On multi-solitons for the energy-critical wave equation in dimension 5

TL;DR

This paper constructs multi-soliton solutions for the energy-critical nonlinear wave equation in five dimensions, demonstrating the existence of solutions that asymptotically resemble a sum of Lorentz-transformed standing solitons with small velocities.

Contribution

It introduces a method to explicitly construct multi-soliton solutions for the 5D energy-critical wave equation with distinct small velocities.

Findings

Existence of multi-soliton solutions in 5D energy-critical wave equation.

Solutions asymptotically approach a sum of Lorentz-transformed solitons.

Explicit smallness condition on soliton velocities.

Abstract

In this paper, we construct $K$-solitons of the focusing energy-critical nonlinear wave equation in five-dimensional space, i.e. solutions $u$ of the equation such that \begin{equation*} \|\nabla_{t,x}u(t)-\nabla_{t,x}\big(\sum_{k=1}^{K}W_{k}(t)\big)\|_{L^{2}}\to 0\quad \mathrm{as}\ t\to \infty, \end{equation*} where for any $k\in \{1,\dots,K\}$, $W_{k}$ is Lorentz transform of the explicit standing soliton $W(x)=(1+|x|^{2}/15)^{-3/2}$, with any speed $\boldsymbol{\ell}_{k}\in \mathbb{R}^{5}$ ,$|\boldsymbol{\ell}_{k}|<1$ ($\boldsymbol{\ell}_{k'}\ne \boldsymbol{\ell}_{k}$ for $k'\ne k$) satisfying an explicit smallness condition.