Soliton shielding of the focusing Nonlinear Schr\"odinger Equation

TL;DR

This paper investigates a phenomenon called soliton shielding in the focusing nonlinear Schrödinger equation, showing that a large deterministic or stochastic soliton gas can produce a single soliton solution, with robustness across different spectral distributions.

Contribution

It introduces the concept of soliton shielding in the FNLS equation and demonstrates its robustness in both deterministic and stochastic soliton gases, including various spectral distributions.

Findings

Soliton shielding occurs when a large soliton gas yields a single soliton solution.

The phenomenon persists under stochastic distributions such as uniform and Ginibre eigenvalue statistics.

Soliton shielding reduces spectral data to the soliton density, especially in elliptical domains.

Abstract

We first consider a deterministic gas of $N$ solitons for the Focusing Nonlinear Schr\"odinger (FNLS) equation in the limit $N\to\infty$ with a point spectrum chosen to interpolate a given spectral soliton density over a bounded domain of the complex spectral plane. We show that when the domain is a disk and the soliton density is an analytic function, then the corresponding deterministic soliton gas surprisingly yields the one-soliton solution with point spectrum the center of the disk. We call this effect {\it soliton shielding}. We show that this behaviour is robust and survives also for a {\it stochastic} soliton gas: indeed, when the $N$ soliton spectrum is chosen as random variables either uniformly distributed on the circle, or chosen according to the statistics of the eigenvalues of the Ginibre random matrix the phenomenon of soliton shielding persists in the limit $N\to \infty$. When the domain is an ellipse, the soliton shielding reduces the spectral data to the soliton density concentrating between the foci of the ellipse. The physical solution is asymptotically step-like oscillatory, namely, the initial profile is a periodic elliptic function in the negative $x$--direction while it vanishes exponentially fast in the opposite direction.