# Project A12 • Dynamics of the Gross–Pitaevskii equation

### Principal investigators

 JProf. Dr. Xian Liao (1/2020 - ) Prof. Dr. Guido Schneider (1/2020 - )

### Project summary

The Gross–Pitaevskii equations (especially the nonlinear Schrödinger equations) arise naturally in numerous physical fields, such as Bose-Einstein condensation, deep water waves, nonlinear optics, etc. We will focus on the following one-dimensional Gross–Pitaevskii equation $$$\label{GP}\tag{GP} \mathrm{i}\partial_t q+\partial_{xx} q=2(|q|^2-1)q, \quad t,x\in\mathbb{R}, \quad q=q(t,x):\mathbb{R}\times\mathbb{R}\mapsto \mathbb{C},$$$ where the wave function $q$ satisfies the nonzero boundary condition at infinity: $|q(x)|\rightarrow 1\text{ as }|x|\rightarrow\infty.$ This changes dramatically the dynamical behavior of the wave function compared to the zero boundary condition case: $|\psi(x)|\rightarrow 0\text{ as }|x|\rightarrow\infty$, for solutions $\psi$ of the cubic nonlinear defocusing Schrödinger equation $$$\label{NLS}\tag{NLS} \mathrm{i}\partial_t \psi+\partial_{xx} \psi=2|\psi|^2\psi, \quad t,x\in\mathbb{R},\quad \psi=\psi(t,x):\mathbb{R}\times\mathbb{R}\mapsto \mathbb{C}.$$$ Indeed, the \eqref{GP} equation describes the phenomena of the dark solitons in nonlinear optics, which preserve the initial energy (without scattering) for all times. The \eqref{NLS} equation has been extensively studied in the literature, while the \eqref{GP} equation still remains much less well understood. We propose to study the dynamics of the \eqref{GP} equation, including the stability issues for soliton solutions, the long time dynamics under long-wave weak perturbation, as well as the separation of the internal and interaction dynamics. The solutions of low regularity and with high oscillations will also be taken into account.