Validity of Whitham's modulation equations for dissipative systems with a conservation law -- Phase dynamics in a generalized Ginzburg-Landau system --
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866914954896998400 |
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| author | Haas, Tobias de Rijk, Björn Schneider, Guido |
| author_facet | Haas, Tobias de Rijk, Björn Schneider, Guido |
| contents | It is well-established that Whitham's modulation equations approximate the dynamics of slowly varying periodic wave trains in dispersive systems. We are interested in its validity in dissipative systems with a conservation law. The prototype example for such a system is the generalized Ginzburg-Landau system that arises as a universal amplitude system for the description of a Turing-Hopf bifurcation in spatially extended pattern-forming systems with neutrally stable long modes. In this paper we prove rigorous error estimates between the approximation obtained through Whitham's modulation equations and true solutions to this Ginzburg-Landau system. Our proof relies on analytic smoothing, Cauchy-Kovalevskaya theory, energy estimates in Gevrey spaces, and a local decomposition in Fourier space, which separates center from stable modes and uncovers a (semi)derivative in front of the relevant nonlinear terms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_04993 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Validity of Whitham's modulation equations for dissipative systems with a conservation law -- Phase dynamics in a generalized Ginzburg-Landau system -- Haas, Tobias de Rijk, Björn Schneider, Guido Analysis of PDEs Pattern Formation and Solitons 35A35, 35B10, 35A10 It is well-established that Whitham's modulation equations approximate the dynamics of slowly varying periodic wave trains in dispersive systems. We are interested in its validity in dissipative systems with a conservation law. The prototype example for such a system is the generalized Ginzburg-Landau system that arises as a universal amplitude system for the description of a Turing-Hopf bifurcation in spatially extended pattern-forming systems with neutrally stable long modes. In this paper we prove rigorous error estimates between the approximation obtained through Whitham's modulation equations and true solutions to this Ginzburg-Landau system. Our proof relies on analytic smoothing, Cauchy-Kovalevskaya theory, energy estimates in Gevrey spaces, and a local decomposition in Fourier space, which separates center from stable modes and uncovers a (semi)derivative in front of the relevant nonlinear terms. |
| title | Validity of Whitham's modulation equations for dissipative systems with a conservation law -- Phase dynamics in a generalized Ginzburg-Landau system -- |
| topic | Analysis of PDEs Pattern Formation and Solitons 35A35, 35B10, 35A10 |
| url | https://arxiv.org/abs/2101.04993 |