Soliton Condensates for the Focusing Nonlinear Schr\"odinger Equation: a Non-Bound State Case

TL;DR

This paper explores the spectral theory of soliton condensates for the focusing nonlinear Schrödinger equation, providing explicit solutions and analyzing large-scale dynamics, rarefaction, shock waves, and conserved quantities.

Contribution

It introduces the spectral theory of soliton condensates, solves the kinetic equation explicitly, and studies large-scale dynamics and wave phenomena for the focusing NLS.

Findings

Explicit solution of the kinetic equation for circular condensates

Analysis of rarefaction and dispersive shock waves

Calculation of conserved quantities and kurtosis

Abstract

In this paper, we study the spectral theory of soliton condensates - a special limit of soliton gases - for the focusing NLS (fNLS). In particular, we analyze the kinetic equation for the fNLS circular condensate, which represents the first example of an explicitly solvable fNLS condensate with nontrivial large scale space-time dynamics. Solution of the kinetic equation was obtained by reducing it to Whitham type equations for the endpoints of spectral arcs. We also study the rarefaction and dispersive shock waves for circular condensates, as well as calculate the corresponding average conserved quantities and the kurtosis. We want to note that one of the main objects of the spectral theory - the nonlinear dispersion relations - is introduced in the paper as some special large genus (thermodynamic) limit the Riemann bilinear identities that involve the quasimomentum and the quasienergy meromorphic differentials.