Construction of excited multi-solitons for the 5D energy-critical wave equation

TL;DR

This paper constructs multi-soliton solutions with excited states for the 5D energy-critical wave equation, extending previous methods to include collinear speeds and excited states.

Contribution

It introduces a new construction of excited multi-solitons with collinear speeds for the 5D energy-critical wave equation, using energy methods and coercivity properties.

Findings

Existence of excited multi-solitons with collinear speeds.

Extension of previous soliton construction techniques.

Application of energy and coercivity methods to this setting.

Abstract

For the 5D energy-critical wave equation, we construct excited $N$-solitons with collinear speeds, i.e. solutions $u$ of the equation such that \begin{equation*} \lim_{t\to+\infty}\bigg\|\nabla_{t,x}u(t)-\nabla_{t,x}\bigg(\sum_{n=1}^{N}Q_{n}(t)\bigg)\bigg\|_{L^{2}}=0, \end{equation*} where for $n=1,\ldots,N$, $Q_n(t,x)$ is the Lorentz transform of a non-degenerate and sufficiently decaying excited state, each with different but collinear speeds. The existence proof follows the ideas of Martel-Merle and C\^ote-Martel developed for the energy-critical wave and nonlinear Klein-Gordon equations. In particular, we rely on an energy method and on a general coercivity property for the linearized operator.