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A dense focusing Ablowitz-Ladik soliton gas and its asymptotics

Meisen Chen, Engui Fan, Zhaoyu Wang, Yiling Yang, Lun Zhang

Abstract

In this paper, we propose a soliton gas solution for the focusing Ablowitz-Ladik system. This solution is defined as the large N limit of the N-soliton solution, and arises from a continuous spectrum of poles that accumulate within two disjoint intervals on the imaginary axis. We show that this gas solution admits a Fredholm determinant representation. By further exploring its Riemann-Hilbert characterization, we are able to establish the large-space asymptotics at t = 0 and large-time asymptotics of the gas solution.

A dense focusing Ablowitz-Ladik soliton gas and its asymptotics

Abstract

In this paper, we propose a soliton gas solution for the focusing Ablowitz-Ladik system. This solution is defined as the large N limit of the N-soliton solution, and arises from a continuous spectrum of poles that accumulate within two disjoint intervals on the imaginary axis. We show that this gas solution admits a Fredholm determinant representation. By further exploring its Riemann-Hilbert characterization, we are able to establish the large-space asymptotics at t = 0 and large-time asymptotics of the gas solution.
Paper Structure (39 sections, 6 theorems, 240 equations, 15 figures)

This paper contains 39 sections, 6 theorems, 240 equations, 15 figures.

Table of Contents

  1. Introduction and main results
  2. Main results
  3. Notations
  4. The soliton gas solution and its RH characterization
  5. Fredholm determinant representation of $q_n$
  6. The Fredholm determinant representation of the $N$-soliton solution
  7. Proof of Theorem \ref{['thm1']}
  8. Large-$n$ asymptotic analysis of the RH problem for $Z(\lambda; n,0)$
  9. The auxiliary functions
  10. Lenses opening
  11. Global parametrix
  12. Local parametrices
  13. The small-norm RH problem
  14. Large-$t$ asymptotic analysis of the RH problem for $Z(\lambda;n+1,t)$
  15. The auxiliary functions
  16. ...and 24 more sections

Key Result

Theorem 1.1

Suppose that $r(\lambda)$ is a positive and continuous function on $[{\rm i}\eta_1,{\rm i}\eta_2]$ and let $\mathcal{K}: L^2(\Sigma_1)\to L^2(\Sigma_1)$ be an integral operator defined by where is the associated kernel. Then, is the soliton gas solution for the focusing AL system e1.2. In addition, we have

Figures (15)

  • Figure 1: Five different asymptotic regions given in Definition \ref{['def region']}.
  • Figure 2: The genus-1 Riemann surface $\mathcal{R}$ with homology basis $\{\mathfrak{a}, \mathfrak{b}\}$ associated with $R(\lambda)$ in \ref{['eRn']}, where Sheet I is the principal sheet, and the curves $\Sigma_1$ and $\Sigma_2$ are the branch cuts.
  • Figure 3: The signature table of $\operatorname{Re} [g]$. $\operatorname{Re} [g]>0$ in the green region, while $\operatorname{Re}[g]<0$ in the white region, and $\operatorname{Re}[g]=0$ on the blue curve.
  • Figure 4: The regions $\Omega_{i,\pm}$, $i=1,2$, and the jump contour $\Sigma^{(1)}$ of $Z^{(1)}$.
  • Figure 5: The jump contour $\Sigma^E$ of $E$.
  • ...and 10 more figures

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Proposition 2.3
  • proof
  • Proposition 4.1
  • proof
  • Proposition 5.1
  • ...and 1 more