Papers › Constructive proofs of existence and stability of solitary waves in the Whitham and...
Constructive proofs of existence and stability of solitary waves in the Whitham and capillary-gravity Whitham equations
Matthieu Cadiot
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
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 ũ 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 ũ, 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}.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections