The formation of a soliton gas condensate for the focusing Nonlinear Schrödinger equation

TL;DR

This work analyzes the focusing NLS equation in the condensate scaling, where $N$-soliton eigenvalues accumulate on two horizontal spectral segments with norming constants bounded away from zero. By formulating and asymptotically solving a sequence of Riemann-Hilbert problems, the authors show that the $N$-soliton condensate converges, on compact sets in $(x,t)$, to a rapidly oscillatory elliptic wave whose modulus is time-invariant and whose phase is governed by a discrete- versus continuum-spectral balance. The leading-order solution is written explicitly in terms of Jacobi elliptic functions (sd) involving $|A|$, $ heta$, and a logarithmic in $N$ term, with corrections of order $1/\ln N$; a global and local parametrix construction plus a small-norm RH analysis provides a rigorous justification. The paper also establishes a deterministic connection to the kinetic theory of solitons by incorporating a tracer soliton into the condensate and deriving its effective velocity, consistent with kinetic predictions for dense soliton gases. Overall, the results rigorously bridge multi-soliton dynamics, soliton-gas condensates, and kinetic theory, offering a precise elliptic-wave description and validating soliton-kinetics in this deterministic setting.

Abstract

In this work, we carry out a rigorous analysis of a multi-soliton solution of the focusing nonlinear Schrödinger equation as the number, $N$, of solitons grows to infinity. We discover configurations of $N$-soliton solutions which exhibit the formation (as $N \to \infty$) of a soliton gas condensate. Specifically, we show that when the eigenvalues of the Zakharov - Shabat operator for the NLS equation accumulate on two bounded horizontal segments in the complex plane with norming constants bounded away from $0$, then, asymptotically, the solution is described by a rapidly oscillatory elliptic-wave with constant velocity, on compact subsets of $(x,t)$. We then consider more complex solutions with an extra soliton component, and provide rigorous justification of the predictions of the kinetic theory of solitons in this deterministic setting. This is to be distinguished from previous analyses of soliton gasses where the norming constants were tending to zero with $N$, and the asymptotic description only included elliptic waves in the long-time asymptotics.

Figures (11)

• Figure 1: Poles $\lambda_j$ and their conjugates
• Figure 2: Example of contour $\Gamma_1,\Gamma_2$
• Figure 3: Solution to the NLS equation \ref{['eq:NLS']} in assumptions \ref{['ass:sym']}. Here $A=1+i$, and $N$ is specified in the plots
• Figure 4: Contours $\Gamma_1,\Gamma_2,\eta_1,\eta_2$
• Figure 5: Non-analyticity contour for the function $C(z)$

Theorems & Definitions (44)

• Remark 2.1
• Theorem 2.5
• Theorem 2.6
• Remark 2.2
• Proposition 2.7
• Corollary 2.8
• Proposition 3.6
• Remark 3.1
• Proposition 3.7
• Remark 3.2