Periodic Wave Solution to the （3＋1）-Dimensional Boussinesq Equation

One- and two-periodic wave solutions for （3＋l）-dimensional Boussinesq equation are presented by means of Hirota＇s bilinear method and the Riemann theta function. The soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure.     

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 homoelinic test method. Moreover, the mechanical feature of the wave solution is investigated and the phenomenon of homoelinic 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.     

Assessment of two analytical approaches in some nonlinear problems arising in engineering sciences

In this research, two powerful analytical methods are introduced to handle nonlinear good Boussinesq, heat transfer and coupled Burgers' equations. One is the homotopy-perturbation method (HPM) and the other is the variational iteration method (VIM). VIM is used to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory. HPM converts a difficult problem into a simple one, which can be easily handled. The results attained in this paper confirm the idea that HPM and VIM are powerful mathematical tools and that they can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.     

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.     

N-Soliton Solution of a Generalized Hirota-Satsuma Coupled KdV Equation and Its Reduction

Based on the Hirota method and the perturbation technique, the N-soliton solution of a generalized Hirota-Satsuma coupled KdV equation is obtained. Further, the N-soliton solution of a complex coupled KdV equation is given by reducing.     

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.     

New block matrix spectral problem and Hamiltonian structure of the discrete integrable coupling system

In [W.X. Ma, J. Phys. A: Math. Theor. 40 (2007) 15055], Prof. Ma gave a beautiful result (a discrete variational identity). In this Letter, based on a discrete block matrix spectral problem, a new hierarchy of Lax integrable lattice equations with four potentials is derived. By using of the discrete variational identity, we obtain Hamiltonian structure of the discrete soliton equation hierarchy. Finally, an integrable coupling system of the soliton equation hierarchy and its Hamiltonian structure are obtained through the discrete variational identity.     

Integrable coupling system of fractional soliton equation hierarchy

In this Letter, we consider the derivatives and integrals of fractional order and present a class of the integrable coupling system of the fractional order soliton equations. The fractional order coupled Boussinesq and KdV equations are the special cases of this class. Furthermore, the fractional AKNS soliton equation hierarchy is obtained.     

A simple and direct method for generating travelling wave solutions for nonlinear equations

We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.     

New Periodic Solution to Jacobi Elliptic Functions of a (2+1)-Dimensional BKP Equation and a Generalized Klein-Gordon Equation

With the help of the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to obtain the Jacobi doubly periodic wave solutions of the (2+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation and the generalized Klein-Gordon equation. The method is also valid for other (1+1)-dimensional and higher dimensional systems.     

Cross Soliton-Like Waves for the （2＋1）-Dimensional Breaking Soliton Equation

We construct a two-soliton-like solution for the （2＋1）-dimensionai breaking soliton equation. The obtained solution contains two arbitrary functions and hence can model various cross soliton-like waves including the two-solitary waves. We show the evolution of some special cross soliton-like waves diagrammatically.     

A constrained variational principle for heat conduction

The heat equation is re-studied in this Letter in view of variational theory. By the semi-inverse method, a variational principle for the heat conduction is obtained, which is first appeared in the literature. The physical understanding of the obtained variational principle still needs further explanation.     

Boussinesq方程组的广义变分原理

半反推法是何吉欢为了寻求物理问题的变分原理而提出的,可避免由拉氏乘子法引起的临界变分现象. 应用半反推法分别获得了描述水波运动的两类Boussinesq方程组的一族广义变分原理,并验证了它们的正确性. 关键词： 半反推法 广义变分原理 Boussinesq方程组     

Small Amplitude Solitons in Bose--Einstein Condensates with External Perturbation

By developing a small amplitude soliton approximation method, we study analytically weak nonlinear excitations in cigar-shaped condensates with repulsive interatomic interaction under consideration of external perturbation potential. It is shown that matter wave solitons may exist and travel over a long distance without attenuation and change in shape by properly adjusting the strength of interatomic interaction to compensate for the effect of external perturbation potential.     

Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation

The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter ε tends to zero. Assuming natural initial data having the profile of a moving −2π kink at time zero, we analytically calculate the scattering data of this completely integrable Cauchy problem for all ε>0 sufficiently small, and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small ε. This sequence of exact solutions is analogous to that of the well-known N-soliton (or higher-order soliton) solutions of the focusing nonlinear Schrödinger equation. We then use plots obtained from a careful numerical implementation of the inverse-scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit. In the limit ε↓0 one observes the appearance of nonlinear caustics, i.e. curves in space-time that are independent of ε but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases.In the appendices, we give a self-contained account of the Cauchy problem from the perspectives of both inverse scattering and classical analysis (Picard iteration). Specifically, Appendix A contains a complete formulation of the inverse-scattering method for generic L¹-Sobolev initial data, and Appendix B establishes the well-posedness for L^p-Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).     

A hierarchy of Liouville integrable discrete Hamiltonian equations

Based on a discrete four-by-four matrix spectral problem, a hierarchy of Lax integrable lattice equations with two potentials is derived. Two Hamiltonian forms are constructed for each lattice equation in the resulting hierarchy by means of the discrete variational identity. A strong symmetry operator of the resulting hierarchy is given. Finally, it is shown that the resulting lattice equations are all Liouville integrable discrete Hamiltonian systems.     

Solving linear variational inequalities by projection neural network with time-varying delays

A projection neural network with time-varying delays is proposed for solving linear variational inequalities. By the theory of functional differential equation and the linear matrix inequality technique, the proposed neural network is proved to be globally exponentially stable. The obtained results complement and extend the previously known work.     

A short remark on fractional variational iteration method

This Letter compares the classical variational iteration method with the fractional variational iteration method. The fractional complex transform is introduced to convert a fractional differential equation to its differential partner, so that its variational iteration algorithm can be simply constructed.     

Generalized Variational Iteration Solution of Soliton for Disturbed KdV Equation

The corresponding solution for a class of disturbed KdV equation is considered using the analytic method. From the generalized variational iteration theory, the problem of solving soliton for the corresponding equation translates into the problem of variational iteration. And then the approximate solution of the soliton for the equation is obtained.