The Generalized Dressing Method And Algebraic Curve Method And Their Applications In Soliton System

The thesis is mainly divided into two parts.On the one hand,a hierarchy of integrable variable-coefficient nonlinear Schr(o|¨)dinger equations,integrable variable-coefficient Dirac system and integrable variable-coefficient Toda lattice equations are discussed by using a generalized version of the dressing method,further,their explicit solutions and Lax pairs are given.On the other hand,some multi-dimensional integrable discrete systems are studied and their quasi-periodic solutions are given by utilizing algebra curve approaches.The generalized version of the dressing method was presented by Dai and Jeffery in 1990s,which is a extension of the original dressing method.The generalization provides a procedure not only for construction of integrable variable-coefficient nonlinear evolution equations,but also giving their explicit solutions and Lax pairs.It was based on the problem of factorization of an integral operator F on the line into the product of two Volterra type integral operators K_±,from which the Gel'fand-Levitan -Marchenko(GLM) is obtained.These Volterra operators are then used to construct dressed operators((?),(?)) starting from a pair of initial variable-coefficient operators(M₁,M₂).Integrable variable-coefficient nonlinear evolution equations are obtained from the compatibility condition of the dressed operators.In order to derive the solutions of the equations,it is necessary to explicitly construct the kernel F of integral operator F from the commutative relation between integral operator F and initial operators(M₁,M₂).Then the kernel K of Volterra operators is obtained from the GLM equation,that is,the solution of the integrable nonlinear evolution equation can be described.As an application,two problems are mainly discussed in the paper.Firstly,with the aid of n×n AKNS matrix isospectrum (n=2,3,N+1,2N+1) and Dirac system,integrable variable-coefficient coupled cylin- drical NLS equations and mKdV equation;integrable variable-coefficient coupled Hirota equation and Manakov equation;a hierarchy of integrable variable-coefficient N-coupled NLS equations and integrable variable-coefficient defocusing NLS equation, are discussed,respectively.Secondly,the generalized dressing method is parallel to the discrete system from the continuous system,from which the integrable variable-coefficient Toda lattice equations are presented.Further,some solutions and Lax pairs of the equations are given explicitly.The nonlinearization approach of eigenvalue problem was presented by Cao in 1989,which is generalized to obtain quasi-periodic solutions of multi-dimensional soliton equations,The new scheme is further shown to be very effective.This can be realized through three steps:decomposition→straightening→inversion. In 6th chapter of the paper,two discrete spectrum are studied by using the nonlinearization scheme.Firstly,a new discrete spectral is proposed,and nonlinear differential difference equations of the corresponding hierarchy are obtained,it is interesting to derive a 2+1-dimensional discrete derivation NLS model.Then,under Bargmann constraint,the soliton equations are decomposed into finite-dimensional systems and a new integrable symplectic map.And then,the generating function method is applied to the study of their integrability structure,by which it is easy to prove the involutivity and independence of integrals of motion.Introducing the elliptic coordinates and Abel-Jacobi coordinates,discrete flow and continuous flow are straightened.Finally,quasi-periodic solutions of the soliton equations in the original coordinate are obtained by Riemann-θfunction and Abel-Jacobi inversion. Secondly,in a similar way,a new 2+1-dimensional discrete model is proposed,and some rich conclusions are discussed.Finite-order expansion of the solution matrix of Lax equation is derived by Geng,which is also a strong powerful approach for obtaining the solutions of multidimensional soliton equations.Using decomposition technique,a 2+1-dimensional discrete model is decomposed into two compatible ordinary differential equations and discrete flow inversion.With the aid of Lax equation matrix of characteristic function satisfying,elliptical variables are introduced reasonably.And using algebra-geometric,it is easy to construct the Riemann surface.Infinite dimensional integrable systems and discrete system are straightened by introducing Abel-Jacobi coordinate.In 5th chapter of the paper,two semi-discrete systems are studied.In the first section,semi-discrete Kaup-Newell system is discussed,it is interesting that the continuous limits of a 2+1-dimensional discrete model is exactly 2+1-dimensional Chen-Lee-Liu equation.Furthermore,quasi-periodic solution of the equation is described by introducing the elliptic coordinates and Abel-Jacobi inversion. In the second section,semi-discrete Chen-Lee-Liu system is studied in detail. With the help of Lenard's gradient sequence,a hierarchy of nonlinear differential-difference equation are given.Moreover,a well-known 2+1-dimensional derivation Toda lattice equation is derived.Similarly,we obtain the quasi-periodic solution of the corresponding equation in the original coordinates.