Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.03027 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910814886166528 |
|---|---|
| author | Rybalko, Yan Shepelsky, Dmitry Tian, Shou-Fu |
| author_facet | Rybalko, Yan Shepelsky, Dmitry Tian, Shou-Fu |
| contents | We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation \[ \I q_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \] subject to the step-like initial data: $q(x,0)\to0$ as $x\to-\infty$ and $q(x,0)\simeq Ae^{2\I Bx}$ as $x\to\infty$, where $A>0$ and $B\in\mathbb{R}$. The goal is to study the long-time asymptotic behavior of the solution of this problem assuming that $q(x,0)$ is close, in a certain spectral sense, to the ``step-like'' function $q_{0,R}(x)= \begin{cases} 0, &x\leq R,\\ Ae^{2\I Bx}, &x>R, \end{cases}$ with $R>0$. A special attention is paid to how $B\ne0$ affects the asymptotics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03027 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The integrable nonlocal nonlinear Schrödinger equation with oscillatory boundary conditions: long-time asymptotics Rybalko, Yan Shepelsky, Dmitry Tian, Shou-Fu Analysis of PDEs We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation \[ \I q_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \] subject to the step-like initial data: $q(x,0)\to0$ as $x\to-\infty$ and $q(x,0)\simeq Ae^{2\I Bx}$ as $x\to\infty$, where $A>0$ and $B\in\mathbb{R}$. The goal is to study the long-time asymptotic behavior of the solution of this problem assuming that $q(x,0)$ is close, in a certain spectral sense, to the ``step-like'' function $q_{0,R}(x)= \begin{cases} 0, &x\leq R,\\ Ae^{2\I Bx}, &x>R, \end{cases}$ with $R>0$. A special attention is paid to how $B\ne0$ affects the asymptotics. |
| title | The integrable nonlocal nonlinear Schrödinger equation with oscillatory boundary conditions: long-time asymptotics |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.03027 |