A new auxiliary function method for solving the generalized coupled Hirota–Satsuma KdV system

In this letter, a new auxiliary function method is presented for constructing exact travelling wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions of the elliptic equation to construct exact travelling wave solutions of nonlinear partial differential equations. More new exact travelling wave solutions are obtained for the generalized coupled Hirota–Satsuma KdV system.     

On soliton solutions for a generalized Hirota–Satsuma coupled KdV equation

The extended homogeneous balance method is used to construct exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation are successfully obtained, which contain soliton-like and periodic-like solutions This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

New travelling wave solutions of the generalized coupled Hirota–Satsuma KdV system

In this paper, by the application of hyperbolic function, triangle function and symbolic computation, we devise a new method to seek the exact travelling wave solutions of the nonlinear partial differential equations in mathematical physics. The generalized coupled Hirota–Satsuma KdV system is chosen to illustrate the approach. As a consequence, abundant new solitary and periodic solutions are obtained.     

Bifurcations of traveling wave solutions for the generalized coupled Hirota–Satsuma KdV system

In this paper, the bifurcations of solitary, kink and periodic waves for the generalized coupled Hirota–Satsuma KdV system are studied by using the bifurcation theory of planar dynamical systems. Bifurcation parameter sets are shown. Under given parameter conditions, explicit formulas for solitary wave solutions, kink wave solutions and periodic wave solutions are obtained.     

New Darboux transformation for Hirota–Satsuma coupled KdV system

A new Darboux transformation (DT) is presented for the Hirota–Satsuma coupled KdV system. It is shown that this DT can be constructed by means of two methods: Painlevé analysis and reduction of a binary DT. By iteration of the DT, the Grammian type solutions are found for the coupled KdV system.     

New generalized Jacobi elliptic function expansion method

A new generalized Jacobi elliptic function expansion method is described and used for constructing many new exact travelling wave solutions for nonlinear partial differential equations (PDEs) in a unified way. We obtain many new Jacobi and Weierstrass double periodic elliptic function solutions for (3 + 1)-dimensional Kadmtsev–Petviashvili (KP) equation. This method can be applied to many other equations.     

New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV system

In this paper, an generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used for constructing more new exact Jacobi elliptic functions solutions of the generalized coupled Hirota-Satsuma KdV system. As a result, eight families of new doubly periodic solutions are obtained by using this method, some of these solutions are degenerated to solitary wave solutions and triangular functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the applied method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.     

New generalized Jacobi elliptic function rational expansion method

In this work, a new generalized Jacobi elliptic function rational expansion method is based upon twenty-four Jacobi elliptic functions and eight double periodic Weierstrass elliptic functions, which solve the elliptic equation ?^′2=r+p?²+q?⁴, is described. As a consequence abundant new Jacobi-Weierstrass double periodic elliptic functions solutions for (3+1)-dimensional Kadmtsev-Petviashvili (KP) equation are obtained by using this method. We show that the new method can be also used to solve other nonlinear partial differential equations (NPDEs) in mathematical physics.     

White noise functional solutions of Wick-type stochastic generalized Hirota–Satsuma coupled KdV equations

Generalized Hirota–Satsuma coupled KdV equations with variable coefficients and Wick-type stochastic generalized Hirota–Satsuma coupled KdV equations are investigated. White noise functional solutions are shown by Hermite transform, homogeneous balance principle and F-expansion method.     

Jacobian elliptic function expansion solutions for the Wick-type stochastic coupled KdV equations

The Jacobian elliptic function expansion approach is extended and applied to construct exact solutions of the nonlinear Wick-type stochastic coupled KdV equations and some new exact solutions are obtained via the approach and Hermite transformation.     

The generalized dressing method with applications to variable-coefficient coupled Kadomtsev–Petviashvili equations

A generalized version of the dressing method is applied to solve the variable-coefficient coupled Kadomtsev–Petviashvili (cKP) equations. The compatibility of the dressed operators leads to two sets of equations, which are reduced to the variable-coefficient cKP equations. With the help of the bilinear transformation method, the bilinearized variable-coefficient cKP equations are obtained. Further, the N-soliton solution of the variable-coefficient cKP equations is given by using Pfaffian.     

New exact solutions for two generalized Hirota–Satsuma coupled KdV systems

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact solutions for nonlinear evolution equations. By this method two generalized Hirota–Satsuma coupled KdV systems are investigated and new exact solutions are explicitly obtained with the aid of symbolic computation.     

The generalized Kupershmidt deformation for the fifth-order coupled KdV equations hierarchy

Based on the Kupershmidt deformation, we propose the generalized Kupershmidt deformation (GKD) to construct new systems from integrable bi-Hamiltonian system. As applications, the generalized Kupershmidt deformation of the fifth-order coupled KdV equations hierarchy with self-consistent sources and its Lax representation are presented.     

A modified extended method to find a series of exact solutions for a system of complex coupled KdV equations

In this paper an algebraic method is devised to uniformly construct a series of complete new exact solutions for general nonlinear equations. For illustration, we apply the modified proposed method to revisit a complex coupled KdV system and successfully construct a series of new exact solutions including the soliton solutions and elliptic doubly periodic solutions.     

A new method for a generalized Hirota-Satsuma coupled KdV equation

In this paper, differential transform method (DTM), which is one of the approximate methods is implemented for solving the nonlinear Hirota-Satsuma coupled KdV partial differential equation. A variety of initial value system is considered, and the convergence of the method as applied to the Hirota-Satsuma coupled KdV equation is illustrated numerically. The obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient. Numerical examples are illustrated the pertinent features of the proposed algorithm.     

The extension of the Jacobi elliptic function rational expansion method

Using a new ansätz, we extend the Jacobi elliptic function rational expansion method and apply it to the asymmetric Nizhnik–Novikov–Veselov equations and the Davey–Stewartson equations. With the aid of symbolic computation, we construct more new Jacobi elliptic doubly periodic wave solutions and the corresponding solitary wave solutions and triangular functional (singly periodic) solutions.     

Construction of periodic and solitary wave solutions by the extended Jacobi elliptic function expansion method

In this paper, an extended Jacobi elliptic function expansion method is used with a computerized symbolic computation for constructing the exact periodic solutions of some polynomials or nonlinear evolution equations. The validity and reliability of the method is tested by its applications on a class of nonlinear evolution equations of special interest in nonlinear mathematical physics. As a result, many exact travelling wave solutions are obtained which include new solitary or shock wave solution and envelope solitary and shock wave solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

Nonlinear transform and Jacobi elliptic function solutions of nonlinear equations

The nondegenerative elliptic function solutions of some nonlinear equations are obtained by a nonlinear transform, which names the Jacobi elliptic function expansion. When taking particular parameters, the elliptic function solutions can degenerate as solitary wave solutions and singularity solutions.     

Exact elliptic compactons in generalized Korteweg–De Vries equations

Using the action principle, and assuming a solitary wave of the generic form u(x,t) = AZ(β(x + q(t)), we derive a general theorem relating the energy, momentum, and velocity of any solitary wave solution of the generalized Korteweg‐De Vries equation K*(l,p). Specifically we find that , where l,p are nonlinearity parameters. We also relate the amplitude, width, and momentum to the velocity of these solutions. We obtain the general condition for linear and Lyapunov stability. We then obtain a two‐parameter family of exact solutions to these equations, which include elliptic and hyper‐elliptic compacton solutions. For this general family we explicitly verify both the theorem and the stability criteria. © 2006 Wiley Periodicals, Inc. Complexity 11: 30–34, 2006