Scalar behavior for a complex multi-soliton arising in blow-up for a semilinear wave equation

TL;DR

This paper investigates the blow-up behavior of complex-valued semilinear wave equations in one dimension, revealing that near blow-up points, solutions resemble real-valued multi-solitons with a novel analysis of an associated Toda system.

Contribution

It demonstrates that complex solutions decompose into decoupled solitons similar to the real case, and introduces a new approach to analyze the governing Toda system for phases and positions.

Findings

Solutions decompose into finite sums of solitons with alternating signs.

Near blow-up, solutions behave like real-valued solutions up to a complex rotation.

A new method resolves the complex Toda system governing soliton dynamics.

Abstract

This paper deals with blow-up for the complex-valued semilinear wave equation with power nonlinearity in dimension 1. Up to a rotation of the solution in the complex plane, we show that near a characteristic blow-up point, the solution behaves exactly as in the real-valued case. Namely, up to a rotation in the complex plane, the solution decomposes into a sum of a finite number of decoupled solitons with alternate signs. The main novelty of our proof is a resolution of a complex-valued first order Toda system governing the evolution of the positions and the phases of the solitons.