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arXiv · 2006.04262

Singular Soliton Molecules of the Nonlinear Schrodinger Equation

Khelifa Mohammed Elhadj·L. Al Sakkaf·U. Al Khawaja·Abdelaali Boudjemaa
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Abstract

We derive an exact solution to the local nonlinear Schrödinger 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.

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Khelifa Mohammed Elhadj, L. Al Sakkaf, U. Al Khawaja, Abdelaali Boudjemaa. 2020-06-07. Singular Soliton Molecules of the Nonlinear Schrodinger Equation. https://doi.org/10.1103/physreve.101.042221

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