Constructive proofs of existence and stability of solitary waves in the Whitham and capillary-gravity Whitham equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913521569103872 |
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| author | Cadiot, Matthieu |
| author_facet | Cadiot, Matthieu |
| contents | 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}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_18718 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructive proofs of existence and stability of solitary waves in the Whitham and capillary-gravity Whitham equations Cadiot, Matthieu Analysis of PDEs Dynamical Systems Functional Analysis 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}. |
| title | Constructive proofs of existence and stability of solitary waves in the Whitham and capillary-gravity Whitham equations |
| topic | Analysis of PDEs Dynamical Systems Functional Analysis |
| url | https://arxiv.org/abs/2403.18718 |