Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{ö}dinger equation
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arXiv
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| Format: | Preprint |
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2022
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| author | Nakamura, Gen sarkar, Tanmay Vashisth, Manmohan |
| author_facet | Nakamura, Gen sarkar, Tanmay Vashisth, Manmohan |
| contents | We study an inverse problem related to the dynamical Schr{ö}dinger equation in a bounded domain of $\Rb^n,n\geq 2$. Since the concerned non-linear Schrödinger equation possesses a trivial solution, we linearize the equation around the trivial solution. Demonstrating the well-posedness of the direct problem under appropriate conditions on initial and boundary data, it is observed that the solution admits $\eps$-expansion. By taking into account the fact that the terms $\Oh(|\nabla u(t,x)|^3)$ are negligible in this context, we shall reconstruct the time-dependent coefficients such as electric potential and vector-valued function associated with quadratic nonlinearity from the knowledge of input-output map using the geometric optics solution and Fourier inversion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_09809 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{ö}dinger equation Nakamura, Gen sarkar, Tanmay Vashisth, Manmohan Analysis of PDEs Mathematical Physics We study an inverse problem related to the dynamical Schr{ö}dinger equation in a bounded domain of $\Rb^n,n\geq 2$. Since the concerned non-linear Schrödinger equation possesses a trivial solution, we linearize the equation around the trivial solution. Demonstrating the well-posedness of the direct problem under appropriate conditions on initial and boundary data, it is observed that the solution admits $\eps$-expansion. By taking into account the fact that the terms $\Oh(|\nabla u(t,x)|^3)$ are negligible in this context, we shall reconstruct the time-dependent coefficients such as electric potential and vector-valued function associated with quadratic nonlinearity from the knowledge of input-output map using the geometric optics solution and Fourier inversion. |
| title | Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{ö}dinger equation |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2201.09809 |