Soliton surfaces for complex modified Korteweg–de Vries equation

Abstract

In mathematics and physics, one of the main tasks is to relate differential geometry and non-linear differential equations, which means that the study of particular cases of subvarieties, curves, and surfaces are of great importance. Soliton surfaces associated with the integrable system play an essential role in many problems with the physical application. In this paper, we study the complex modi ed Korteweg de Vries (cmKdV) equation. It is well known that the cmKdV equation is a very important integrable equation. We present the relationship between an integrable system and soliton surfaces and namely Lax representation of the cmKdV equation was used to obtain the rst and second fundamental forms, surface area and curvature.

Soliton Geometry Using the Lax Pair of Isomonodromic Deformation

“…The construction of the surface of isomonodromic deformations became possible in many respects after the proof of several theorems about the properties of systems of linear differential equations. The method of isomonodromic deformations [4], [5] is used to study nonlinear equations, its idea is to implement a nonlinear equation as an isomonodromic condition of a certain system, and this interpretation provides essential information about the nonlinear equation. Under the surface, we consider the set Σ at points ( )…”