Xian Da-Quan

Homoclinic Breather-Wave with Convective Effect for the (1+1)-Dimensional Boussinesq Equation

A new type of two-wave solution, i.e. a homoclinic breather-wave solution with convective effect, for the (1+1)-dimensional Boussinesq equation is obtained using the extended homoclinic test method. Moreover, the mechanical feature of the wave solution is investigated and the phenomenon of homoclinic convection of the two-wave is exhibited on both sides of the equilibrium. These results enrich the dynamical behavior of (1+1)-dimensional nonlinear wave fields.

Symmetry reductions and rational non-traveling wave solutions for the (2+1)-D Ablowitz-Kaup-Newell-Segur equation

Purpose The purpose of this paper is to find out some new rational non-traveling wave solutions and to study localized structures for (2+1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation. Design/methodology/approach Along with some special transformations, the Lie group method and the rational function method are applied to the (2+1)-dimensional AKNS equation. Findings Some new non-traveling wave solutions are obtained, including generalized rational solutions with two arbitrary functions of time variable. Research limitations/implications As a typical nonlinear evolution equation, some dynamical behaviors are also discussed. Originality/value With the help of the Lie group method, special transformations and the rational function method, new non-traveling wave solutions are derived for the AKNS equation by Maple software. These results are much useful for investigating some new localized structures and the interaction of waves in high-dimensional models, and enrich dynamical features of solutions for the higher dimensional systems.

Non-traveling wave solutions for the (2+1)-D Caudrey-Dodd-Gibbon-Kotera-Sawada equation

– The purpose of this paper is to find new non-traveling wave solutions and study its localized structure of Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation. , – The authors apply the Lie group method twice and combine with the Exp-function method and Riccati equation mapping method to the (2+1)-dimensional CDGKS equation. , – The authors have obtained some new non-traveling wave solutions with two arbitrary functions of time variable. , – As non-linear evolution equations is characterized by rich dynamical behavior, the authors just found some of them and others still to be found. , – These results may help the authors to investigate some new localized structure and the interaction of waves in high-dimensional models. The new non-traveling wave solutions with two arbitrary functions of time variable are obtained for CDGKS equation using Lie group approach twice and combining with the Exp-function method and Riccati equation mapping method by the aid of Maple.

Breather-Type Periodic Soliton Solutions for (1+1)-Dimensional Sinh-Poisson Equation

In this paper, by using bilinear form and extended homoclinic test approach, we obtain new breather-type periodic soliton solutions of the (1+1)-dimensional Sinh-Poisson equation. These results demonstrate that the nonlinear evolution equation has rich dynamical behavior even if it is (1+1)-dimensional.

Symmetry Reduced and New Exact Nontraveling Wave Solutions of (2+1)-Dimensional Potential Boiti-Leon-Manna-Pempinelli Equation

With the aid of Maple symbolic computation and Lie group method, ()-dimensional PBLMP equation is reduced to some ()-dimensional PDE with constant coefficients. Using the homoclinic test technique and auxiliary equation methods, we obtain new exact nontraveling solution with arbitrary functions for the PBLMP equation.

New Exact Non-traveling Wave Solutions for the (3+1)-dimensional KP Equation

In this paper the (3+l)-dimensional nonlinear Kadomtsev-Petviashvili equation is described as: (u,+auux+ ßuxxx)x-ruyy-Su:!=0. (1) where u:Rx xRyxR:xR* R,a, β,γ,δ e R . When a = β = \,δ = 0 and γ = ±1, equation (1) becomes usual KP equation [1], which arises in a number of remarkable non-linear problems both in physics and mathematics [2]. The solutions of KP equation have been studied extensively in various papers [3-10]. It was verified that higher dimensional non-linear wave fields have richer behavior than the one-dimensional. To clarify the role of a soliton in the variety of the dynamics, we must seek exact solitary-wave solution and non-traveling wave solution and analyze its singularity. In Ref. [11], a new type of special soliton solutions including singular and non-singular periodic solitons was found with generalized Hirota method. To find exact solutions of nonlinear PDE, it is well known that many effective methods [12-21] have been presented such as Lie group method [12-14], variational iteration method [15], homotopy perturbation method [16], F-expansion method [17], auxiliary equation method [18], Adomian decomposition method [19], Jacobi elliptic function expansion method [20], Exp-function method [21,22] and so on. In this paper, we will discover the exact non-traveling wave solutions of (3+l)-dimensional KP equation by using the Lie group method combined with other methods.