Completely Integrable Models in the Domain Walls and Interphases Theories

Abstract

Many problems of nonlinear physics lead to necessity of studying completely integrable dynamical systems. For instance, the integrable models in the theory of interphases for the media with several order parameters (magnetic or segnetoelectric media) allow us to give an exhaustive classification of the interphases and to study how the number and the kind of interphases may vary due to continuous variation of structural parameters of a model [l]. The construction of completely integrable Hamiltonian systems with the potentials associated with the Landau phase transitions theory with several order parameters (such as potential, permitting a continuous transition from the potential pit with one global minimum to that with several local minima) gives us new possibilities to develop the models of phase transition (also in terms of soliton states). The existence of such classical models means that the corresponding quantum model has a coordinate system in which the Schrodinger equation variables become separated. The method of transfer matrix for quasi-one-dimensional systems is known [2] to lead it the thermodynamical approach to the eigenvalue problem for the Schrodinger type operator. Here the changes of the structure of the phase space of classical problem (for example, those connected with a change of the type and number of singular points of a Hamiltonian system) correspond to characteristic changes of the dependence of eigenvalues on the model’s structural parameters. The opportunity of thorough analysis of both classical and quantum completely integrable Hamiltonian systems is of considerable interest for statistical physics of quasi-one-dimensional structures with several order parameters.