Nonsingular Travelling Complexiton Solutions to a Coupled Korteweg--de Vries Equation

A class of novel nonsingular travelling complexiton solutions to a coupled Korteweg-de Vries （KdV） equation is presented via the first step Darboux transformation of the complex KdV equation with nonzero seed solution. Furthermore, the properties of the nonsingular solutions are discussed.     

A Discrete Lax-Integrable Coupled System Related to Coupled KdV and Coupled mKdV Equations

A modified Korteweg-de Vries （mKdV） lattice is found to be also a discrete Korteweg-de Vries （KdV） equation. A discrete coupled system is derived from the single lattice equation and its Lax pair is proposed. The coupled system is shown to be related to the coupled KdV and coupled mKdV systems which are widely used in physics.     

N-Soliton Solution in Wronskian Form for a Generalized Variable-Coefficient Korteweg--de Vries Equation

We concentrate on finding exact solutions for a generalized variable-coefficient Korteweg-de Vries equation of physically significance. The analytic N-soliton solution in Wronskian form for such a model is postulated and verified by direct substituting the solution into the bilinear form by virtue of the Wronskian technique. Additionally, the bilinear auto-Backlund transformation between the （ N - 1）- and N-soliton solutions is verified.     

A Variational Iteration Solving Method for a Class of Generalized Boussinesq Equations

We study a generalized nonlinear Boussinesq equation by introducing a proper functional and constructing the variational iteration sequence with suitable initial approximation. The approximate solution is obtained for the solitary wave of the Boussinesq equation with the variational iteration method.     

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.     

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.     

Interaction between Soliton and Periodic Wave

A truncation for the Laurent series in the Padnleve analysis of the KdV equation is restudied. When the truncation occurs the singular manifold satisfies two compatible fourth-order PDEs, which are homogeneous of degree 3. Both of the PDEs can be factored in the operator sense. The common factor is a third-order PDE, which is homogeneous of degree 2. The first few Invariant manifolds of the third-order PDE are studied. We find that the invariant manifolds of the third-order PDE can be obtained by factoring the invariant manifolds of the KdV equation. A numerical solution of the third-order PDE facts about the third-order PDE. is also presented. The solution reveals some interesting     

N-Soliton Solutions of Non-Isospectral Derivative Nonlinear Schrödinger Equation

Bilinear forms of the non-isospectral derivative nonlinear Schrǒdinger equation are derived. The N-soliton solutions of this equation are obtained by Hirota＇s method.     

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.     

Higher-Dimensional KdV Equations and Their Soliton Solutions

A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of bell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.     

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 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.     

New interaction solutions to the KdV equation

In this Letter, a few new types of interaction solutions to the KdV equation are obtained through a constructed Wronskian form expansion method. The method takes advantage of the forms and structures of Wronskian solutions to the KdV equation, and the functions used in the Wronskian determinants don't satisfy the systems of linear partial differential equations.     

Two-component description of dynamical systems that can be approximated by solitons: The case of the ion acoustic wave equations of plasma physics

A new approach to the perturbative analysis of dynamical systems, which can be described approximately by soliton solutions of integrable non-linear wave equations, is employed in the case of small-amplitude solutions of the ion acoustic wave equations of plasma physics. Instead of pursuing the traditional derivation of a perturbed KdV equation, the ion velocity is written as a sum of two components: elastic and inelastic. In the single-soliton case, the elastic component is the full solution. In the multiple-soliton case, it is complemented by the inelastic component. The original system is transformed into two evolution equations: An asymptotically integrable Normal Form for ordinary KdV solitons, and an equation for the inelastic component. The zero-order term of the elastic component is a single-soliton or multiple-soliton solution of the Normal Form. The inelastic component asymptotes into a linear combination of single-soliton solutions of the Normal Form, with amplitudes determined by soliton interactions, plus a second-order decaying dispersive wave. Satisfaction of a conservation law by the inelastic component and of mass conservation by the disturbance to the ion density is determined solely by the initial data and/or boundary conditions imposed on the inelastic component. The electrostatic potential is a first-order quantity. It is affected by the inelastic component only in second order. The charge density displays a triple-layer structure. The analysis is carried out through the third order.     

From single- to multiple-soliton solutions of the perturbed KdV equation

The solution of the perturbed KdV equation (PKDVE), when the zero-order approximation is a multiple-soliton wave, is constructed as a sum of two components: elastic and inelastic. The elastic component preserves the elastic nature of soliton collisions. Its perturbation series is identical in structure to the series-solution of the PKDVE when the zero-order approximation is a single soliton. The inelastic component exists only in the multiple-soliton case, and emerges from the first order and onwards. Depending on initial data or boundary conditions, it may contain, in every order, a plethora of inelastic processes. Examples are given of sign-exchange soliton-anti-soliton scattering, soliton-anti-soliton creation or annihilation, soliton decay or merging, and inelastic soliton deflection. The analysis has been carried out through third order in the expansion parameter, exploiting the freedom in the expansion to its fullest extent. Both elastic and inelastic components do not modify soliton parameters beyond their values in the zero-order approximation. When the PKDVE is not asymptotically integrable, the new expansion scheme transforms it into a system of two equations: The Normal Form for ordinary KdV solitons, and an auxiliary equation describing the contribution of obstacles to asymptotic integrability to the inelastic component. Through the orders studied, the solution of the latter is a conserved quantity, which contains the dispersive wave that has been observed in previous works.     

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.     

Exact solutions to a coupled modified KdV equations with non-uniformity terms

The bilinear form of a coupled modified KdV equations with non-uniformity terms is given and a few soliton solutions are obtained. Furthermore, the multisoliton of the coupled system is expressed by Pfaffian.     

N-Soliton Solutions of Modified KdV Equation under Bargmann Constraint

By means of Hirota method, N-soliton solutions of the modified KdV equation under the Bargmann constraint are obtained through solving the Bargmann constraint and the related Lax pair and conjugate Lax pair of the modified KdV equation.     

The Bilinear Integrability,N-soliton and Riemann-theta function solutions of B-type KdV Equation

In this paper, the bilinear integrability for B-type KdV equation have been explored. According to the relation to tau function, we find the bilinear transformation and construct the bilinear form with an auxiliary variable of the B-type KdV equation. Based on the truncation form, the Bäcklund transformation has been constructed. Furthermore, the N-soliton solutions and Riemann-theta function 1-periodic solutions of the B-type KdV equation are obtained.     

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.