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Infinite-order multisoliton solutions to the Benjamin--Ono equation and soliton resolution

Louise Gassot, Patrick Gérard

Abstract

We construct a class of infinite-order multisoliton solutions of the Benjamin-Ono equation on the line, for which the initial data exhibits slow spatial decay. We prove that in the long-time asymptotics, such a solution decouples as an infinite superposition of independent soliton solutions with different velocities and no radiation term.

Infinite-order multisoliton solutions to the Benjamin--Ono equation and soliton resolution

Abstract

We construct a class of infinite-order multisoliton solutions of the Benjamin-Ono equation on the line, for which the initial data exhibits slow spatial decay. We prove that in the long-time asymptotics, such a solution decouples as an infinite superposition of independent soliton solutions with different velocities and no radiation term.
Paper Structure (5 sections, 9 theorems, 74 equations)

This paper contains 5 sections, 9 theorems, 74 equations.

Table of Contents

  1. Introduction and main result
  2. Explicit formula for the Benjamin-Ono equation
  3. Eigenfunctions of the Lax operator
  4. Construction of the infinite-order multisoliton
  5. Soliton resolution for the infinite-order multisoliton

Key Result

Theorem 1.2

The initial data $u_0$ given by eq:u0 under the condition eq:norm belong to $H^s(\mathbb{R})$ for every $s\in \mathbb{R}$. Moreover, there is a sequence $(p_j^{\infty})_{j\geq 1}$ of complex numbers in $\mathbb{C}_+$ with increasing imaginary parts $0<\mathrm{Im}(p_1^\infty)<\mathrm{Im}(p_2^\infty)< Furthermore, for every $t\in \mathbb{R}$, the formula defines a continuous function $u_{\mathrm{so

Theorems & Definitions (19)

  • Definition 1.1: Infinite-order multisoliton
  • Theorem 1.2: Soliton resolution for the infinite-order multisoliton
  • Theorem 2.1: Explicit formula, see Gerard22
  • Proposition 3.1: Lax eigenfunctions belong to the domain of $X^*$
  • proof
  • Corollary 3.2
  • proof
  • Lemma 4.1: Finite $H^s$ norm for $s\geq 0$
  • proof
  • Remark 4.2
  • ...and 9 more