Constructive proofs of existence and stability of solitary waves in the Whitham and capillary–gravity Whitham equations
Bibliographic record
Abstract
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 ).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".