A COUPLED SYSTEM OF INTEGRAL EQUATIONS IN REFLEXIVE BANACH SPACES

We present an existence theorem for at least one weak solution for a coupled system of integral equations of Volterra type in a reflexive Banach spaces relative to the weak topology.

Coupled Extremal Quasi-solutions to Nonlinear Impulsive Fredholm Integral Equations in Banach Spaces

By means of the method of coupled lower and upper quasisolutio ns, the paper applies, instead of establishing the comparison theorem, a new ite rative technique to nonlinear impulsive Fredholm integral equations in Banach sp aces and proves the existence theorem on their coupled extremal quasisolutions . Finally, an infinite system of nonlinear impulsive integral equations is provi ded to demonstrate the obtained results.

The periodic solutions for coupled integrable dispersionless equations

By using the Jacobi elliptic-function method, this paper obtains the periodic solutions for coupled integrable dispersionless equations. The periodic solutions include some kink and anti-kink solitons.

N-soliton solution of a coupled integrable dispersionless equation

A new coupled integrable dispersionless equation is presented by considering a spectral problem. A Darboux transformation for the resulting coupled integrable dispersionless equation is constructed with the help of spectral problems. As an application, the N-soliton solution of the coupled integrable dispersionless equation is explicitly given.

Existence of Solution for a Coupled System of Fractional Integro-Differential Equations on an Unbounded Domain

We present the existence of solution for a coupled system of fractional integro-differential equations. The differential operator is taken in the Caputo fractional sense. We combine the diagonalization method with Arzela-Ascoli theorem to show a fixed point theorem of Schauder.

Conservation Laws and Analytic Soliton Solutions for Coupled Integrable Dispersionless Equations with Symbolic Computation

Under investigation in this paper are two coupled integrable dispersionless (CID) equations modelingthe dynamics of the current-fed string within an external magnetic field.Through a set of the dependent variabletransformations, the bilinear forms for the CID equations are derived.Based on the Hirota method and symboliccomputation, the analytic N-soliton solutions are presented.Infinitely many conservation laws for the CID equationsare given through the known spectral problem.Propagation characteristics and interaction behaviors of the solitons areanalyzed graphically.

New Exact Solutions and Special Coherent Structures for Coupled Integrable Dispersionless Equation

Using improved homogeneous balance method, we obtain new exact solutions for the coupled integrable dispersionless equation. On the basis of these exact solutions, we find some new interesting coherent structures by selecting arbitrary functions appropriately.

N-Rotating Loop-Soliton Solution of the Coupled Integrable Dispersionless Equation

In this paper, we investigate the Rotating N Loop-Soliton solution of the coupled integrable dispersionless equation (CIDE) that describes a current-fed string within an external magnetic field in 2D-space. Through a set of independent variable transformation, we derive the bilinear form of the CIDE Equation. Based on the Hirota’s method, Perturbation technique and Symbolic computation, we present the analytic N-rotating loop soliton solution and proceed to some illustrations by presenting the cases of three- and four-soliton solutions.

Soliton Solutions to Coupled Integrable Dispersionless Equations

In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless （CID） equations is derived. It will be shown that this bilineax form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions axe derived.

AN INTEGRATION BY PARTS FORMULA FOR STOCHASTIC HEAT EQUATIONS WITH FRACTIONAL NOISE

In this paper,we establish the integration by parts formula for the solution of fractional noise driven stochastic heat equations using the method of coupling.As an application,we also obtain the shift Harnack inequalities.

Multi-component Dirac equation hierarchy and its multi-component integrable couplings system

A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M- dimensional loop algebra ~X is produced. By taking advantage of ~X a new isospectral problem is established and then by making use of the Tu scheme the multi-component Dirac equation hierarchy is obtained. Finally, an expanding loop algebra ~FM of the loop algebra ~X is presented. Based on the ~FM, the multi-component integrable coupling system of the multi-component Dirac equation hierarchy is investigated. The method in this paper can be applied to other nonlinear evolution equation hierarchies.

A New Nonlinear Integrable Couplings of Yang Equations Hierarchy and Its Hamiltonian Structure

Based on a kind of non-semisimple Lie algebras, we establish a way to construct nonlinear continuous integrable couplings. Variational identities over the associated loop algebras are used to furnish Hamiltonian structures of the resulting continuous couplings.As an illustrative example of the scheme is given nonlinear continuous integrable couplings of the Yang hierarchy.

Integrable Coupling System of JM Equations Hierarchy with Self-Consistent Sources

We propose a systematic method for generalizing the integrable couplings of soliton eqhations hierarchy with self-consistent sources associated with s/（4）. The JM equations hierarchy with self-consistent sources is derived. Furthermore, an integrable couplings of the JM soliton hierarchy with self-consistent sources is presented by using of the loop algebra sl（4）.

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.

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.

A Higher Dimensional Loop Algebra and Integrable Couplings System of Evolution Equations Hierarchy

An extension of the Lie algebra A_~n-1 has been proposed [Phys. Lett. A, 2003, [STHZ]310:19-24]. In this paper, the new Lie algebra was used to construct a new higher dimensional loop algebra [AKG～]. Based on the loop algebra [AKG～], the integrable couplings system of the NLS-MKdV equations hierarchy was obtained. As its reduction case, generalized nonlinear NLS-MKdV equations were obtained. The method proposed in this letter can be applied to other hierarchies of evolution equations.

Applications of the first integral method to nonlinear evolution equations

In this paper, we establish travelling wave solutions for some nonlinear evolution equations. The first integral method is used to construct the travelling wave solutions of the modified Benjamin-Bona-Mahony and the coupled Klein-Gordon equations. The obtained results include periodic and solitary wave solutions. The first integral method presents a wider applicability to handling nonlinear wave equations.

(2+2)-Dimensional Discrete Soliton Equations and Integrable Coupling System

In this paper, we extend a （2＋2）-dimensional continuous zero curvature equation to （2＋2）-dimensional discrete zero curvature equation, then a new （2＋2）-dimensional cubic Volterra lattice hierarchy is obtained. Fhrthermore, the integrable coupling systems of the （2＋2）-dimensional cubic Volterra lattice hierarchy and the generalized Toda lattice soliton equations are presented by using a Lie algebraic system sl（4）.

A New Method to Construct Integrable Coupling System for Burgers Equation Hierarchy by Kronecker Product

It is shown that the Kronecker product can be applied to construct a new integrable coupling system of soliton equation hierarchy in this paper. A direct application to the Burgers spectral problem leads to a novel soliton equation hierarchy of integrable coupling system. It indicates that the Kronecker product is an efficient and straightforward method to construct the integrable couplings.

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.