c − 2k − 1) n 2 (n + 1)(c − 2k − 1)2 (x + y − ct)2 − ac(1 − n)(2 − n)2

 n1

.

2.4. m = 1 When m = 1, Eq. (3) is identical to Eq. (1). In this situation, Eq. (11) becomes   2µ2 c dV 2 c − 2k − 1 aV 2 = + . V dξ 2 (n + 1)(n + 2) n2

(14)

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2.4.1. c−2k−1 < 0, n > 0 c Solving Eq. (14), we have " r # (2k + 1 − c)(n + 1)(n + 2) 2k + 1 − c 2 nξ V = sec 2a 2µ c

(15)

" r # nξ 2k + 1 − c (2k + 1 − c)(n + 1)(n + 2) V2 = csc2 . 2a 2µ c

(16)

2

or

Using

ξ µ

= x + y − ct, we get that the periodic solutions for Eq. (3) in the case where m = 1 are in the forms (

u=

" r #) 1 n n (2k + 1 − c)(n + 1)(n + 2) 2k + 1 − c sec2 (x + y − ct) 2a 2 c

(17)

" r #) 1 n (2k + 1 − c)(n + 1)(n + 2) 2 n 2k + 1 − c csc (x + y − ct) . 2a 2 c

(18)

and ( u=

> 0, n > 0 2.4.2. c−2k−1 c The soliton solutions of Eq. (3) take the forms ( u=

" r #) 1 n (2k + 1 − c)(n + 1)(n + 2) 2 n c − 2k − 1 sech (x + y − ct) 2a 2 c

(19)

and " r #) 1 n n (2k + 1 − c)(n + 1)(n + 2) c − 2k − 1 u= − . csch2 (x + y − ct) 2a 2 c (

2.4.3. c−2k−1 < 0, n < 0, n 6= −1, −2 c The compacton solutions are given by  " r #)− 1 ( n  n 2k + 1 − c 2a  2   cos (x + y − ct) , u =   (2k + 1 − c)(n + 1)(n + 2) 2 c    r π n 2k + 1 − c     (x + y − ct) < ,   2 c 2    u = 0, otherwise

(20)

(21)

and  " r #)− 1 ( n  n 2k + 1 − c 2a  2   sin (x + y − ct) , u =   (2k + 1 − c)(n + 1)(n + 2) 2 c    r n 2k + 1 − c     (x + y − ct) < π,   2 c    u = 0, otherwise.

(22)

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2.4.4. c−2k−1 > 0, n < 0, n 6= −1, −2 c We obtain the solitary pattern solutions as follows ( u=

" r #)− 1 n 2a 2 n c − 2k − 1 cosh (x + y − ct) (2k + 1 − c)(n + 1)(n + 2) 2 c

(23)

" r #)− 1 n n −2a c − 2k − 1 sinh2 (x + y − ct) . (2k + 1 − c)(n + 1)(n + 2) 2 c

(24)

and ( u=

The solution formulas from (17) to (24) are in full agreement with those presented by Wazwaz [7] in which the sine–cosine method and the tanh technique were used. 3. The generalized Camassa–Holm–KP equation (4) As in Section 2, the variable ξ = µ(x + y − ct) turns Eq. (4) into the following ODE (−c + 2k + 1)u +

au 2n + µ2 c(u m )ξ ξ = 0. 2

dZ Setting (u m )ξ = Z , we have (u m )ξ ξ = Z du m and thus Eq. (25) becomes   au 2n m−1 µ2 c Z dZ = 0, (−c + 2k + 1)u + u du + 2 m

(25)

(26)

where m 6= −1, 0 and 2n + m 6= 0. Integrating (26) and setting the additive constant to zero yield 2(−c + 2k + 1)u m+1 au 2n+m µ2 c 2 + + Z = 0, m+1 2n + m m from which we have u

µ cm 2

m−3 2

du

!2

dξ

=

au 2n−1 2(c − 2k − 1) − . m+1 2n + m

2

Setting u = W 2n−1 (n 6= 12 ) transforms the above equation into m−1

W 2n−1 −1 dW dξ

!2

1 = 2 µ cm



2n − 1 2

2 

 2(c − 2k − 1) aW 2 − . m+1 2n + m

(27)

