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Large-time existence results for the nonlocal NLS around ground state solutions

Hideo Takaoka, Toshihiro Tamaki

TL;DR

This work analyzes the Cauchy problem for the focusing nonlocal NLS $i\partial_t u-\partial_x^2 u = u^2 u^{\star}$ in one spatial dimension, focusing on long-time behavior near the ground-state soliton $Q$. The authors develop a modulation framework around a ground-state, perform a symplectic decomposition into even/odd fluctuations, and reduce the linearized dynamics to a pair of matrix Schrödinger problems with well-understood spectral structure. Through a bootstrap argument that controls modulation parameters and residuals, they prove the solution exists up to time $T_{\varepsilon}=c\log(1/\varepsilon)$ and remains within $O(\varepsilon)$ of the ground-state orbit. This establishes large-time stability near ground-state solitons for the nonlocal NLS using hyperbolic-type dynamics near the soliton rather than standard linear stability analysis, with potential implications for related nonlocal dispersive models.

Abstract

This paper discusses about solutions of the nonlocal nonlinear Schrodinger equation. We prove that the solution remains close to the orbit of the soliton for a large-time, if the initial data is close to the ground state solitons. The proof uses the hyperbolic dynamics near ground state, which exhibits properties of local structural stability of solutions with respect to the flows of the nonlocal nonlinear Schrodinger equation.

Large-time existence results for the nonlocal NLS around ground state solutions

TL;DR

This work analyzes the Cauchy problem for the focusing nonlocal NLS in one spatial dimension, focusing on long-time behavior near the ground-state soliton . The authors develop a modulation framework around a ground-state, perform a symplectic decomposition into even/odd fluctuations, and reduce the linearized dynamics to a pair of matrix Schrödinger problems with well-understood spectral structure. Through a bootstrap argument that controls modulation parameters and residuals, they prove the solution exists up to time and remains within of the ground-state orbit. This establishes large-time stability near ground-state solitons for the nonlocal NLS using hyperbolic-type dynamics near the soliton rather than standard linear stability analysis, with potential implications for related nonlocal dispersive models.

Abstract

This paper discusses about solutions of the nonlocal nonlinear Schrodinger equation. We prove that the solution remains close to the orbit of the soliton for a large-time, if the initial data is close to the ground state solitons. The proof uses the hyperbolic dynamics near ground state, which exhibits properties of local structural stability of solutions with respect to the flows of the nonlocal nonlinear Schrodinger equation.
Paper Structure (9 sections, 6 theorems, 165 equations)

This paper contains 9 sections, 6 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminary
  4. Reformulate the setup of the problem
  5. Linearized operator
  6. Parameter choices in evolutionary computation
  7. Computations on $a_e(t),~b_e(t),~a_o(t),~b_o(t),~\dot{\theta}(t),~\dot{\alpha}(t)$
  8. Computations on $\eta_e(t),~\eta_o(t)$
  9. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.3

There exists $\varepsilon_0>0$ such that for any $0< \varepsilon\le \varepsilon_0$, if $u_0\in\mathcal{S}_{\varepsilon}$, then there exists a unique solution $u(t)$ to eq:nnls on $[-T_{\varepsilon},T_{\varepsilon}]$, where $T_{\varepsilon}=c \log(1/\varepsilon)$ with $0<c\ll 1$ independent of $\vare

Theorems & Definitions (22)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 3.1: cazenavetsutsumi
  • Lemma 3.2
  • proof
  • Proposition 3.3: changweinstein1
  • ...and 12 more