Solitary waves in a stochastic parametrically forced nonlinear Schrödinger equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917192898969600 |
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| author | Gnann, Manuel V. Westdorp, Rik W. S. van Winden, Joris |
| author_facet | Gnann, Manuel V. Westdorp, Rik W. S. van Winden, Joris |
| contents | We study a parametrically forced nonlinear Schrödinger (PFNLS) equation, driven by multiplicative translation-invariant noise. We show that a solitary wave in the stochastic equation is orbitally stable on a timescale which is exponential in the inverse square of the noise strength. We give explicit expressions for the phase shift and fluctuations around the shifted wave which are accurate to second order in the noise strength. This is done by developing a new perspective on the phase-lag method introduced by Krüger and Stannat. Additionally, we show well-posedness of the equation in the fractional Bessel space $H^{s}$ for any $s \in [0,\infty)$, demonstrating persistence of regularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_04625 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solitary waves in a stochastic parametrically forced nonlinear Schrödinger equation Gnann, Manuel V. Westdorp, Rik W. S. van Winden, Joris Dynamical Systems Analysis of PDEs Probability 37H30, 35C08, 35Q55, 35Q60, 35R60, 60H15 We study a parametrically forced nonlinear Schrödinger (PFNLS) equation, driven by multiplicative translation-invariant noise. We show that a solitary wave in the stochastic equation is orbitally stable on a timescale which is exponential in the inverse square of the noise strength. We give explicit expressions for the phase shift and fluctuations around the shifted wave which are accurate to second order in the noise strength. This is done by developing a new perspective on the phase-lag method introduced by Krüger and Stannat. Additionally, we show well-posedness of the equation in the fractional Bessel space $H^{s}$ for any $s \in [0,\infty)$, demonstrating persistence of regularity. |
| title | Solitary waves in a stochastic parametrically forced nonlinear Schrödinger equation |
| topic | Dynamical Systems Analysis of PDEs Probability 37H30, 35C08, 35Q55, 35Q60, 35R60, 60H15 |
| url | https://arxiv.org/abs/2403.04625 |