# 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.