A direct method of solution for the Fokas-Lenells derivative nonlinear Schr\"odinger equation: I. Bright soliton solutions

TL;DR

This paper introduces a direct algebraic method to derive bright soliton solutions of the Fokas-Lenells derivative nonlinear Schr"odinger equation, avoiding inverse scattering, and explores their properties including collision phase shifts.

Contribution

It presents a novel determinant-based algebraic approach for constructing bright soliton solutions, providing new insights into their interaction properties.

Findings

Derived explicit formulas for bright soliton solutions as ratios of determinants.

Identified new features of soliton interactions, including phase shifts.

Provided two different determinant expressions for the solutions.

Abstract

We develop a direct method of solution for finding the bright $N$-soliton solution of the Fokas-Lenells derivative nonlinear Schr\"odinger equation. The construction of the solution is performed by means of a purely algebraic procedure using an elementary theory of determinants and does not rely on the inverse scattering transform method. We present two different expressions of the solution both of which are expressed as a ratio of determinants. We then investigate the properties of the solutions and find several new features. Specifically, we derive the formula for the phase shift caused by the collisions of bright solitons.