Existence, Uniqueness and Asymptotic Dynamics of Nonlinear Schrödinger Equations With Quasi-Periodic Initial Data: I. The Standard NLS
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866911330231910400 |
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| author | Damanik, David Li, Yong Xu, Fei |
| author_facet | Damanik, David Li, Yong Xu, Fei |
| contents | This is the first part of a two-paper series studying nonlinear Schrödinger equations with quasi-periodic initial data. In this paper, we consider the standard nonlinear Schrödinger equation. Under the assumption that the Fourier coefficients of the initial data obey a power-law 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. In addition, for the nonlinear Schrödinger equation with small nonlinearity, within the time scale, as the small parameter of nonlinearity tends to zero, we prove that the nonlinear solution converges asymptotically to the linear solution with respect to both the sup-norm $\|\cdot\|_{L_x^\infty(\mathbb R)}$ and the Sobolev-norm $\|\cdot\|_{H^s_x(\mathbb R)}$.
The proof proceeds via a consideration of an associated infinite system of coupled ordinary differential equations for the Fourier coefficients and a combinatorial analysis of the resulting tree expansion of the coefficients. For this purpose, we introduce a Feynman diagram for the Picard iteration and $\ast^{[\cdot]}$ to denote the complex conjugate label. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19583 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence, Uniqueness and Asymptotic Dynamics of Nonlinear Schrödinger Equations With Quasi-Periodic Initial Data: I. The Standard NLS Damanik, David Li, Yong Xu, Fei Analysis of PDEs Mathematical Physics Classical Analysis and ODEs This is the first part of a two-paper series studying nonlinear Schrödinger equations with quasi-periodic initial data. In this paper, we consider the standard nonlinear Schrödinger equation. Under the assumption that the Fourier coefficients of the initial data obey a power-law 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. In addition, for the nonlinear Schrödinger equation with small nonlinearity, within the time scale, as the small parameter of nonlinearity tends to zero, we prove that the nonlinear solution converges asymptotically to the linear solution with respect to both the sup-norm $\|\cdot\|_{L_x^\infty(\mathbb R)}$ and the Sobolev-norm $\|\cdot\|_{H^s_x(\mathbb R)}$. The proof proceeds via a consideration of an associated infinite system of coupled ordinary differential equations for the Fourier coefficients and a combinatorial analysis of the resulting tree expansion of the coefficients. For this purpose, we introduce a Feynman diagram for the Picard iteration and $\ast^{[\cdot]}$ to denote the complex conjugate label. |
| title | Existence, Uniqueness and Asymptotic Dynamics of Nonlinear Schrödinger Equations With Quasi-Periodic Initial Data: I. The Standard NLS |
| topic | Analysis of PDEs Mathematical Physics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2405.19583 |