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

TL;DR

This paper constructs multi-soliton solutions for the energy-critical wave equation in five dimensions, addressing the challenge of strong interactions due to slow decay, and extends the results to multiple collinear solitons.

Contribution

It provides the first construction of multi-solitons for the energy-critical wave equation in 5D, including cases with more than two solitons and arbitrary speeds.

Findings

Constructed 2-solitons with any speed in 5D energy-critical wave equation.

Extended construction to K-solitons with collinear speeds.

Addressed the challenge of strong soliton interactions due to slow decay.

Abstract

We construct 2-solitons of any speed of the focusing energy-critical nonlinear wave equation in dimension 5. The existence result also holds for the case of K-solitons, for any K >2, assuming that the speeds are collinear. The main difficulty of the construction is the strong interaction between the solitons due to the slow spatial decay of the single soliton. This is in contrast with previous constructions of multi-solitons for other nonlinear models (like generalized KdV and nonlinear Schrodinger equations in energy subcritical cases), where the interactions are exponentially small in time due to the exponential decay of the solitons.