The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2 + 1)-dimensional Boiti–Leon–Pempinelle equation

The tanh method is used to find travelling wave solutions to various wave equations. In this paper, the extended tanh function method is further improved by the generalizing Riccati equation mapping method and picking up its new solutions. In order to test the validity of this approach, the (2 + 1)-dimensional Boiti–Leon–Pempinelle equation is considered. As a result, the abundant new non-travelling wave solutions are obtained.     

New exact solutions of the Broer–Kaup equations in (2 + 1)-dimensional spaces

With the aid of Maple, several new kinds of exact solutions for the Broer–Kaup equations in (2 + 1)-dimensional spaces are obtained by using a new ansätz. This approach can also be applied to other nonlinear evolution equations.     

Two hierarchies of new nonlinear evolution equations associated with 3 × 3 matrix spectral problems

Two hierarchies of new nonlinear evolution equations associated with 3 × 3 matrix spectral problems are proposed. The generalized bi-Hamiltonian structures for one of the two hierarchies are derived with the aid of the trace identity. Some explicit solutions of a typical nonlinear evolution equation in the hierarchy are obtained, which include soliton and periodic solutions.     

Explicit exact solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we construct new explicit exact solutions for the coupled the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation (KD equation) by using a improved mapping approach and variable separation method. By means of the method, new types of variable-separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) for the KD system are successfully obtained. The improved mapping approach and variable separation method can be applied to other higher-dimensional coupled nonlinear evolution equations.     

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.     

A new generalized algebra method and its application in the (2 + 1) dimensional Boiti–Leon–Pempinelli equation

In the present paper, some types of general solutions of a first-order nonlinear ordinary differential equation with six degree are given and a new generalized algebra method is presented to find more exact solutions of nonlinear differential equations. As an application of the method and the solutions of this equation, we choose the (2 + 1) dimensional Boiti Leon Pempinelli equation to illustrate the validity and advantages of the method. As a consequence, more new types and general solutions are found which include rational solutions and irrational solutions and so on. The new method can also be applied to other nonlinear differential equations in mathematical physics.     

Chaos, solitons and fractals in (2 + 1)-dimensional KdV system derived from a periodic wave solution

With the help of an extended mapping method and a linear variable separation method, new types of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with two arbitrary functions for (2 + 1)-dimensional Korteweg–de Vries system (KdV) are derived. Usually, in terms of solitary wave solutions and rational function solutions, one can find some important localized excitations. However, based on the derived periodic wave solution in this paper, we find that some novel and significant localized coherent excitations such as dromions, peakons, stochastic fractal patterns, regular fractal patterns, chaotic line soliton patterns as well as chaotic patterns exist in the KdV system as considering appropriate boundary conditions and/or initial qualifications.     

Construction of a series of travelling wave solutions to nonlinear equations

In this paper, based on new auxiliary ordinary differential equation with a sixth-degree nonlinear term, we study the (1 + 1)-dimensional combined KdV–MKdV equation, Hirota equation and (2 + 1)-dimensional Davey–Stewartson equation. Then, a series of new types of travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions. The method used here can be also extended to many other nonlinear partial differential equations.     

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.     

The extended Jacobi elliptic function method to solve a generalized Hirota–Satsuma coupled KdV equations

In this paper, we extend the Jacobi elliptic function rational expansion method by using a new generalized ansätz. With the help of symbolic computation, we construct more new explicit exact solutions of nonlinear evolution equations (NLEEs). We apply this method to a generalized Hirota–Satsuma coupled KdV equations and gain more general solutions. The general solutions not only contain the solutions by the existing Jacobi elliptic function expansion methods but also contain many new solutions. When the modulus of the Jacobi elliptic functions m → 1 or 0, the corresponding solitary wave solutions and triangular functional (singly periodic) solutions are also obtained.     

Bounded travelling wave solutions for the (n + 1)-dimensional sine- and sinh-Gordon equations

For the (n + 1)-dimensional sine- and sinh-Gordon equations, by using the approach of dynamical systems to a class of travelling wave solutions, in 21 different regions of a five-parameter space, all possible 21 bounded solutions ( solitary breather solutions, kink and anti-kink breather solutions and double-periodic wave solutions) are obtained. Explicit exact parametric representations are given.     

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.     

Symbolic computation and new families of exact non-travelling wave solutions of (2 + 1)-dimensional Konopelchenko–Dubrovsky equations

The improved tanh function method [Chaos, Solitons & Fractals 2005;24:257] is further improved by constructing new ansatz solution of the considered equation. As its application, the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations are considered and abundant new exact non-travelling wave solutions are obtained.     

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 new families of exact solutions to the (2 + 1)-dimensional dispersive long-wave equations

In this Letter, We present a further generalized algebraic method to the (2 + 1)-dimensional dispersive long-wave equations (DLWS), As a result, we can obtain abundant new formal exact solutions of the equation. The method can also be applied to solve more (2 + 1)-dimensional (or (3 + 1)-dimensional) nonlinear partial differential equations (NPDEs).     

A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation

In this paper, we present a new Riccati equation rational expansion method to uniformly construct a series of exact solutions for nonlinear evolution equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions. The solutions obtained in this paper include rational triangular periodic wave solutions, rational solitary wave solutions and rational wave solutions. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equation.     

Classification of qt=P(x, t, q, qx, qxx)

The classification of q_t = P ( x ,  t ,  q ,  q_x ,  q_xx ) is given by using linearization. Two types of integrable equations which are generalizations of the integrable equations of Fokas and Svinolupov are given. Results are compared with Fokas's symmetry and the Mikhailov–Shabat–Sokolov formal symmetry approaches.     

Solitons, chaos and fractals in the (2 + 1)-dimensional dispersive long wave equation

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of a projective Riccati equation approach, the paper obtains several types of exact solutions to the (2 + 1)-dimensional dispersive long wave (DLW) equation which include multiple soliton solution, periodic soliton solution and Weierstrass function solution. Subsequently, several multisolitons are derived and some novel features are revealed by introducing lower-dimensional patterns.     

Rational formal solutions of differential-difference equations

In this paper, we study rational formal solutions of differential-difference equations by using a generalized ansätz. With the help of symbolic computation Maple, we obtain many explicit exact solutions of differential-difference equations(DDEs). The solutions contain solitary wave solutions and periodic wave solutions. The (2 + 1)-dimensional Toda lattice equation, relativistic Toda lattice equation and the discrete mKdV equation are chosen to illustrate our algorithm.     

A new compound Riccati equations rational expansion method and its application to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Vesselov system

In this paper, based on a new general ansätze and symbolic computation, a new compound Riccati equations rational expansion method is proposed. Being concise and straightforward, it is applied to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Vesselov system. It is shown that more complexiton solutions can be found by this new method. The method can be applied to other nonlinear partial differential equations in mathematical physics.