##### What's Hot

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.

Share.