Singular Soliton Molecules of the Nonlinear Schrodinger Equation

TL;DR

This paper derives exact solutions for the nonlinear Schrödinger equation, revealing singular soliton molecules with finite binding energy, and explores interactions, nonlocal variants, and their dynamics.

Contribution

It introduces a novel exact solution describing singular soliton molecules and analyzes their interactions and stability, extending to nonlocal and time-nonlocal NLSE variants.

Findings

Identified singular soliton molecules with diverging peaks.

Calculated interaction forces showing molecular-like behavior.

Derived solutions for nonlocal and time-nonlocal NLSE variants.

Abstract

We derive an exact solution to the local nonlinear Schr\"odinger equation (NLSE) using the Darboux transformation method. The new solution describes the profile and dynamics of a two-soliton molecule. Using an algebraically-decaying seed solution, we obtain a two-soliton solution with diverging peaks, which we denote as singular soliton molecule. We find that the new solution has a finite binding energy. We calculate the force and potential of interaction between the two solitons, which turn out to be of molecular-type. The robustness of the bond between the two solitons is also verified. Furthermore, we obtain a new solution to the nonlocal NLSE using the same method and seed solution. The new solution in this case corresponds to an elastic collision of a soliton, a breather soliton on flat background, and a breather soliton on a background with linear ramp. Finally, we consider an NLSE which is nonlocal in time rather than space. Although we did not find a Lax pair to this equation, we derive three exact solutions.