The first and second fundamental forms for the Camassa-Holm equation

Abstract

One of the actual problems of mathematical physics is the study of nonlinear partial differential equations.
Research in this direction is very important, as the results are theoretical and practical application. There are different approaches to solving these equations. Methods of soliton theory allow us to construct solutions of nonlinear partial differential equations. One of the methods for solving the above equations is the method of the inverse scattering problem. The purpose of this work is to determine the first and second fundamental forms of the surface corresponding to the soliton solution of the nonlinear
Camassa-Holm equation. According to this approach, in the (1+1)-dimensional case, nonlinear partial differential equations are given as zero curvature conditions and are a condition of compatibility of the system of linear equations. It is well known that the integrable nonlinear Camassa-Holm equations play an important role in the study of wave propagation. Various soliton solutions of the Camassa-Holm equation can be found using known transformation methods. In this paper, the first and second fundamental forms of the surface are determined using the Sym-Tafel formula for the nonlinear Camassa-Holm equation. The
obtained result can be used for further investigation of the multicomponent generalized Camassa-Holm equation.