Nonlinear Evolution Equations and Dynamical Systems

PREFACE

CONTENTS

Using the nonstandard and standard truncations of a modified Conte's invariant Painlevé expansion for the dispersive long wave equation system, two types of soliton excitations without any dispersive relations are found. Periodic waves expressed by Jacobi elliptic functions are found by the truncations of a special extended Painlevé expansion. The soliton solutions are special cases of the corresponding two of the given periodic solutions. The dispersion relations of the solutions are crucially dependent on the boundary conditions.

This paper is devoted to the global bifurcation phenomena from the eigenvalues of some semilinear biharmonic equations in an open bounded domain that contains the origin with homogeneous Dirichlet boundary condition. Our main tools are based on the operator theory, the Leray-Schauder degree theory and the Rabinowitz's global bifurcation theory.

The KP equation with self-consistent sources is derived through the linear problem of the KP system. The Multisoliton solutions for the KP equation with self-consistent sources are presented by using Hirota method and Wronskian technique. The coincidence of these solutions is shown by direct computation. The novel multisoliton solutions of the KP equation with a self-consistent source are also obtained by Hirota method.

A direct extension of Lie algebra A_n−1 is presented. For the sake of simplicity, we consider a special case gl(2, C). For which, a new commutator is defined, which becomes an extension form of Lie algebra A₁. A type of new loop algebra is constructed. It follows that a kind of hierarchy of evolution equations is obtained, which is an extension formalism of the well-known AKNS hierarchy. As its reduction case, a generalized nonlinear Schrödinger equationis shown. At last, the Hamiltonian structure of the hierarchy obtained is presented.

In this paper we discuss some aspects of gauge-gravity correspondence as part of open-closed string duality. In such a correspondence people find a super-gravity dual of a super-symmetric gauge theory. Some physically interesting theories such as SUSY Yang-Mills admit such a SUGRA dual. It then gives very useful information on the side of gauge theory, even in the regime of strong coupling.

Integrable semi-discretizations of two model equations for shallow water waves are investigated. As a result, one integrable differential-difference version for the AKNS equation and three integrable differential-difference versions for the Hirota-Satsuma equation are found. These four differential-difference versions are transformed into bilinear forms. Bäcklund transformations, soliton solutions and Lax pairs for these differential-difference equations are presented.

We discuss the recent results of the author on the existence of systems of differential equations for chiral genus-zero and genus-one correlation functions in conformal field theories.

Lie symmetries has been introduced by Sophus Lie to study differential equations. It has been one of the most efficient way for obtaining exact analytic solution of differential equations. Here we extend this technique to the case of differential difference and difference equations.

Perturbation methods has proved very successful in analyzing classes of partial differential equations, pinning out integrable cases and providing informations on their long time behavior. Here we present the state of art of this approach in the case of differential equations defined on a lattice. In particular we present the results of Leon and Manna on the multiscale analysis of discrete nonlinear evolution equations and show that their resulting equation has not the same integrability property as Ablowitz and Ladik Discrete Nonlinear Schrödinger equation.

The purpose of this note is to give a general overview of the theory of p-harmonic maps, and its applications.

A simple and systematic approach of finding conservation laws for the Lax integrable (1+1)-dimensional differential-difference systems is described. The conservation laws of the systems associated with two kinds of (1+1)-dimensional spectral problems are considered. By introducing the generalized Riccati equation related to pseudo-difference operator, we generalize this procedure to the high dimensional differential-difference systems. The conservation laws of the two-dimensional Toda lattice hierarchy are discussed.

This lecture was given in UIA at Antwerp, and in USTC at Hefei, aiming at outlining a construction of non-commutative, non-cocommutative pointed Hopf algebras via quivers, given by Cibils and Rosso³. We thank Sen Hu for his interest to include it in this proceeding. To save the space, we omit details here.

A theory of bidirectional solitons on water has been developed by using the classical Boussinesq equation. A solid mathematical foundation of the bidirectional water wave interaction has been well-established with our theory.

A BKK soliton hierarchy is proposed by means of a 3 × 3 Lenard operator pairs. Through a natural nonlinearization of the BKK eigenvalue problems, a finite-dimensional Hamiltonian system is obtained. Some interesting solutions of the system are also given.

In this paper, two model equations for shallow water waves are investigated. By using Hirota's bilinear method, the novel N-soliton solutions are obtained. Finally, we analysis the singularities of the novel solution.

A new isospectral problem, which is an expanding form of the KN spectral problem, is designed. Using Tu-model leads to a new integrable hierarchy with bi-Hamiltonian structure. Then, a new loop algebra , which is a subalgebra of loop algebra Ã₂, is constructed. It follows that a new isospectral problem with multi-potentials is designed from . Again using Tu-model obtains expanding integrable system of the derived system above.

No abstract received.

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