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

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 $\tilde{u}$ 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 $\tilde{u}$, 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 \cite{julia_cadiot}.