Construction of excited multi-solitons for the focusing 4D cubic wave equation

TL;DR

This paper proves the existence of multi-soliton solutions for the focusing 4D cubic wave equation, where each soliton is an excited state generated by Lorentz transforms, despite strong interactions in four dimensions.

Contribution

It constructs multi-soliton solutions involving excited states for the 4D cubic wave equation, extending techniques from higher-dimensional and related dispersive equations.

Findings

Existence of multi-solitons with excited states in 4D

Analysis of asymptotic behavior of excited states

Understanding of the linearized operator's kernel

Abstract

Consider the focusing 4D cubic wave equation \[ \partial_{tt}u-\Delta u-u^{3}=0,\quad \mbox{on}\ (t,x)\in [0,\infty)\times \mathbb{R}^{4}.\] The main result states the existence in energy space $\dot{H}^{1}\times L^{2}$ of multi-solitary waves where each traveling wave is generated by Lorentz transform from a specific excited state, with different but collinear Lorentz speeds. The specific excited state is deduced from the non-degenerate sign-changing state constructed in Musso-Wei [34]. The proof is inspired by the techniques developed for the 5D energy-critical wave equation and the nonlinear Klein-Gordon equation in a similar context by Martel-Merle [30] and C\^ote-Martel [6]. The main difficulty originates from the strong interactions between solutions in the 4D case compared to other dispersive and wave-type models. To overcome the difficulty, a sharp understanding of the asymptotic behavior of the excited states involved and of the kernel of their linearized operator is needed.