On the nontrivial non-travelling wave profiles of nonlinear evolution and wave equations

The overall aim of the present paper is to find and analyze the new non-travelling wave solutions of the nonlinear evolution and wave equations. With the aid of symbolic computation and based on the generalized extended tanh-function method, we propose the newly extended tanh-function expansion algorithm and get many new non-travelling wave solutions of the (2 + 1)-dimensional Broer–Kaup–Kupershmidt equations. The solutions which we obtain are more abundant than the solutions which the generalized extended tanh-function method gets. At the same time, the solutions contain arbitrary functions which may be helpful to explain some complex phenomena. We also give some figures to describe the property of these solutions. In additions, the method can also be successfully applied to other nonlinear evolution and wave equations.     

A generalized F-expansion method to find abundant families of Jacobi Elliptic Function solutions of the (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation

In the present paper, a generalized F-expansion method is proposed by further studying the famous extended F-expansion method and using a generalized transformation to seek more types of solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations to illustrate the validity and advantages of the method. As a result, abundant new exact solutions are obtained including Jacobi Elliptic Function solutions, soliton-like solutions, trigonometric function solution etc. The method can be also applied to other nonlinear partial differential equations.     

New families of exact soliton-like solutions and dromion solutions to the (2+1)-dimensional higher-order Broer–Kaup equations

Using homogeneous balance method we obtain Bäcklund transformation (BT) and a linear partial differential equation of higher-order Broer–Kaup equations. As a result, new soliton-like solutions and new dromion solution and other exact solutions of (2 + 1)-dimensional higher-order Broer–Kaup equations are given. By analyzing a soliton-like solution, we get some dromions solutions. This method, which can be generalized to some (2 + 1)-dimensional nonlinear evolution equations, is simple and powerful.     

A series of new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation

In this paper, we extend the algebraic method proposed by Fan (Chaos, Solitons & Fractals 20 (2004) 609) and the improved extended tanh method by Yomba (Chaos, Solitons and Fractals 20 (2004) 1135) to uniformly construct a series of soliton-like solutions and double-like periodic solutions for nonlinear partial differential equations (NPDE). Some new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation are obtained.     

Notes on the equivalence of different variable separation approaches for nonlinear evolution equations

The equivalence of multilinear variable separation approach, the extended projective Ricatti equation method and the improved tanh-function method is firstly reported when these three popular methods are used to realize variable separation for nonlinear evolution equations. We take the (2 + 1)-dimensional modified Broer–Kaup system for an example to illustrate this point. All solutions obtained by the extended projective Ricatti equation method and the improved tanh-function method coincide with the one obtained by the multilinear variable separation approach. Moreover, based on one of variable separation solutions, we also find that although abundant localized coherent structures can be constructed for a special component, we must pay our attention to the solution expression of the corresponding other component for the same equation lest many un-physical related structures might be obtained.     

Variable-coefficient projective Riccati equation method and its application to a new (2 + 1)-dimensional simplified generalized Broer–Kaup system

In this paper, based on a new intermediate transformation, a variable-coefficient projective Riccati equation method is proposed. Being concise and straightforward, it is applied to a new (2 + 1)-dimensional simplified generalized Broer–Kaup (SGBK) system. As a result, several new families of exact soliton-like solutions are obtained, beyond the travelling wave. When imposing some condition on them, the new exact solitary wave solutions of the (2 + 1)-dimensional SGBK system are given. The method can be applied to other nonlinear evolution equations in mathematical physics.     

The stochastic soliton-like solutions of stochastic KdV equations

By means of a generalized method and symbolic computation, we consider a stochastic KdV equation U_t + f(t)U ♢ U_x + g(t)U_xxx = W(t) ♢ R^♢(t, U, U_x, U_xxx). We construct new and more general formal solutions. At the same time, we recover all the solutions found by Xie [Phys. Lett. A 310 (2003) 161]. The solutions obtained include the nontravelling wave and coefficient function’s stochastic soliton-like solutions, singular stochastic soliton-like solutions, stochastic triangular functions solutions.     

An improved generalized F-expansion method and its application to the (2 + 1)-dimensional KdV equations

An improved generalized F-expansion method is proposed to seek exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2 + 1)-dimensional KdV equations to illustrate the validity and advantages of the proposed method. Many new and more general non-travelling wave solutions are obtained, including single and combined non-degenerate Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions, each of which contains two arbitrary functions.     

New variable separation solutions and nonlinear phenomena for the (2+1)-dimensional modified Korteweg–de Vries equation

Variable separation approach, which is a powerful approach in the linear science, has been successfully generalized to the nonlinear science as nonlinear variable separation methods. The (2 + 1)-dimensional modified Korteweg–de Vries (mKdV) equation is hereby investigated, and new variable separation solutions are obtained by the truncated Painlevé expansion method and the extended tanh-function method. By choosing appropriate functions for the solution involving three low-dimensional arbitrary functions, which is derived by the truncated Painlevé expansion method, two kinds of nonlinear phenomena, namely, dromion reconstruction and soliton fission phenomena, are discussed.     

