# Prescribing the center of mass of a multi-soliton solution for a perturbed semilinear wave equation

## TL;DR

This paper constructs finite-time blow-up solutions for a perturbed semilinear wave equation, featuring multiple solitons with prescribed center of mass, and introduces new methods to handle the lack of Lorentz invariance.

## Contribution

It develops a novel approach to prescribe the center of mass in multi-soliton blow-up solutions for perturbed wave equations, extending and simplifying previous unperturbed case methods.

## Key findings

- Constructed solutions with prescribed center of mass.
- Extended methods to perturbed equations lacking Lorentz invariance.
- Simplified the proof for the unperturbed case.

## Abstract

We construct a finite-time blow-up solution for a class of strongly perturbed semilinear wave equation with an isolated characteristic point in one space dimension. Given any integer $k\ge 2$ and $\zeta_0 \in \mathbb{R}$, we construct a blow-up solution with a characteristic point $a$, such that the asymptotic behavior of the solution near $(a,T(a))$ shows a decoupled sum of $k$ solitons with alternate signs, whose centers (in the hyperbolic geometry) have $\zeta_0$ as a center of mass, for all times. Although the result is similar to the unperturbed case in its statement, our method is new. Indeed, our perturbed equation is not invariant under the Lorentz transform, and this requires new ideas. In fact, the main difficulty in this paper is to prescribe the center of mass $\zeta_0 \in \mathbb{R}$. We would like to mention that our method is valid also in the unperturbed case, and simplifies the original proof by C\^ote and Zaag \cite{CZcpam13}, as far as the center of mass prescription is concerned.