2024/03/27 by Matthieu Cadiot, Cadiot, Matthieu · 4 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Gas Dynamics and Kinetic Theory #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2403.18718
openalex publication_date 2024/03/27 · openalex created_date 2024/03/29 · openalex updated_date 2026/07/28
In this manuscript, we present a method to prove constructively the existence and spectral stability of solitary waves in both the Whitham and the capillary-gravity Whitham equations. By employing Fourier series analysis and computer-aided techniques, we successfully approximate the Fourier multiplier operator in this equation, allowing the construction of an approximate inverse for the linearization around an approximate solution u0. Then, using a Newton-Kantorovich approach, we provide a sufficient condition under which the existence of a unique solitary wave u in a ball centered at u0 is obtained. The verification of such a condition is established combining analytic techniques and rigorous numerical computations. Moreover, we derive a methodology to control the spectrum of the linearization around u, enabling the study of spectral stability of the solution. As an illustration, we provide a (constructive) computer-assisted proof of existence of stable solitary waves in both the case with capillary effects (T>0) and without capillary effects (T=0). Moreover, we provide an existence proof for a branch of solitary waves in the case T=0 via a rigorous continuation in the wave velocity. The methodology presented in this paper can be generalized and provides a new approach for addressing the existence and spectral stability of solitary waves in nonlocal nonlinear equations. All computer-assisted proofs, including the requisite codes, are accessible on GitHub at \citejuliacadiot.