Two remarks on solutions of Gross-Pitaevskii equations on Zhidkov spaces

WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity

We consider the semi-classical limit for the Gross-Pitaevskii equation. In order to consider non-trivial boundary conditions at infinity, we work in Zhidkov spaces rather than in Sobolev spaces. For the usual cubic nonlinearity, we obtain a point-wise description of the wave function as the Planck constant goes to zero, so long as no singularity appears in the limit system. For a cubic-quintic nonlinearity, we show that working with analytic data may be necessary and sufficient to obtain a similar result.

全空间上格罗斯-皮塔耶夫斯基方程分裂方法及涡旋核化

Splitting methods for the Gross-Pitaevskii equation on the full space and vortex nucleation

本文提出并验证了全空间上格罗斯-皮塔耶夫斯基方程的分裂方法，探讨了非零边界条件下涡旋核化的实验相关设置。

Semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition

In this paper, we justify the semiclassical limit of Gross-Pitaevskii equation with Dirichlet boundary condition on the 3-D upper space under the assumption that the leading order terms to both initial amplitude and initial phase function are sufficiently small in some high enough Sobolev norms. We remark that the main difficulty of the proof lies in the fact that the boundary layer appears in the leading order terms of the amplitude functions and the gradient of the phase functions to the WKB expansions of the solutions. In particular, we partially solved the open question proposed in \cite{C…

围绕度数为1的涡旋线性化的Gross-Pitaevskii演化

On the Gross-Pitaevskii evolution linearized around the degree-one vortex

本文研究了在等变对称性下围绕度为1的Ginzburg-Landau涡旋线性化的Gross-Pitaevskii演化，确定了线性化算子的谱，揭示了与零能量共振相关的$L^2$范数增长现象，并构建了小能量下的扭曲傅里叶变换。

Gross-Pitaevskii方程$N$-孤子的有效动力学

Effective dynamics for $N$-solitons of the Gross-Pitaevskii equation

研究在缓慢变化外场中多个孤子的有效动力学，证明将哈密顿量限制在$N$-孤子流形上得到的有效动力学与实际孤子动力学误差为$h^2$量级，并推广到多孤子相互作用。

Solvable cubic resonant systems

Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian systems with cubic nonlinearities in the equations of motion, these resonant systems admit special analytic solutions, which furthermore display periodic perfect energy returns to the initial configurations. Here, we construct a very large class of resonant systems that shares these properties that have so far been seen in specific examples emerging from a fe…