Optimal solutions of a (3 + 1)-dimensional B-Kadomtsev-Petviashvii equation

We explored and specialized new Lie infinitesimals for the (3 + 1)-dimensional B-Kadomtsev-Petviashvii (BKP) using the commutation product, which results a system of nonlinear ODEs manually solved. Through two stages of Lie symmetry reduction, (3 + 1)-dimensional BKP equation is reduced to nonsolvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the integration and Riccati equation methods, we investigate new analytical solutions for these ODEs. Back substituting to the original variables generates new solutions for BKP. Some selected solutions illustrated through three-dimensional plots.     

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.     

Exact solitary wave solutions of the generalized (2 + 1) dimensional Boussinesq equation

Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the generalized (2 + 1) dimensional Boussinesq equation. Numbers of solitary waves are given for each parameter condition. Under some parameter conditions, exact solitary wave solutions are obtained.     

New (3+1)‐dimensional nonlinear equations with KdV equation constituting its main part: multiple soliton solutions

In thiswork,we present two new(3+1)‐dimensional nonlinear equationswith Korteweg‐de Vries equation constituting its main part. We show that the dispersive relation is distinct for each model, whereas the phase shift remains the same. We determine multiple solitons solutions, with distinct physical structures, for each established equation. The architectures of the simplified Hirota's method is implemented in this paper. The constraint conditions that fall out which must remain valid in order for themultiple solitons to exist are derived.Copyright © 2015 John Wiley & Sons, Ltd.     

Dressing the boundary: On soliton solutions of the nonlinear Schrödinger equation on the half‐line

Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin, we provide a direct method for computing soliton solutions of the focusing nonlinear Schrödinger equation on the half‐line. The integrable BCs at the origin are represented by constraints of the Lax pair, and our method lies on dressing the Lax pair by preserving those constraints in the Darboux‐dressing process. The method is applied to two classes of solutions: solitons vanishing at infinity and self‐modulated solitons on a constant background. Half‐line solitons in both cases are explicitly computed. In particular, the boundary‐bound solitons, which are static solitons bounded at the origin, are also constructed. We give a natural inverse scattering transform interpretation of the method as evolution of the scattering data determined by the integrable BCs in space.     

Consistent Riccati expansion for finding interaction solutions of (2+1)‐dimensional modified dispersive water‐wave system

A consistent Riccati expansion (CRE) is proposed to solve the (2+1)‐dimensional modified dispersive water‐wave (MDWW) system. It is proved that the MDWW system is CRE solvable. Furthermore, new exact interaction solutions, namely, soliton‐trigonometric waves, trigonometric waves‐soliton, soliton‐cosine periodic waves, and soliton‐cnoidal waves are explicitly derived.     

Finding general and explicit solutions (2 + 1) dimensional Broer-Kaup-Kupershmidt system nonlinear equation by exp-function method

In this work, we implement a relatively new analytical technique, the exp-function method, for solving nonlinear special form of generalized nonlinear (2 + 1) dimensional Broer-Kaup-Kupershmidt equation, which may contain high nonlinear terms. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations which applied in engineering mathematics. Some numerical examples are presented to illustrate the efficiency and reliability of exp method. It is predicted that exp-function method can be found widely applicable in engineering.     

Jacobi elliptic function solutions of the (1 + 1)‐dimensional dispersive long wave equation by Homotopy Perturbation Method

In this article, we try to obtain approximate Jacobi elliptic function solutions of the (1 + 1)‐dimensional long wave equation using Homotopy Perturbation Method. This method deforms a difficult problem into a simple problem which can be easily solved. In comparison with HPM, numerical methods leads to inaccurate results when the equation intensively depends on time, while He's method overcome the above shortcomings completely and can therefore be widely applicable in engineering. As a result, we obtain the approximate solution of the (1 + 1)‐dimensional long wave equation with initial conditions. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

A (3 + 1)-dimensional nonlinear evolution equation with multiple soliton solutions and multiple singular soliton solutions

In this work, a (3 + 1)-dimensional nonlinear evolution equation is investigated. The Hirota’s bilinear method is applied to determine the necessary conditions for the complete integrability of this equation. Multiple soliton solutions are established to confirm the compatibility structure. Multiple singular soliton solutions are also derived. The resonance phenomenon does not exist for this model.     

