On Coupled KdV Equations with Self-consistent Sources

The coupled Korteweg-de Vries （CKdV） equation with self-consistent sources （CKdVESCS） and its Lax representation are derived. We present a generalized binary Darboux transformation （GBDT） with an arbitrary time- dependent function for the CKdVESCS as well as the formula for the N-times repeated GBDT. This GBDT provides non-auto-Biicklund transformation between two CKdVESCSs with different degrees of sources and enables us to construct more generM solutions with N arbitrary t-dependent functions. We obtain positon, negaton, complexiton, and negaton- positon solutions of the CKdVESCS.

New approximate solution for time-fractional coupled KdV equations by generalised differential transform method

In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial. An illustrative example shows that the genera]ised two-dimensional DTM is effective for the coupled equations.

EXPLICIT SOLUTIONS TO THE COUPLED KdV EQUATIONS WITH VARIABLE COEFFICIENTS

By means of sn-function expansion method and cn-function expansion method, several kinds of explicit solutions to the coupled KdV equations with variable coefficients are obtained, which include three sets of periodic wave-like solutions. These solutions degenerate to solitary wave-like solutions at a certain limit. Some new solutions are presented.

Coupled KdV Equations and Their Explicit Solutions Through Two-Dimensional Hamiltonian System with a Quartic Potential

Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.

Similarity Reductions and Painlevé Property of Coupled KdV Equations

Using Ablowitz-Ramani Segur algorithm, the coupled KdV systems are reclassified under the Painlevé integrable sense while the similarity reductions of the model are obtained by using the Clarkson and Kruskal＇s direct method. Some new types of Painlevé integrable models including a model with different dispersion relations for two layer fluids are found.

Average vector field methods for the coupled Schrdinger KdV equations

The energy preserving average vector field （AVF） method is applied to the coupled Schr6dinger-KdV equations. Two energy preserving schemes are constructed by using Fourier pseudospectral method in space direction discretization. In order to accelerate our simulation, the split-step technique is used. The numerical experiments show that the non-splitting scheme and splitting scheme are both effective, and have excellent long time numerical behavior. The comparisons show that the splitting scheme is faster than the non-splitting scheme, but it is not as good as the non-splitting scheme in preserving the invariants.

The fractional coupled KdV equations: Exact solutions and white noise functional approach

Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the mod- ified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types.

New Complex Solutions of the Coupled KdV Equations

In this paper,new infinite sequence complex solutions of the coupled Kd V equations are constructed with the help of function transformation and the second kind of elliptic equation.First of all,according to the function transformation,the coupled Kd V equations are changed into the second kind of elliptic equation.Secondly,the new solutions and Bäcklund transformation of the second kind of elliptic equation are applied to search for new infinite sequence complex solutions of the coupled Kd V equations.These solutions include new infinite sequence complex solutions composed by Jacobi elliptic function,hyperbolic function and triangular function.

Multi-symplectic method for the coupled Schrdinger–KdV equations

In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schrtidinger-KdV equations （CS＇KE） with periodic boundary conditions. Then we develop a novel multi-symplectic Fourier pseudospectral （MSFP） scheme for the CSKE. In numerical experiments, we compare the MSFP method with the Crank-Nicholson （CN） method. Our results show high accuracy, effectiveness, and good ability of conserving the invariants of the MSFP method.

An element-free Galerkin(EFG) method for numerical solution of the coupled Schrdinger-KdV equations

The present paper deals with the numerical solution of the coupled Schrodinger-KdV equations using the elementfree Galerkin （EFG） method which is based on the moving least-square approximation. Instead of traditional mesh oriented methods such as the finite difference method （FDM） and the finite element method （FEM）, this method needs only scattered nodes in the domain. For this scheme, a variational method is used to obtain discrete equations and the essential boundary conditions are enforced by the penalty method. In numerical experiments, the results are presented and compared with the findings of the finite element method, the radial basis functions method, and an analytical solution to confirm the good accuracy of the presented scheme.

Lie Point Symmetries and Exact Solutions of Couple KdV Equations

The simple Lie point symmetry reduction procedure is used to obtain infinitely many symmetries to a new integrable system of coupled KdV equations. Using some symmetry subalgebra of the equations, five types of the significant similarity reductions are obtained by virtue of the Lie group approach, and obtain abundant solutions of the coupled KdV equations, such as the solitary wave solution, exponential solution, rational solution, polynomial solution, etc.

