Constructive proofs of existence and stability of solitary waves in the Whitham and capillary–gravity Whitham equations

Abstract 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 u 0. 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 u 0 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 (CAP) 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 CAPs, including the requisite codes, are accessible on GitHub at Cadiot (2023 https://github.com/matthieucadiot/WhithamSoliton.jl).