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Existence of multi-solitons with any parameters for the 5D energy critical wave equation

Yvan Martel, Frank Merle

TL;DR

This work proves the existence of multi-soliton solutions for the energy-critical focusing wave equation in dimension five with arbitrary numbers of solitons and parameter choices, addressing the challenge of algebraic soliton decay through a dimension-aware energy method. A refined approximate solution, together with a modulation and bootstrap framework, yields precise control of soliton interactions and the asymptotic behavior $\nabla u(t) \to \sum_k \nabla W_k^\infty$ as $t\to+\infty$. The authors also extend inelasticity results to general multi-soliton configurations under a non-vanishing interaction condition, employing a channel-of-energy analysis for the radial linear wave equation to show non-purity of the solution backward in time. Together, these results advance the understanding of multi-soliton dynamics, stability, and potential soliton-resolution phenomena in high-dimensional critical wave equations.

Abstract

For the focusing, energy critical wave equation in dimension 5, we construct multi-solitons with any number of solitons, any choice of signs, speeds, scaling parameters and translation parameters. This requires to revisit in depth previous constructions of multi-solitons based on a unidirectional approach, to fully take into account the dimension of the space and the possibility for solitons to move in any direction. Then, as a consequence of this more general construction and of the arguments developed in a previous article, the inelastic nature of any collision of solitons is proved under a non-cancellation assumption on the parameters.

Existence of multi-solitons with any parameters for the 5D energy critical wave equation

TL;DR

This work proves the existence of multi-soliton solutions for the energy-critical focusing wave equation in dimension five with arbitrary numbers of solitons and parameter choices, addressing the challenge of algebraic soliton decay through a dimension-aware energy method. A refined approximate solution, together with a modulation and bootstrap framework, yields precise control of soliton interactions and the asymptotic behavior as . The authors also extend inelasticity results to general multi-soliton configurations under a non-vanishing interaction condition, employing a channel-of-energy analysis for the radial linear wave equation to show non-purity of the solution backward in time. Together, these results advance the understanding of multi-soliton dynamics, stability, and potential soliton-resolution phenomena in high-dimensional critical wave equations.

Abstract

For the focusing, energy critical wave equation in dimension 5, we construct multi-solitons with any number of solitons, any choice of signs, speeds, scaling parameters and translation parameters. This requires to revisit in depth previous constructions of multi-solitons based on a unidirectional approach, to fully take into account the dimension of the space and the possibility for solitons to move in any direction. Then, as a consequence of this more general construction and of the arguments developed in a previous article, the inelastic nature of any collision of solitons is proved under a non-cancellation assumption on the parameters.
Paper Structure (19 sections, 19 theorems, 289 equations)

This paper contains 19 sections, 19 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. Soliton interactions
  3. Non homogeneous linearized equation
  4. Approximate multi-soliton
  5. Modulation
  6. Bootstrap estimates
  7. Energy estimates
  8. Technical estimates
  9. Energy functional
  10. Heuristics
  11. Coercivity of the energy
  12. Variation of the energy
  13. Existence
  14. Bootstrap setting
  15. Closing the bootstrap estimates
  16. ...and 4 more sections

Key Result

Theorem 1

Let $K\geq 2$. For all $k\in \{1,\ldots,K\}$, let $\lambda_k^\infty>0,$${\boldsymbol y}_k^\infty\in \mathbb{R}^5$, $\epsilon_k\in \{\pm1\}$ and ${\boldsymbol\ell}_k\in \mathbb{R}^5$ with $|{\boldsymbol\ell}_k|<1$ such that for any $k\neq m$, ${\boldsymbol\ell}_k\neq {\boldsymbol\ell}_m$. Let Then, there exist $T>0$ and a solution $u$ of eq:NW on $[T,+\infty)$, satisfying

Theorems & Definitions (37)

  • Theorem 1: Multi-solitons with any set of parameters
  • Corollary 1: Inelasticity and dispersion as $t\to-\infty$
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: MaM2
  • Remark 2.5
  • ...and 27 more