Two-Soliton Solutions and Interactions for the Generalized Complex Coupled Kortweg-de Vries Equations

Kortweg-de Vries （KdV）-typed equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Generalized complex coupled KdV （GCCKdV） equations are investigated in this paper. Through the dependent variable transformations and symbolic computation, GCCKdV equations are transformed into their bilinear forms, based on which the one- and two-soliton solutions are obtained. Through the interactions of two solitons, the regular elastic collision are shown. When the wave numbers are complex, three kinds of solitonie collisions are presented： （i） two solitons merge and separate from each other periodically; （ii） two solitons exhibit the attraction and repulsion nearly twice, and finally separate from each other after such type of interaction; （iii） two solitons are ftuctuant in the central region of the collision. Propagation features of solitons are investigated with the effects of the coefficients in the GCCKdV equations considered. Velocity of soliton increase with the a increasing. Amplitude of v increase with the a increasing and decrease with the β increasing.

Exact Solutions for a Nonisospectral and Variable-Coefficient KdV Equation

The bilinear form for a nonisospectral and variable-coefficient KdV equation is obtained and some exact soliton solutions are derived through Hirota method and Wronskian technique. We also derive the bilmear transformation from its Lax pairs and End solutions with the help of the obtained bilinear transformation.

A New Coupled KdV Equation： Painlevé Test

A new type of coupled Korteweg de-Vries equation is found to be Painlevé-integrable. The new model is a special case which can be used to describe two-layer fluids with different dispersion relations.

Nonsingular Positon Solutions of a Variable-Coefficient Modified KdV Equation

The determinant representation of three-fold Darboux transformation for a variable-coefficient modified KdV equation is displayed based on the technique used to solve Ablowitz-Kaup-Newell-Segur system. Additionally, the nonsingular positon solutions of the variable-coefficient modified KdV equation are firstly discovered analytically and graphically.

Approximate analytic solutions for a generalized Hirota—Satsuma coupled KdV equation and a coupled mKdV equation

This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.

Homotopy Perturbation Method for the Generalized Hirota-Satsuma Coupled KdV Equation

In this paper, we consider the homotopy perturbation method (HPM) to obtain the exact solution of Hirota-Satsuma Coupled KdV equation. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear mathematical problems.

Darboux Transformation and Soliton Solutions for Inhomogeneous Coupled Nonlinear Schr(o|¨)dinger Equations with Symbolic Computation

With the aid of computation, we consider the variable-coefficient coupled nonlinear Schrodinger equations with the effects of group-velocity dispersion, self-phase modulation and cross-phase modulation, which have potential applications in the long-distance communication of two-pulse propagation in inhomogeneous optical fibers. Based on the obtained nonisospectral linear eigenvalue problems （i.e. Lax pair）, we construct the Darboux transformation for such a model to derive the optical soliton solutions. In addition, through the one- and two-soliton-like solutions, we graphically discuss the features of picosecond solitons in inhomogeneous optical fibers.

Painlevé Property and Complexiton Solutions of a Special Coupled KdV Equation

A special coupled KdV equation is proved to be the Painleve property by the Kruskal＇s simplification of WTC method. In order to search new exact solutions of the coupled KdV equation, Hirota＇s bilinear direct method and the conjugate complex number method of exponential functions are applied to this system. As a result, new analytical eomplexiton and soliton solutions are obtained synchronously in a physical field. Then their structures, time evolution and interaction properties are further discussed graphically.

一类广义Hirota-Satsuma Coupled KdV系统的新精确行波解

利用新近提出的(G′/G)-展开法,借助齐次平衡方法的思想原则和吴文俊消元法,对一类Hirota-Sat-suma Coupled KdV方程进行研究,获得了该系统的含有多参数的双曲函数、三角函数和有理函数表示的三种形式的精确通解.从求方程组精确行波解的过程看,此方法不仅是求解非线性发展方程行波解的一种简捷和有效的方法,而且也可以用来求解其他的许多非线性发展方程的精确行波解.