Soliton-like and periodic form solutions to (2 + 1)-dimensional Toda equation

In this paper, we present a further extended tanh method for constructing exact solutions to nonlinear difference-differential equation(s) (NDDEs) and Lattice equations. By using this method via symbolic computation system MAPLE, we obtain abundant soliton-like and period-form solutions to the (2 + 1)-dimensional Toda equation. Solitary wave solutions are merely a special case in one family. This method can also be used to other nonlinear difference differential equations.     

Symbolic computation and construction of soliton-like solutions for a breaking soliton equation

Based on the symbolic computation system––Maple and a Riccati equation, by introducing a new more general ansätz than the ansätz in the tanh method, extended tanh-function method, modified extended tanh-function method, generalized tanh method and generalized hyperbolic-function method, we propose a generalized Riccati equation expansion method for searching for exact soliton-like solutions of nonlinear evolution equations and implemented in computer symbolic system––Maple. Making use of our method, we study a typical breaking soliton equation and obtain new families of exact solutions, which include the nontravelling wave’ and coefficient function’ soliton-like solutions, singular soliton-like solutions and periodic solutions. The arbitrary functions of some solutions are taken to be some special constants or functions, the known solutions of this equation can be recovered.     

Chaotic behaviors in the (2 + 1)-dimensional breaking soliton system

Starting from the extended tanh-function method based on mapping method, the variable separation solutions of the (2 + 1)-dimensional breaking soliton system are derived. By further studying, we find that these variable separation solutions, which seem independent, actually depend on each other. Based on the derived variable separation solution, chaotic behaviors, i.e. periodic solution with chaotic behavior and chaotic peaked and compact line solitons, are investigated.     

Relation among nonlinear evolution equations and their reductions

The LCZ soliton hierarchy is presented, and their generalized Hamiltonian structures are deduced. From the compatibility of soliton equations, it is shown that this soliton hierarchy is closely related to the Burger equation, the mKP equation and a new (2 + 1)-dimensional nonlinear evolution equation (NEE). Resorting to the nonlinearization of Lax pairs (NLP), all the resulting NEEs are reduced into integrable Hamiltonian systems of ordinary differential equations (ODEs). As a concrete application, the solutions for NEEs can be derived via solving the corresponding ODEs.     

New exact solutions to MKDV-Burgers equation and (2 + 1)-dimensional dispersive long wave equation via extended Riccati equation method

In this paper, with the aid of symbolic computation and a general ansätz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansätz. The method can also be applied to other nonlinear partial differential equations.     

New exact travelling wave solutions using modified extended tanh-function method

Based on computerized symbolic computation and modified extended tanh-function method for constructing a new exact travelling wave solutions of nonlinear evolution equations (NEEs) is presented and implemented in a computer algebraic system. Applying this method, with the aid of Maple, we consider some (NEEs) with mathematical physics interests. As a results, we can successfully recover the previously known solitary wave solutions that had been found by the tanh-function method and other more sophisticated methods.     

A new Jacobi elliptic function rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation

With the aid of computerized symbolic computation, a new elliptic function rational expansion method is presented by means of a new general ansätz and is very powerful to uniformly construct more new exact doubly-periodic solutions in terms of rational formal Jacobi elliptic function of nonlinear evolution equations (NLEEs). As an application of the method, we choose a (1 + 1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.     

Trilinearization and localized coherent structures and periodic solutions for the (2 + 1) dimensional K-dV and NNV equations

In this paper, using a novel approach involving the truncated Laurent expansion in the Painlevé analysis of the (2 + 1) dimensional K-dV equation, we have trilinearized the evolution equation and obtained rather general classes of solutions in terms of arbitrary functions. The highlight of this method is that it allows us to construct generalized periodic structures corresponding to different manifolds in terms of Jacobian elliptic functions, and the exponentially decaying dromions turn out to be special cases of these solutions. We have also constructed multi-elliptic function solutions and multi-dromions and analysed their interactions. The analysis is also extended to the case of generalized Nizhnik–Novikov–Veselov (NNV) equation, which is also trilinearized and general class of solutions obtained.     

New exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation

The repeated homogeneous balance method is used to construct new exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain soliton-like and periodic-like solutions. This method is straightforward and concise, and it can be also applied to other nonlinear evolution equations.     

New exact solutions to the Zakharov–Kuznetsov equation and its generalized form

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional Zakharov–Kuznetsov (ZK) equation and its generalized form. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the solitons solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.     

New abundant exact solutions for the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov system

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov (GNNV) system. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the soliton solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.