The generalized dressing method and algebra curves method and their applications to integrable equations

Abstract

The thesis is mainly divided into two parts. In the first part, a hierarchy of integrable variable-coefficient nonlinear SchrÄodinger equations, integrable variable-coefficient Dirac system and Toda Lattice equations are discussed by using a generalized version of the dressing method. In the second part, some 2+1-dimensional integrable discrete systems are studied by utilizing algebra curves approaches. The generalized version of the dressing method was presented by Dai and Jeffrey in 1990s, which is an 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) equation is obtained. These Volterra operators are then used to construct dressed operators (N1;N2) 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 these equations, it is necessary to explicitly construct the kernel F of the integral operator F from the commutative relation between the 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 discussed in this thesis. Firstly, with the aid of n £ n AKNS matrix isospectrum (n=2, 3, N+1, 2N+1) and Dirac system, integrable variable-coefficient coupled cylindrical 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 applied to the discrete system, from which 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 1988, which is linked to algebra-geometric curves. The new scheme is further shown to be a very powerful tool, through which quasi-periodic solutions of multi-dimensional continuous and discrete soliton equations can be obtained by using decomposition technique. With the help of the constraint between potential and characteristic function, the continuous and discrete spectrums are nonlinearized. In the 6th chapter of this thesis, two discrete spectrums are studied by using the nonlinearization scheme. Firstly, a new discrete spectrum is proposed, and nonlinear differential difference equations of the corresponding hierarchy are obtained. We derive an interesting 2+1- dimensional discrete NLS model. Then, under the Bargmann constraint, the soliton equations are decomposed into certain finite-dimensional systems and a new integrable symplectic map. And then, the generating function method is applied to the study of integrability, 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 coordinates are obtained by Riemann-theta function and Abel-Jacobi inversion. Secondly, in a similar way, a new 2+1-dimensional discrete model is proposed, and some interesting conclusions are drawn. Finite-order expansion of the solution matrix of Lax equation is derived by Geng, which is also a strong powerful tool for obtaining the solutions of multi-dimensional soliton equations. This can be realized through three steps: decomposition!straightening !inversion. 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, elliptic variables are introduced. And using algebra-geometric curves, it is easy to construct the Riemann surface. Infinite dimensional integrable system and discrete system are straightened by introducing the Abel- Jacobi coordinate. In the 5th chapter, two semi-discrete systems are studied. In the first section, semi-discrete Kaup-Newell system is discussed. It is interesting that the continuous limit of a 2+1-dimensional discrete model is exactly a 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 is 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. Keywords: Dressing method, the nonlinearizaton of Lax pair, quasi-periodic solution, Lax equation, the generating function.