New explicit solitary wave solutions for (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation
Introduction
In recent years, searching for explicit exact solutions, in particular, solitary wave solutions, of nonlinear evolution equations (NEEs) in mathematical physics plays an important role in soliton theory [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [14], [21]. Particularly, various powerful methods have been presented, such as, Backlund transformation, Darboux transformation, Cole–Hopf transformation, tanh method, sine–cosine method, Painlevé method, homogeneous balance method (HBM), Hirota method [12], Lie group analysis, similarity reduced method and so on. Based upon the well-known Riccati equation, homogeneous balance method (HBM) proposed by Wang et al. [6], [7] is to find exact solutions of certain nonlinear PDEs. Fan and Zhang [13], [15] improved considerably the key steps of the HBM. Particularly, more general ansatz have been proposed in order to obtain new form of solutions. Recently, Senthilvelan [16] studied the travelling wave solutions for (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation by homogeneous balance method (HBM) and explored certain new solution of the equations. In this Letter, we would like to discuss further (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation by our improved method, in which we presented a new generalized transformation [17]. As a result, more new exact solutions, which include the solutions obtained by Senthilvel [16], are obtained.
Section snippets
Method
Our method is summed up as follows.
For the given nonlinear evolution equations, say, in three variables, x,y,t we seek the following formal travelling wave solutions where α,β,λ are all constants to be determined later. Then (2) reduces to a nonlinear ordinary differential equation where “” denotes . In order to seek the travelling wave solutions of (3), we take the following transformations
(2+1)-dimensional Boussinesq equation
Let us consider a (2+1)-dimensional generalization of Boussinesq equation [20] According to the above steps, we firstly make the following formal travelling wave transformation: where α,β,λ are constants to be determined.
Substituting (9) into (8) and integrating it twice reads According to Step 1 in Section 2, we support that (10) has the following formal solutions and ω=ω(ξ)
Conclusions
In summary, based on the well-known Riccati equation, many new types of exact solutions for both (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation have been derived by a generalized transformation. These solutions contain the known ones [16]. Seven kinds of them are singular soliton solutions. Such solutions develop a singularity at a finite point, i.e., for any fixed t=t0, there exist x0 at which these solutions blow up. There is much current interest in the formation of
Acknowledgements
The work is supported by the National Natural Science Foundation of China under the Grant No. 1007201, the National Key Basic Research Development Project Program under the Grant No. G1998030600 and Doctoral Foundation of China under the Grant No. 98014119.
References (21)
Phys. Lett. A
(1996)Phys. Lett. A
(1996)Phys. Lett. A
(1995)- et al.
Phys. Lett. A
(1999) Int. J. Non-Linear Mech.
(1996)- et al.
Phys. Lett. A
(1998) - et al.
Phys. Lett. A
(1998) Appl. Math. Comput.
(2001)- et al.
Phys. Lett. A
(2001) - et al.
Phys. Lett. A
(1997)
Cited by (132)
On the soliton solutions to the modified Benjamin-Bona-Mahony and coupled Drinfel'd-Sokolov-Wilson models and its applications
2020, Journal of King Saud University - Science