Exact solutions of nonlinear Schrödinger equation by using symbolic computation

The (G′/G,1/G)‐expansion method and (1/G′)‐expansion method are interesting approaches to find new and more general exact solutions to the nonlinear evolution equations. In this paper, these methods are applied to construct new exact travelling wave solutions of nonlinear Schrödinger equation. The travelling wave solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. It is shown that the proposed methods provide a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Copyright © 2015 John Wiley & Sons, Ltd.     

Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method

In this paper, we establish exact solutions for (2 + 1)-dimensional nonlinear evolution equations. The sine-cosine method is used to construct exact periodic and soliton solutions of (2 + 1)-dimensional nonlinear evolution equations. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional Boussinesq, breaking soliton and BKP equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.     

A generalized extended rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation

In this Letter, a generalized extended rational expansion method is used to construct exact solutions of the (1 + 1)-dimensional dispersive long wave equation. As a result, many new and more general exact solutions are obtained, the solutions obtained in this Letter include rational triangular periodic wave solutions, rational solitary wave solutions.     

Dynamical analysis of rational and semi-rational solution for a new extended (3 + 1)-dimensional Kadomtsev-Petviashvili equation

This paper proposes a new extended (3 + 1)-dimensional Kadomtsev-Petviashvili equation that portrays a unique dispersion effect about $x,z$. Its integrability is confirmed via the WTC-Kruskal algorithm in Painlevé sense. $N$-soliton, breather, and $O$-type solitary wave are derived systematically at first. Then, the mixed solution composed of soliton and breather is obtained. In addition, the “long wave” limit is employed to construct rational and semi-rational solution. The rational solution can be classified as rogue wave, $T$-type solitary wave, and lump wave. The semi-rational solution has the form a hybrid of two solitons, a hybrid of rogue wave and soliton, a hybrid of lump and soliton(s), and a hybrid of lump and breather. The results may help simulate complex waves and their interactions in fluid.     

(2+1)维浅水波方程的新精确解

对(2+1)维浅水波方程的现有解进行了推广.应用CK方法对方程进行求解,得到方程的Backlund变换公式,将已知解代入公式,求得一些新的精确解,从而推广了浅水渡方程的解.     

Dynamics of a coupled modified (1 + 1)‐dimensional Toda equation of BKP type

In this paper, we present a new coupled modified (1 + 1)‐dimensional Toda equation of BKP type (Kadomtsev‐Petviashvilli equation of B‐type), which is a reduction of the (2 + 1)‐dimensional Toda equation. Two‐soliton and three‐soliton solutions to the coupled system are derived. Furthermore, the N‐soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two‐soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.     

An extended subequation rational expansion method with symbolic computation and solutions of the nonlinear Schrödinger equation model

To construct exact analytical solutions of nonlinear evolution equations, an extended subequation rational expansion method is presented and used to construct solutions of the nonlinear Schrödinger equation with varing dispersion, nonlinearity, and gain or absorption. As a result, many previous known results of the nonlinear Schrödinger equation can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. With computer simulation, the properties of a new non-travelling wave soliton-like solutions with coefficient functions and some elliptic function solutions are shown by some figures.     

Exact traveling wave solutions to the fourth‐order dispersive nonlinear Schrödinger equation with dual‐power law nonlinearity

In this paper, we investigate exact traveling wave solutions of the fourth‐order nonlinear Schrödinger equation with dual‐power law nonlinearity through Kudryashov method and (G'/G)‐expansion method. We obtain miscellaneous traveling waves including kink, antikink, and breather solutions. These solutions may be useful in the explanation and understanding of physical behavior of the wave propagation in a highly dispersive optical medium. Copyright © 2015 John Wiley & Sons, Ltd.     

A general bilinear form to generate different wave structures of solitons for a (3+1)‐dimensional Boiti‐Leon‐Manna‐Pempinelli equation

In this paper, we investigate a (3+1)‐dimensional Boiti‐Leon‐Manna‐Pempinelli equation (3D‐BMLP). By using bilinear forms under certain conditions, we obtain different wave structures for the 3D‐BMLP. Among these waves, lump waves, breather waves, mixed waves, and multi‐soliton wave solutions are constructed. The propagation and the dynamical behavior of the obtained solutions are discussed for different values of the free parameters.