3.1. m = 4n − 1 If m = 4n − 1, the Eq. (27) becomes       W dW 2 1 2n − 1 2 (c − 2k − 1) aW 2 = 2 − , dξ 2 2n 6n − 1 µ c(4n − 1) where n 6= 0, 16 , 14 , 21 and c − 2k − 1 6= 0. Solving (28), we get the algebraic travelling wave solution given by  u=

  1 1 (6n − 1)(c − 2k − 1) a 2 (2n − 1)2 (x + y − ct)2 2n−1 − . 2a n 2c(4n − 1)(6n − 1)

(28)

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1095

3.2. m = 2n The assumption m = 2n makes Eq. (27) take in the following form 

where

dW dξ

2

c−2k−1 2n+1

1 = 2cnµ2



2n − 1 2

2 

 2(c − 2k − 1) aW 2 − , 2n + 1 4n

6= 0, n 6= 0, 12 .

3.2.1. ac > 0, n > 12 It follows from (29) that the compacton solutions for Eq. (4) have the forms  1 r   2n−1   2n − 1 a 8n(c − 2k − 1)  2   cos (x + y − ct) , u= √   a(2n + 1)  4n 2 c         

r 2n − 1 a π √ < , (x + y − ct) 2 4n 2 c u = 0, otherwise

and  1 r   2n−1   8n(c − 2k − 1) 2 2n − 1 a    sin (x + y − ct) , u= √   a(2n + 1)  4n 2 c  r 2n − 1 a   < π,  (x + y − ct) √   c  4n 2   u = 0, otherwise. 3.2.2. ac < 0, n > 12 The solitary pattern solutions admit the forms  u=

1 r   2n−1 8n(c − 2k − 1) a 2 2n − 1 cosh − (x + y − ct) √ a(2n + 1) c 4n 2

and 1 r   2n−1 8n(c − 2k − 1) a 2 2n − 1 u= − sinh − (x + y − ct) . √ a(2n + 1) c 4n 2



3.2.3. ac > 0, n < 12 The periodic solutions are given by  u=

1 r   1−2n a(2n + 1) 2 2n − 1 a sec (x + y − ct) √ 8n(c − 2k − 1) 4n 2 c

and  u=

1 r   1−2n a(2n + 1) 2 2n − 1 a csc (x + y − ct) . √ 8n(c − 2k − 1) 4n 2 c

(29)

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3.2.4. ac < 0, n < 21 The case allows Eq. (4) to have solitons in the forms  u=

1 r   1−2n a(2n + 1) a 2 2n − 1 sech − (x + y − ct) √ 8n(c − 2k − 1) c 4n 2

and 1 r   1−2n a(2n + 1) a 2 2n − 1 csch u= − − (x + y − ct) . √ 8n(c − 2k − 1) c 4n 2



3.3. m = 1 When m = 1, Eq. (4) becomes Eq. (2). Solving Eq. (27) directly, we can obtain the following travelling wave solutions. 3.3.1. c−2k−1 < 0, n > 21 c It follows from Eq. (27) that the periodic solutions of Eq. (4) take the forms ( u=

" #) 1 r 2n−1 (c − 2k − 1)(2n + 1) 2 2n − 1 2k + 1 − c sec (x + y − ct) a 2 c

(30)

" #) 1 r 2n−1 (c − 2k − 1)(2n + 1) 2 2n − 1 2k + 1 − c csc (x + y − ct) . a 2 c

(31)

and ( u=

> 0, n > 21 3.3.2. c−2k−1 c We obtain the following solitons for Eq. (4) ( u=

" #) 1 r 2n−1 (c − 2k − 1)(2n + 1) 2 2n − 1 c − 2k − 1 sech (x + y − ct) a 2 c

(32)

and " #) 1 r 2n−1 (c − 2k − 1)(2n + 1) 2 2n − 1 c − 2k − 1 csch (x + y − ct) u= − . a 2 c (

3.3.3. c−2k−1 < 0, n < 21 , n 6= − 12 c The compacton solutions are expressed in the forms  #) 1 " ( r 1−2n   2n − 1 2k + 1 − c a u = 2  cos (x + y − ct) ,   (c − 2k − 1)(2n + 1) 2 c    2n − 1 r 2k + 1 − c π     (x + y − ct) < ,   2 c 2    u = 0, otherwise

(33)

(34)

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1097

and  " #) 1 ( r 1−2n   2n − 1 2k + 1 − c a  2  , sin (x + y − ct) u =   (c − 2k − 1)(2n + 1) 2 c    2n − 1 r 2k + 1 − c     (x + y − ct) < π,   2 c    u = 0, otherwise.

