Constructive proofs of existence and stability of solitary waves in the Whitham and capillary–gravity Whitham equations
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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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>u</mml:mi> <mml:mo stretchy="true">~</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>u</mml:mi> <mml:mo stretchy="true">~</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> </mml:math> , 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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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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