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Traveling periodic waves and breathers in the nonlocal derivative NLS equation

Jinbing Chen, Dmitry E. Pelinovsky

TL;DR

The paper analyzes the nonlocal derivative NLS equation $i u_t = u_{xx} + \sigma u \mathcal{P}^+(\bar{u}\cdot)$, addressing both defocusing and focusing regimes. It develops a Lax pair framework and Hirota bilinear reduction to obtain traveling periodic waves on nonzero and zero backgrounds, studies their Lax spectra, and constructs breather solutions, including a closed determinant form for general $N$-breathers. Stability results show linear and nonlinear stability for the nonzero background in the defocusing case and stability under restrictions for the focusing case, with the latter sharing a resonance-driven caveat. The work provides a comprehensive determinant-based toolkit for multi-breathers and clarifies how the Lax-spectrum bands govern breather existence, contributing to dispersive hydrodynamics of nonlocal NLS-type models and linking to deep-fluid and CMS contexts.

Abstract

A nonlocal derivative NLS (nonlinear Schrödinger) equation describes modulations of waves in a stratified fluid and a continuous limit of the Calogero--Moser--Sutherland system of particles. For the defocusing version of this equation, we prove the linear stability of the nonzero constant background for decaying and periodic perturbations and the nonlinear stability for periodic perturbations. For the focusing version of this equation, we prove linear and nonlinear stability of the nonzero constant background under some restrictions. For both versions, we characterize the traveling periodic wave solutions by using Hirota's bilinear method, both on the nonzero and zero backgrounds. For each family of traveling periodic waves, we construct families of breathers which describe solitary waves moving across the stable background. A general breather solution with $N$ solitary waves propagating on the traveling periodic wave background is derived in a closed determinant form.

Traveling periodic waves and breathers in the nonlocal derivative NLS equation

TL;DR

The paper analyzes the nonlocal derivative NLS equation , addressing both defocusing and focusing regimes. It develops a Lax pair framework and Hirota bilinear reduction to obtain traveling periodic waves on nonzero and zero backgrounds, studies their Lax spectra, and constructs breather solutions, including a closed determinant form for general -breathers. Stability results show linear and nonlinear stability for the nonzero background in the defocusing case and stability under restrictions for the focusing case, with the latter sharing a resonance-driven caveat. The work provides a comprehensive determinant-based toolkit for multi-breathers and clarifies how the Lax-spectrum bands govern breather existence, contributing to dispersive hydrodynamics of nonlocal NLS-type models and linking to deep-fluid and CMS contexts.

Abstract

A nonlocal derivative NLS (nonlinear Schrödinger) equation describes modulations of waves in a stratified fluid and a continuous limit of the Calogero--Moser--Sutherland system of particles. For the defocusing version of this equation, we prove the linear stability of the nonzero constant background for decaying and periodic perturbations and the nonlinear stability for periodic perturbations. For the focusing version of this equation, we prove linear and nonlinear stability of the nonzero constant background under some restrictions. For both versions, we characterize the traveling periodic wave solutions by using Hirota's bilinear method, both on the nonzero and zero backgrounds. For each family of traveling periodic waves, we construct families of breathers which describe solitary waves moving across the stable background. A general breather solution with solitary waves propagating on the traveling periodic wave background is derived in a closed determinant form.
Paper Structure (14 sections, 10 theorems, 181 equations, 12 figures)

This paper contains 14 sections, 10 theorems, 181 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Lax pair for the nonlocal derivative NLS equations
  3. Stability of the nonzero constant background
  4. Traveling periodic waves and breathers on the nonzero background
  5. Traveling periodic wave
  6. Lax spectrum of the traveling periodic wave
  7. Breathers on the traveling periodic wave
  8. $N$-breathers on the traveling periodic wave
  9. Traveling periodic waves and breathers on the zero background
  10. Traveling periodic waves
  11. Lax spectrum of the traveling periodic wave
  12. Breathers on the traveling periodic wave
  13. $N$-breathers on the traveling periodic wave
  14. Conclusion

Key Result

Theorem 1

Let $u = 1 + v$ and consider the linearized equations of motion If $\sigma = +1$, then for every initial data $v_0 \in H^s(\mathbb{R})$, $s \geq 0$, the unique solution $v \in C^0(\mathbb{R},H^s(\mathbb{R}))$ to the linearized equation (lin-NLS) with $v |_{t=0} = v_0$ satisfies for some constant $C > 0$. If $\sigma = -1$, then for every $v_0 \in H^s(\mathbb{R}) \cap L^{2,p}(\mathbb{R})$, $s \geq

Figures (12)

  • Figure 1: The profile of $|u|^2$ versus $x$ for $\sigma = +1$, $k_1 = 0.25$, and either $c_1 = -1$ (left) or $c_1 = -0.5$ (right).
  • Figure 2: The profile of $|u|^2$ versus $x$ for $\sigma = -1$, $k_1 = 0.25$, and either $c_1 = 2 + 2k_1$ (left) or $c_1 = 2 + 4k_1$ (right).
  • Figure 3: The Lax spectrum for the breather solutions of Figure \ref{['fig-2']} (left) and Figure \ref{['fig-3']} (right).
  • Figure 4: The solution surface of $|u|^2$ for the breather versus $(x+t,t)$ for $k_1 = 0.25$, $c_1 = -1$, and $c_2 = -0.5$.
  • Figure 5: The solution surface of $|u|^2$ for the breather versus $(x+t,t)$ for $k_1 = 0.25$, $c_1 = -0.5$, and $c_2 = -1$.
  • ...and 7 more figures

Theorems & Definitions (38)

  • Definition 1
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Remark 3
  • ...and 28 more