$N$-soliton solutions of the Fokas-Lenells equation for the plasma ion-cyclotron waves: Inverse scattering transform approach

TL;DR

This paper develops a straightforward inverse scattering transform method to derive various $N$-soliton solutions, including bright, rational, and mixed forms, for the Fokas-Lenells equation modeling plasma ion-cyclotron waves.

Contribution

It introduces a simple, constructive approach to obtain explicit $N$-soliton solutions, including those with continuous spectrum contributions, for the Fokas-Lenells equation.

Findings

Derived explicit $N$-soliton solutions including rational and mixed types.

Presented breather solutions as specific examples.

Provided a general expression incorporating continuous spectrum effects.

Abstract

We present a simple and constructive method to find $N$-soliton solutions of the equation suggested by Davydova and Lashkin to describe the dynamics of nonlinear ion-cyclotron waves in a plasma and subsequently known (in a more general form and as applied to nonlinear optics) as the Fokas-Lenells equation. Using the classical inverse scattering transform approach, we find bright $N$-soliton solutions, rational $N$-soliton solutions, and $N$-soliton solutions in the form of a mixture of exponential and rational functions. Explicit breather solutions are presented as examples. Unlike purely algebraic constructions of the Hirota or Darboux type, we also give a general expression for arbitrary initial data decaying at infinity, which contains the contribution of the continuous spectrum (radiation).