Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-existence of internal mode for small solitary waves of the 1D Zakharov system

Yvan Martel, Guillaume Rialland

TL;DR

The paper proves that small solitary waves of the 1D Zakharov system admit no internal modes in their linearization. By linking the Zakharov system to its NLS limit, the authors perform a detailed spectral analysis using almost-orthogonality, a transformed problem via the factorisation $S^2 L_+ L_- = M^2 S^2$, and localized virial arguments. The main result shows that any time-periodic solution of the linearized system must lie in the kernel corresponding to $Q'$ and $Q$, with all other components vanishing for small $\omega$, thereby excluding internal modes and resonances. This spectral property is expected to facilitate the study of asymptotic stability of solitary waves in this model. The work combines coercivity estimates, strategic decompositions, and virial techniques to achieve a robust nonexistence result in a setting close to the cubic NLS limit.

Abstract

We prove that the linearised operator around any sufficiently small solitary wave of the one-dimensional Zakharov system has no internal mode. This spectral result, along with its proof, is expected to play a role in the study of the asymptotic stability of solitary waves.

Non-existence of internal mode for small solitary waves of the 1D Zakharov system

TL;DR

The paper proves that small solitary waves of the 1D Zakharov system admit no internal modes in their linearization. By linking the Zakharov system to its NLS limit, the authors perform a detailed spectral analysis using almost-orthogonality, a transformed problem via the factorisation , and localized virial arguments. The main result shows that any time-periodic solution of the linearized system must lie in the kernel corresponding to and , with all other components vanishing for small , thereby excluding internal modes and resonances. This spectral property is expected to facilitate the study of asymptotic stability of solitary waves in this model. The work combines coercivity estimates, strategic decompositions, and virial techniques to achieve a robust nonexistence result in a setting close to the cubic NLS limit.

Abstract

We prove that the linearised operator around any sufficiently small solitary wave of the one-dimensional Zakharov system has no internal mode. This spectral result, along with its proof, is expected to play a role in the study of the asymptotic stability of solitary waves.

Paper Structure

This paper contains 11 sections, 4 theorems, 152 equations.

Table of Contents

  1. Introduction and main result
  2. Basic spectral properties
  3. Proof of non-existence of internal mode
  4. Almost orthogonality and resolution of a subsystem
  5. The eigenvalue zero case
  6. Additional almost orthogonality relations
  7. Uniform upper bound on the eigenvalue
  8. Uniform lower bound on the eigenvalue
  9. The transformed problem
  10. Virial arguments on the transformed problem
  11. Non-existence of resonance

Key Result

Theorem 1

Let $\beta \in (-1 \, , 1)$. If $\omega >0$ is sufficiently small then $(U_1 , U_2 , N , V) \in C^0 ( \mathbb{R} , H^1 ( \mathbb{R} )^2 \times L^2 ( \mathbb{R} )^2 )$ is a time-periodic solution of the system LinZ if and only if there exist $a_1,a_2 \in \mathbb{R}$ such that $U_1(s,y) = a_1 Q'(y)$,

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • proof
  • Lemma 2
  • proof