Spectral theory of soliton and breather gases for the focusing nonlinear Schr\"odinger equation

TL;DR

This paper develops an analytical spectral theory for soliton and breather gases in the focusing nonlinear Schrödinger equation, revealing their statistical properties, interactions, and kinetic behavior with broad physical implications.

Contribution

It introduces a thermodynamic limit approach to derive nonlinear dispersion relations and kinetic equations for soliton and breather gases, advancing the understanding of their spectral and statistical characteristics.

Findings

Derived nonlinear dispersion relations for breather and soliton gases.

Established kinetic equations describing the evolution of the density of states.

Demonstrated the theory's applicability through concrete examples in physics.

Abstract

Solitons and breathers are localized solutions of integrable systems that can be viewed as "particles'' of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media these "integrable'' gases present fundamental interest for nonlinear physics. We develop analytical theory of breather and soliton gases by considering a special, thermodynamic type limit of the wavenumber-frequency relations for multi-phase (finite-gap) solutions of the focusing nonlinear Schr\"odinger equation. This limit is defined by the locus and the critical scaling of the band spectrum of the associated Zakharov-Shabat operator and yields the nonlinear dispersion relations for a spatially homogeneous breather or soliton gas, depending on the presence or absence of the "background'' Stokes mode. The key quantity of interest is the density of states defining, in principle, all spectral and statistical properties of a soliton (breather) gas. The balance of terms in the nonlinear dispersion relations determines the nature of the gas: from an ideal gas of well separated, non-interacting breathers (solitons) to a special limiting state, which we term breather (soliton) condensate, and whose properties are entirely determined by the pairwise interactions between breathes (solitons). For a non-homogeneous breather gas, we derive a full set of kinetic equations describing slow evolution of the density of states and of its carrier wave counterpart. The kinetic equation for soliton gas is recovered by collapsing the Stokes spectral band. A number of concrete examples of breather and soliton gases are considered, demonstrating efficacy of the developed general theory with broad implications for nonlinear optics, superfluids and oceanography.