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| Cross-Invariant Sets of the Cubic Nonlinear Schr\"{o}dinger System with Partial Confinement |
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| KeyWord:Bose-Einstein condensates, nonlinear Schr\"{o}dinger system, cross-invariant sets, sharp condition, global existence |
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| Abstract: |
| This paper studies the cubic nonlinear Schr\"{o}dinger system with partial confinement:
\begin{equation*}
\left\{
\begin{split}
&-i\varphi_{t}+(x_{1}^{2}+x_{2}^{2})\varphi=\Delta\varphi+\mu_{1}|\varphi|^{2}\varphi+\beta|\psi|^{2}\varphi,\quad (t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3},\&-i\psi_{t}+(x_{1}^{2}+x_{2}^{2})\psi=\Delta\psi+\mu_{2}|\psi|^{2}\psi+\beta|\varphi|^{2}\psi,\quad (t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3},
\end{split}
\right.
\end{equation*}
which models the Bose-Einstein condensates with multiple states or the propagation of mutually incoherent wave packets in nonlinear optics. The cross-invariant sets of the evolution flow are obtained by constructing the cross-constrained variational problem. Furthermore, the sharp condition for global existence and blowup of the solutions is derived. |
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