The defocusing nonlinear Schrödinger equation with step-like oscillatory initial data
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911805885906944 |
|---|---|
| author | Fromm, Samuel Lenells, Jonatan Quirchmayr, Ronald |
| author_facet | Fromm, Samuel Lenells, Jonatan Quirchmayr, Ronald |
| contents | We study the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation under the assumption that the solution vanishes as $x \to + \infty$ and approaches an oscillatory plane wave as $x \to -\infty$. We first develop an inverse scattering transform formalism for solutions satisfying such step-like boundary conditions. Using this formalism, we prove that there exists a global solution of the corresponding Cauchy problem and establish a representation for this solution in terms of the solution of a Riemann-Hilbert problem. By performing a steepest descent analysis of this Riemann-Hilbert problem, we identify three asymptotic sectors in the half-plane $t \geq 0$ of the $xt$-plane and derive asymptotic formulas for the solution in each of these sectors. Finally, by restricting the constructed solutions to the half-line $x \geq 0$, we find a class of solutions with asymptotically time-periodic boundary values previously sought for in the context of the NLS half-line problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_03714 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The defocusing nonlinear Schrödinger equation with step-like oscillatory initial data Fromm, Samuel Lenells, Jonatan Quirchmayr, Ronald Analysis of PDEs 35Q55, 41A60, 35Q15 We study the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation under the assumption that the solution vanishes as $x \to + \infty$ and approaches an oscillatory plane wave as $x \to -\infty$. We first develop an inverse scattering transform formalism for solutions satisfying such step-like boundary conditions. Using this formalism, we prove that there exists a global solution of the corresponding Cauchy problem and establish a representation for this solution in terms of the solution of a Riemann-Hilbert problem. By performing a steepest descent analysis of this Riemann-Hilbert problem, we identify three asymptotic sectors in the half-plane $t \geq 0$ of the $xt$-plane and derive asymptotic formulas for the solution in each of these sectors. Finally, by restricting the constructed solutions to the half-line $x \geq 0$, we find a class of solutions with asymptotically time-periodic boundary values previously sought for in the context of the NLS half-line problem. |
| title | The defocusing nonlinear Schrödinger equation with step-like oscillatory initial data |
| topic | Analysis of PDEs 35Q55, 41A60, 35Q15 |
| url | https://arxiv.org/abs/2104.03714 |