Lax Matrixes And Their Applications

In this thesis, our main aim is to construct new Lax matrixes and to give their applications. Two class confocal Lax matrixes are discussed and various cases are studied systematically, by which we obtain a lot of new Lax matrixes. In addition, the Bargmann (or Neumann) systems of the famous AKNS, AKNS, KdV, c-KdV, Kaup-Newell, TD. Jaulent-Miodek, Levi and mKdV hierarchies are the special cases of our some results. The studies of the Lax matrixes are very complex and difficult, because they depend on solving eight or nine ordinary differential equations or partial differential equations.Based on the resulting Lax matrixes, we get quite a few new finite-dimensional integrable systems, by which the new eigenvalue problems are proposed resorting to the canonical equations of these finite-dimensional integrable systems. Two new eigenvalue problems and corresponding soliton hierarchies are derived, in which two typical equations are the generalized coupled Korteweg-de Vries (c-KdV) and the generalized (TD) equations. As applications of Riemannian surface and algebraic curve theory in integrable system, Abel-Jacobi coordinates are introduced suitably by the Lax matrix, from which straightening out of the flows are given. The algebraic geometry solutions of these new soliton equations are obtained with the help of Jacobi inversion. As another application of the Lax matrixes . we present a new (2 + l)-dimensional integrable modelwhich is decomposed into two new (1+l)-dimensional soliton equations and a new finite-dimensional integrable Hamiltonian system based on the Lax matrix and nonlinearization of eigenvalue problems in succession. Finally we get the finite band solutions of the (1+l)-dimensional and (2+l)-dimensional integrable models by means of Abel-Jacobi coordinates.