Construction of multi-soliton solutions for the energy critical wave equation in dimension 3

TL;DR

This paper develops a method to construct and refine multi-soliton solutions for the energy-critical wave equation in three dimensions, accounting for strong interactions and slow decay of solitons.

Contribution

It introduces an algorithmic procedure to build approximate multi-soliton solutions with controlled errors that can be corrected to exact solutions.

Findings

Constructed approximate solutions converging to superpositions of solitons.

Achieved solutions with no outgoing radiation.

Error decay rate of solutions can be made arbitrarily fast.

Abstract

We study the energy-critical wave equation in three dimensions, focusing on its ground state soliton, denoted by $W$. Using the Poincar\'e symmetry inherent in the equation, boosting $W$ along any timelike geodesic yields another solution. The slow decay behavior of $W$, $W\sim r^{-1}$, indicates a strong interaction among potential multi-soliton solutions. In this paper, for arbitrary $N\geq0$, we provide an algorithmic procedure to construct approximate solutions to the energy critical wave equation that: (1) converge to a superposition of solitons, (2) have no outgoing radiation, (3) their error to solve the equation decays like $(t-r)^{-N}$. Then, we show that this approximate solution can be corrected to a real solution.