Fokas-Lenells Derivative nonlinear Schr\"odinger equation its associated soliton surfaces and Gaussian curvature

TL;DR

This paper explores the geometric properties of soliton surfaces associated with the Fokas-Lenells derivative nonlinear Schrödinger equation, linking integrable systems with differential geometry through explicit surface construction and curvature analysis.

Contribution

It constructs soliton surfaces related to the Fokas-Lenells equation using Lax pairs and the Sym-Tafel formula, providing explicit geometric characterizations.

Findings

Derived the first and second fundamental forms of the soliton surfaces

Calculated the Gaussian curvature of the surfaces

Connected integrable equations with differential geometry concepts

Abstract

One of the most important tasks in mathematics and physics is to connect differential geometry and nonlinear differential equations. In the study of nonlinear optics, integrable nonlinear differential equations such as the nonlinear Schr\"odinger equation (NLSE) and higher-order NLSE (HNLSE) play crucial roles. Because of the medium's balance between dispersion and nonlinearity, all of these systems display soliton solutions. The soliton surfaces, or manifolds, connected to these integrable systems hold significance in numerous areas of mathematics and physics. We examine the use of soliton theory in differential geometry in this paper. We build the two-dimensional soliton surface in the three-dimensional Euclidean space by taking into account the Fokas-Lenells Derivative nonlinear Schr\"odinger equation (also known as the gauged Fokas-Lenells equation). The same is constructed by us using the Sym-Tafel formula. The first and second fundamental forms, surface area, and Gaussian curvature are obtained using a Lax representation of the gauged FLE.