(35)

3.3.4. c−2k−1 > 0, n < 12 , n 6= − 12 c This case admits that Eq. (4) has the solitary pattern solutions given by ( u=

" #) 1 r 1−2n a 2 2n − 1 c − 2k − 1 cosh (x + y − ct) (c − 2k − 1)(2n + 1) 2 c

(36)

and " #) 1 r 1−2n a 2 2n − 1 c − 2k − 1 u= − sinh (x + y − ct) . (c − 2k − 1)(2n + 1) 2 c (

(37)

The solutions from (30) to (37) are in full agreement with those presented by Wazwaz [7]. 3.4. m + 2n = 2 The assumption m + 2n = 2 makes Eq. (27) become the following form 

dW W 2 dξ

2 =

1 µ2 c(2 − 2n)



2n − 1 2

2 

 2(c − 2k − 1) a 2 − W , 3 − 2n 2

(38)

where n 6= 21 , 32 and c − 2k − 1 6= 0. From (38), we obtain the algebraic travelling wave solution  u=

4c(1 − n)(3 − 2n)(c − 2k − 1) 2 (c − 2k − 1) (2n − 1)2 (x + y − ct)2 + ac(3 − 2n)2 (1 − n)



1 2n−1

.

4. Conclusion A mathematical method is developed to study the generalized Camassa–Holm–KP equations (3) and (4). For each of the equations, we examined four different cases where exponents m and n satisfy suitable relations. The compactons, solitary patterns, solitons, periodic solutions and algebraic travelling wave solutions are obtained analytically. When m = 1, the results acquired for the two equations are in full agreement with those presented by Wazwaz [7] in which the tanh method and sine–cosine technique are used. The method presented here may be applied to find exact solutions for other partial differential equations which satisfy certain conditions. Acknowledgments The authors are grateful to the referees, whose comments led to a number of meaningful improvements.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

E.G. Fan, H.Q. Zhang, New exact solutions for a system of coupled KdV equation, Phys. Lett. A 245 (1998) 389–392. E.G. Fan, H.Q. Zhang, A note on the homogeneous balance method, Phys. Lett. A 246 (1998) 403–406. E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 212 (2000) 277–284. Y. Chen, B. Li, Hongqing Zhang, New exact solutions for modified nonlinear dispersive equations m K (m, n) in higher dimensional spaces, Math. Comput. Simulation 64 (2004) 549–559. J.P. Boyd, Peakons and cashoidal waves: Travelling wave solutions of the Camassa–Holm equation, Appl. Math. Comput. 81 (2–3) (1997) 173–187. L. Tian, X. Song, New peaked solitary wave solutions of the generalized Camassa–Holm equation, Chaos Solitons Fractals 19 (2004) 621–637. A.M. Wazwaz, The Camassa–Holm–KP equations with compact and noncompact travelling wave solutions, Appl. Math. Comput. 170 (2005) 347–360. A.M. Wazwaz, The sine–cosine and tanh method: Reliable tools for analytic treatment of nonlinear dispersive equations, Appl. Math. Comput. 173 (2006) 150–164. A.M. Wazwaz, Nonlinear variants of the BBM equation with compact and noncompact physical structures, Chaos Solitons Fractals 26 (2005) 767–776. A.M. Wazwaz, Two reliable methods for solving variants of the KdV equation with compact and noncompact structures, Chaos Solitons Fractals 28 (2006) 454–462. A.M. Wazwaz, New solitary wave special solutions with compact support for the nonlinear dispersive K (m, n) equations, Chaos Solitons Fractals 13 (2) (2002) 321–330. A.M. Wazwaz, Compactons in a class of nonlinear dispersive equations, Math. Comput. Modelling 37 (3–4) (2003) 333–341. I. Dag, B. Saka, D. Irk, Galerkin method for the numerical solution of the RLW equation using quintic B-splines, J. Comput. Appl. Math. 190 (2006) 532–547. F.M. Allan, K. Al-Khaled, An approximation of the analytic solution of the shock wave equation, J. Comput. Appl. Math. 192 (2006) 301–309.