Existence, Uniqueness and Asymptotic Dynamics of Nonlinear Schrödinger Equations With Quasi-Periodic Initial Data: II. The Derivative NLS
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912780855017472 |
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| author | Damanik, David Li, Yong Xu, Fei |
| author_facet | Damanik, David Li, Yong Xu, Fei |
| contents | This is the second part of a two-paper series studying the nonlinear Schrödinger equation with quasi-periodic initial data. In this paper, we focus on the quasi-periodic Cauchy problem for the derivative nonlinear Schrödinger equation. Under the assumption that the Fourier coefficients of the initial data obey an exponential upper bound, we establish local existence of a solution that retains quasi-periodicity in space with a slightly weaker Fourier decay. Moreover, the solution is shown to be unique within this class of quasi-periodic functions. Also, we prove that, for the derivative nonlinear Schrödinger equation in a weakly nonlinear setting, within the time scale, as the small parameter of nonlinearity tends to zero, the nonlinear solution converges asymptotically to the linear solution in the sense of both sup-norm and analytic Sobolev-norm.
The proof proceeds via a consideration of an associated infinite system of coupled ordinary differential equations for the Fourier coefficients and an explicit combinatorial analysis for the Picard iteration with the help of Feynman diagrams and the power of $\ast^{[\cdot]}$ labelling the complex conjugate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence, Uniqueness and Asymptotic Dynamics of Nonlinear Schrödinger Equations With Quasi-Periodic Initial Data: II. The Derivative NLS Damanik, David Li, Yong Xu, Fei Analysis of PDEs Mathematical Physics Classical Analysis and ODEs This is the second part of a two-paper series studying the nonlinear Schrödinger equation with quasi-periodic initial data. In this paper, we focus on the quasi-periodic Cauchy problem for the derivative nonlinear Schrödinger equation. Under the assumption that the Fourier coefficients of the initial data obey an exponential upper bound, we establish local existence of a solution that retains quasi-periodicity in space with a slightly weaker Fourier decay. Moreover, the solution is shown to be unique within this class of quasi-periodic functions. Also, we prove that, for the derivative nonlinear Schrödinger equation in a weakly nonlinear setting, within the time scale, as the small parameter of nonlinearity tends to zero, the nonlinear solution converges asymptotically to the linear solution in the sense of both sup-norm and analytic Sobolev-norm. The proof proceeds via a consideration of an associated infinite system of coupled ordinary differential equations for the Fourier coefficients and an explicit combinatorial analysis for the Picard iteration with the help of Feynman diagrams and the power of $\ast^{[\cdot]}$ labelling the complex conjugate. |
| title | Existence, Uniqueness and Asymptotic Dynamics of Nonlinear Schrödinger Equations With Quasi-Periodic Initial Data: II. The Derivative NLS |
| topic | Analysis of PDEs Mathematical Physics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2406.02512 |