We consider the 3D Gross-Pitaevskii equation \begin{equation}\nonumber i\partial_t ψ+Δψ+(1-|ψ|^2)ψ=0 \text{ for } ψ:\mathbb{R}\times \mathbb{R}^3 \rightarrow \mathbb{C} \end{equation} and construct traveling waves solutions to this equation. These are solutions of the form $ψ(t,x)=u(x_1,x_2,x_3-Ct)$ with a velocity $C$ of order $\varepsilon|\log\varepsilon|$ for a small parameter $\varepsilon>0$. We build two different types of solutions. For the first type, the functions $u$ have a zero-set (vortex set) close to an union of $n$ helices for $n\geq 2$ and near these helices $u$ has degree 1. For the second type, the functions $u$ have a vortex filament of degree $-1$ near the vertical axis $e_3$ and $n\geq 4$ vortex filaments of degree $+1$ near helices whose axis is $e_3$. In both cases the helices are at a distance of order $1/(\varepsilon\sqrt{|\log \varepsilon|)}$ from the axis and are solutions to the Klein-Majda-Damodaran system, supposed to describe the evolution of nearly parallel vortex filaments in ideal fluids. Analogous solutions have been constructed recently by the authors for the stationary Gross-Pitaevskii equation, namely the Ginzburg-Landau equation. To prove the existence of these solutions we use the Lyapunov-Schmidt method and a subtle separation between even and odd Fourier modes of the error of a suitable approximation.
Planar Gross--Pitaevskii traveling waves at every subsonic speed
University of Macau（澳门大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；The Chinese University of Hong Kong（香港中文大学）
Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, II
在空间维$ N \geq 2$中证明了一类无穷远非零条件下的非线性薛定谔方程（包括Gross-Pitaevskii方程和所谓的“五次”方程）的非平凡有限能量行波解的存在性.我们表明，在固定动量的能量最小化可以使用时，相关的非线性势是非负的，它给出了一组轨道稳定的行波，而在恒定动能的作用最小化可以在所有情况下使用。我们还探讨了由不同方法得到的行波族之间的关系，并证明了小能量行波的不存在性。
Co-rotating nearly parallel helical vortices with small cross-section in 3D incompressible Euler equations
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Beijing Institute of Technology（北京理工大学）
