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

Asma Azaiez, Jacek Jendrej, Hatem Zaag

TL;DR

This work analyzes blow-up for the complex-valued semilinear wave equation $u_{tt}=u_{xx}+|u|^{p-1}u$ in one space dimension and shows that near characteristic blow-up points the solution, up to a complex rotation, decomposes into a finite sum of decoupled solitons. It introduces a complex-valued first-order Toda system that governs the solitons’ centers and phases, and develops a modulation framework with spectral analysis of the linearized operators to derive a finite-dimensional ODE system for modulation parameters. The authors prove asymptotically real-valued behavior, isolate characteristic points, and obtain sharp slope estimates for the blow-up graph, by reducing to the real-valued theory in the asymptotic regime. Overall, the work extends the real-valued blow-up theory to the complex setting, revealing a novel complex Toda dynamics governing multi-soliton interactions and providing precise asymptotics for the blow-up structure and its graph.

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.

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

TL;DR

This work analyzes blow-up for the complex-valued semilinear wave equation in one space dimension and shows that near characteristic blow-up points the solution, up to a complex rotation, decomposes into a finite sum of decoupled solitons. It introduces a complex-valued first-order Toda system that governs the solitons’ centers and phases, and develops a modulation framework with spectral analysis of the linearized operators to derive a finite-dimensional ODE system for modulation parameters. The authors prove asymptotically real-valued behavior, isolate characteristic points, and obtain sharp slope estimates for the blow-up graph, by reducing to the real-valued theory in the asymptotic regime. Overall, the work extends the real-valued blow-up theory to the complex setting, revealing a novel complex Toda dynamics governing multi-soliton interactions and providing precise asymptotics for the blow-up structure and its graph.

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.
Paper Structure (22 sections, 23 theorems, 266 equations, 2 figures)

This paper contains 22 sections, 23 theorems, 266 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Approaching a sum of decoupled solitons
  3. Decomposition into a sum of decoupled solitons
  4. Dynamics of the equation around the sum of solitons
  5. The linearized operator around one single soliton
  6. Spectral properties of $\check L_{d}$.
  7. Spectral properties of $\tilde{L}_{d}$.
  8. Decomposition and modulation of the solution
  9. Dynamics of the solution near the multi-soliton
  10. Projection of the PDE on the different components
  11. Differential inequalities on the different components
  12. Refined equations on the modulation parameters
  13. Asymptotically real-valued behavior in the characteristic case
  14. Asymptotic behavior of the modulation parameters
  15. Preliminaries
  16. ...and 7 more sections

Key Result

Theorem 1

If $x_0\in {\cal S}$, then and $E(w_{x_0}(s), \partial_s w_{x_0}(s))\rightarrow \bar{k}(x_0) E(\kappa_0,0)$ as $s\rightarrow \infty$, with $d_j(s)=-\tanh (\bar{\zeta}_j(s)+\bar{\zeta}(x_0)) \in (-1,1)$ for $j=1,...,\bar{k}(x_0)$, for some

Figures (2)

  • Figure 1: $x_0$ is a non-characteristic point ($N=1$).
  • Figure 2: $x_0$ is a characteristic point ($N=1$).

Theorems & Definitions (60)

  • Theorem 1: Description of the set of characteristic points and the blow-up behavior at characteristic points
  • Remark
  • Theorem 2: Regularity of the blow-up graph near characteristic points
  • Remark
  • Proposition 2.1: Decomposition of the solution into a sum of decoupled solitons
  • Remark
  • proof
  • Lemma 2.2: A blow-up criterion for equation \ref{['eqw']}
  • proof
  • Lemma 2.3: Characterization of all stationary solutions of equation \ref{['eqw']} in the energy space $\mathcal{H}_0$ \ref{['10']}
  • ...and 50 more