Soliton Geometry Using the Lax Pair of Isomonodromic Deformation

Abstract

Physical processes are described using mathematical models. Many of them are nonlinear in nature. For this reason, the theory of a nonlinear medium is relevant and very extensive. From a mathematical point of view, the subject of physics of nonlinear phenomena is systems described by nonlinear partial differential equations that have partial solutions - solitons. A traveling wave that rapidly decreases at infinity is called a solitary wave or soliton. Soliton theory has many fundamental methods for detailed analysis of processes. One of these methods is the geometric interpretation of the physical process. This paper is devoted to the study of the Lax pair of isomonodromic deformation. The isomonodromy condition is equivalent to the existence of a compatible pair of linear equations, the Lax pair. In this pair, one of the equations undergoes deformation, and the other describes the deformation. Isomonodromic deformation is the theory of isomonodromy (that is, monodromy conservation) of the deformation of ordinary differential equations. This method was used to obtain expressions for the coordinate angle. It is proved that the deformation of a system is isomonodromic if and only if the first and second fundamental forms that define this deformation satisfy the integrability condition. It is shown that, similarly, the area of a soliton surface is represented as a semi-saddle graph.