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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.13038 |
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| _version_ | 1866910749741285376 |
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| author | Keeler, Jack Alberello, Alberto Humphries, Ben Parau, Emilian |
| author_facet | Keeler, Jack Alberello, Alberto Humphries, Ben Parau, Emilian |
| contents | The higher-order nonlinear Schrodinger equation (Dysthe's equation in the context of water-waves) models the time evolution of the slowly modulated amplitude of a wave-packet in dispersive partial differential equations (PDE). These systems, of which water-waves are a canonical example, require the presence of a small-valued ordering parameter so that a multi-scale expansion can be performed. However, often the resulting system itself contains the small-ordering parameter. Thus, these models are difficult to interpret from a formal asymptotics perspective. This paper derives a parameter-free, higher-order evolution equation for a generic infinite-dimensional dispersive PDE with weak linear damping and/or forcing. Instead of focusing on the water-wave problem or another specific problem, our procedure avoids the complicated algebra by placing the PDE in an infinite-dimensional Hilbert space and Taylor expanding with Frechet derivatives. An attractive feature of this procedure is that it can be used in many different physical settings, including water-waves, nonlinear optics and any dispersive system with weak dissipation or forcing. The paper concludes by discussing two specific examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13038 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parameter-free higher-order Schrodinger systems with weak dissipation and forcing Keeler, Jack Alberello, Alberto Humphries, Ben Parau, Emilian Analysis of PDEs Exactly Solvable and Integrable Systems The higher-order nonlinear Schrodinger equation (Dysthe's equation in the context of water-waves) models the time evolution of the slowly modulated amplitude of a wave-packet in dispersive partial differential equations (PDE). These systems, of which water-waves are a canonical example, require the presence of a small-valued ordering parameter so that a multi-scale expansion can be performed. However, often the resulting system itself contains the small-ordering parameter. Thus, these models are difficult to interpret from a formal asymptotics perspective. This paper derives a parameter-free, higher-order evolution equation for a generic infinite-dimensional dispersive PDE with weak linear damping and/or forcing. Instead of focusing on the water-wave problem or another specific problem, our procedure avoids the complicated algebra by placing the PDE in an infinite-dimensional Hilbert space and Taylor expanding with Frechet derivatives. An attractive feature of this procedure is that it can be used in many different physical settings, including water-waves, nonlinear optics and any dispersive system with weak dissipation or forcing. The paper concludes by discussing two specific examples. |
| title | Parameter-free higher-order Schrodinger systems with weak dissipation and forcing |
| topic | Analysis of PDEs Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2412.13038 |