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

2024/09/09 by István Kádár, Kadar, Istvan
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2409.05267

openalex publication_date 2024/09/09 · openalex created_date 2024/10/22 · openalex updated_date 2026/07/28

Abstract

We study the energy-critical wave equation in three dimensions, focusing on its ground state soliton, denoted by W. Using the Poincaré symmetry inherent in the equation, boosting W along any timelike geodesic yields another solution. The slow decay behavior of W, W∼ r^-1, indicates a strong interaction among potential multi-soliton solutions. In this paper, for arbitrary N≥0, 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.

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