Quasi-periodic and non-periodic waves in (2+1) dimensions

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. By means of the extended mapping approach, new exact quasi-periodic and non-periodic solutions for the (2+1)-dimensional nonlinear systems are displayed both analytically and graphically.     

The study of soliton fission and fusion in (2+1)-dimensional nonlinear system

By means of a variable separation approach and an extended homogeneous balance method, a general variable separation excitation of a (2+1)-dimensional nonlinear system is derived. Based on the derived solution with arbitrary functions, we reveal soliton fission and fusion phenomena in the (2+1)-dimensional soliton system.     

Painlevé integrability and multi-dromion solutions of the 2+1 dimensional AKNS system

The Painlevé integrability of the 2+1 dimensional AKNS system is proved. Using the standard truncated Painlevé expansion which corresponds to a special B?cklund transformation, some special types of the localized excitations like the solitoff solutions, multi-dromion solutions and multi-ring soliton solutions are obtained. Received 31 January 2001 and Received in final form 15 May 2001     

He's variational iteration method applied to the solution of the prey and predator problem with variable coefficients

In this Letter, a mathematical model of the problem of prey and predator is presented and He's variational iteration method is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. The results are compared with the results obtained by Adomian decomposition method and homotopy perturbation method. Comparison of the methods show that He's variational iteration method is a powerful method for obtaining approximate solutions to nonlinear equations and their systems.     

Similarity Reductions and Similarity Solutions of the （3＋1）-Dimensional Kadomtsev-Petviashvili Equation

Employing the compatibility method, we obtain the symmetries of the （3＋1）-dimensional Kadomtsev Petviashvili （KP） equation. Four types of similarity reductions of the KP equation are obtained by solving the corresponding characteristic equations associated with symmetry equations. In addition, a lot of similarity solutions to the KP equation are obtadned.     

Noncontinuous Solitary Wave Solutions for Double sine-Gordon Equation

Using the tanh method and a variable separated ordinary difference equation method to solve the double sineGordon equation, we derive some new exact travelling wave solutions, especially a new type of noncontinuous solitary wave solutions. These noncontinuous solitary wave solutions are verified by using the conservation law theory.     

A Bilinear Backlund Transformation and Explicit Solutions for a （3＋1）-Dimensional Soliton Equation

Considering the bilinear form of a （3＋1）-dimensional soliton equation, we obtain a bilinear Backlund transformation for the equation. As an application, soliton solution and stationary rational solution for the （3＋1）- dimensional soliton equation are presented.     

Decomposition of the generalized KP, cKP and mKP and their exact solutions

The generalized (2+1)-dimensional KP, cKP and mKP are decomposed into the known (1+1)-dimensional soliton equations. Then, we show that the (1+1)-dimensional soliton equations give rise to the explicit soliton solutions of the generalized KP, cKP and mKP.     

Bäcklund transformation, nonlinear superposition formula and solutions of the Calogero equation

In this Letter, the Bäcklund transformation for the (2+1)-Calogero equation is presented in the bilinear form. Furthermore, a nonlinear superposition formula related to the transformation is proved rigorously. By the way, the Wronskian determinant solution is also derived and verified completely.     

The variational iteration method for exact solutions of Laplace equation

In this work, we introduce a framework for analytic treatment of Laplace equation with Dirichlet and Neumann boundary conditions. Exact solutions are developed by using the He's variational iteration method (VIM). The work confirms the power of the method in reducing the size of calculations.     

New positon, negaton and complexiton solutions for the Hirota-Satsuma coupled KdV system

New positon, negaton and complexiton solutions for the Hirota-Satsuma coupled KdV system are constructed by means of the Darboux transformation with zero seed solution. The new positon, negaton and complexiton solutions are singular and given out both analytically and graphically.     

Dispersive optical solitons by Kudryashov's method

This paper studies the dynamics of dispersive optical solitions that is modeled by the fourth order nonlinear Schrödinger's equation and Schrödinger–Hirota equation, the latter of which is considered with power law nonlinearity. Kudryashov's method is applied to obtain soliton solutions to the model equations. These results and the solution methodology makes a profound impact in the study of optical solitons.     

A Darboux Transformation for the Coupled Kadomtsev--Petviashvili Equation

The coupled Kadomtsev-Petviashvili equation is considered. It is shown that a Darboux transformation can be constructed by means of an elementary approach.     

Painlevé analysis and exact solutions of the resonant Davey-Stewartson system

It is shown that the resonant Davey-Stewartson (RDS) system can pass the Painlev test. By truncating the Laurent series to a constant level term, a dependent variable transformation is naturally derived, which leads to the bilinear forms of the RDS system. From the bilinear equations, through making suitable assumptions, some new soliton solutions are obtained. Some representative profiles of the solitary waves are graphically displayed including the two-line soliton solution, “Y” soliton solution, “V” soliton solution, solitoff, etc. The solutions might be useful to describe the nonlinear phenomena in Madelung fluids, capillarity fluids, and so on.     

A hierarchy of Hamiltonian lattice equations associated with the relativistic Toda type system

By considering a discrete iso-spectral problem, a hierarchy of bi-Hamiltonian relativistic Toda type lattice equations are revisited. After introducing a semi-direct sum Lie algebras of four by four matrices, integrable coupling system associated with the relativistic Toda type lattice are derived. It is shown that the resulting lattice soliton hierarchy possesses Hamiltonian structures and infinitely many common commuting symmetries as well infinitely many conserved functions. The Liouville integrability of the resulting system is then demonstrated.     

New positon, negaton and complexiton solutions for the Bogoyavlensky-Konoplechenko equation

New positon, negaton and complexiton solutions for the Bogoyavlensky-Konoplechenko equation are constructed by means of the Darboux transformation with constant seed solution. The new positon, negaton and complexiton solutions are analytical or singular and given out both analytically and graphically.     

Novel homoclinic and heteroclinic solutions for the 2D complex cubic Ginzburg-Landau equation

Homoclinic and heteroclinic solutions are two important concepts that are used to investigate the complex properties of nonlinear evolutionary equations. In this Letter, we perform hyperbolic and linear stability analysis, and prove the existence of homoclinic and heteroclinic solutions for two-dimensional cubic Ginzburg-Landau equation with periodic boundary condition and even constraint. Then, using the Hirota's bilinear transformation, we find the closed-form homoclinic and heteroclinic solutions. Moreover, we find that the homoclinic tubes and two families of heteroclinic solutions are asymptotic to a periodic cycle in one dimension.     

Multi-Soliton Solutions of the Levi Equations

The multisoliton solutions of the Levi equations are derived with the Hirota method and Wronskian technique respectively.     

Abundant multisoliton structures of the Konopelchenko-Dubrovsky equation

Using the homogenous balance method, the nonlinear transformations of the (2+1)-dimensional integrable Konopelchenko-Dubrovsky equation are given, and then some new special types of single solitary wave solution and the multisoliton solutions are constructed. The project is supported by the Natural Science Foundation of Shandong Province in China and the Natural Science Foundation of Liaocheng University.     

Binary Darboux Transformation for the Modified Kadomtsev--Petviashvili Equation

Via the elementary Darboux transformation (DT) of the modified Kadomtsev--Petviashvili (mKP) equation, a binary Darboux transformation (BDT) of the mKP equation is constructed.