Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability of multi-solitons for the Benjamin-Ono equation

Yang Lan, Zhong Wang

TL;DR

This work proves the dynamical stability of arbitrary $m$-solitons for the Benjamin-Ono equation by combining a variational Lyapunov framework with a detailed spectral theory for nonlocal, Hilbert-transform-based operators. The authors establish that the $m$-soliton profile $U^{(m)}$ is a constrained minimizer of $H_{m+1}$ under $m$ conservation constraints, and derive the Hessian $\mathcal{L}_m$’s inertia via a novel use of the BO recursion structure and squared eigenfunctions, despite the lack of an explicit recursion operator. They show that the linearized dynamics around $m$-solitons is isoinertial in time and that as $t\to\infty$, the spectrum of $\mathcal{L}_m$ converges to the union of the spectra of the single-soliton Hessians $L_{m,j}$, yielding exactly $\big[\frac{m+1}{2}\big]$ negative eigenvalues and an $m$-dimensional kernel. Consequently, small perturbations in $H^{\frac{m}{2}}(\mathbb{R})$ remain close to the $m$-soliton manifold modulo translations, establishing orbital stability in $H^{\frac{m}{2}}$ and, in particular, providing a unified approach that includes the known double-soliton result as a special case.

Abstract

This paper is concerned with the dynamical stability of the $m$-solitons of the Benjamin-Ono (BO) equation. This extends the work of Neves and Lopes [41], which was restricted to $m=2$ the double solitons case. By constructing a suitable Lyapunov functional, it is found that the multi-solitons are non-isolated constrained minimizers satisfying a suitable variational nonlocal elliptic equation. The stability issue is reduced to the spectral analysis of higher order nonlocal operators consist of the Hilbert transform. Such operators are isoinertial and the negative eigenvalues of which are fully classified. Our approach in the spectral analysis consists of an invariance for the multi-solitons and new operator identities motivated by the bi-Hamiltonian structure of the BO equation. Since the BO equation is more likely a two dimensional integrable system, its recursion operator is not explicit which makes our analysis more involved. The key ingredient in the spectral analysis is to employ the completeness in $L^2$ of the squared eigenfunctions of the eigenvalue problem for the BO equation. It is demonstrated here that orbital stability of soliton in $H^{\frac{1}{2}}$ implies that all $m$-solitons are dynamically stable in $H^{\frac{m}{2}}$.

Stability of multi-solitons for the Benjamin-Ono equation

TL;DR

This work proves the dynamical stability of arbitrary -solitons for the Benjamin-Ono equation by combining a variational Lyapunov framework with a detailed spectral theory for nonlocal, Hilbert-transform-based operators. The authors establish that the -soliton profile is a constrained minimizer of under conservation constraints, and derive the Hessian ’s inertia via a novel use of the BO recursion structure and squared eigenfunctions, despite the lack of an explicit recursion operator. They show that the linearized dynamics around -solitons is isoinertial in time and that as , the spectrum of converges to the union of the spectra of the single-soliton Hessians , yielding exactly negative eigenvalues and an -dimensional kernel. Consequently, small perturbations in remain close to the -soliton manifold modulo translations, establishing orbital stability in and, in particular, providing a unified approach that includes the known double-soliton result as a special case.

Abstract

This paper is concerned with the dynamical stability of the -solitons of the Benjamin-Ono (BO) equation. This extends the work of Neves and Lopes [41], which was restricted to the double solitons case. By constructing a suitable Lyapunov functional, it is found that the multi-solitons are non-isolated constrained minimizers satisfying a suitable variational nonlocal elliptic equation. The stability issue is reduced to the spectral analysis of higher order nonlocal operators consist of the Hilbert transform. Such operators are isoinertial and the negative eigenvalues of which are fully classified. Our approach in the spectral analysis consists of an invariance for the multi-solitons and new operator identities motivated by the bi-Hamiltonian structure of the BO equation. Since the BO equation is more likely a two dimensional integrable system, its recursion operator is not explicit which makes our analysis more involved. The key ingredient in the spectral analysis is to employ the completeness in of the squared eigenfunctions of the eigenvalue problem for the BO equation. It is demonstrated here that orbital stability of soliton in implies that all -solitons are dynamically stable in .
Paper Structure (12 sections, 18 theorems, 162 equations)

This paper contains 12 sections, 18 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Background results for the BO equation
  3. Some properties of the Hilbert transform
  4. Eigenvalue problem and conservation laws
  5. The Euler-Lagrange equation of the $m$-solitons profile
  6. Bi-Hamiltonian formation of \ref{['eq: BO']}
  7. Spectral Analysis
  8. The spectrum of $L_{1,c}$
  9. The spectrum of the recursion operator around the BO one soliton
  10. The spectrum of linearized operators $\mathcal{J}L_n$, $L_n\mathcal{J}$ and $L_n$
  11. The spectrum of linearized operator around the BO $m$-solitons
  12. Proof of the main results

Key Result

Theorem 1.1

Given $m\in\mathbb N,$$m\geq1$, a collection of wave speeds ${\mathbf c}=(c_1,\cdot\cdot\cdot,c_m)$ with $0<c_1<\cdot\cdot\cdot<c_m$ and a collection of space transitions $\mathbf x=(x_1,\cdot\cdot\cdot,x_m)\in\mathbb{R}^m$, let $U^{ (m)}(\cdot,\cdot;{\mathbf c},{\mathbf x})$ be the corresponding mu then for any $t\in\mathbb{R}$ the corresponding solution $u$ of eq: BO verifies

Theorems & Definitions (40)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.1
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 30 more