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 \[\begin{equation}\label{GP}\tag{GP} \mathrm{i}\partial_t q+\partial_{xx} q=2(|q|^2-1)q, \quad t,x\in\mathbb{R}, \quad q=q(t,x):\mathbb{R}\times\mathbb{R}\mapsto \mathbb{C}, \end{equation}\] 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 \[\begin{equation}\label{NLS}\tag{NLS} \mathrm{i}\partial_t \psi+\partial_{xx} \psi=2|\psi|^2\psi, \quad t,x\in\mathbb{R},\quad \psi=\psi(t,x):\mathbb{R}\times\mathbb{R}\mapsto \mathbb{C}.\end{equation}\] 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.

Preprints

  1. , , , , and . Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials. CRC 1173 Preprint 2022/67, Karlsruhe Institute of Technology, November . [bibtex]

  2. , , and . Validity of the Whitham approximation for a complex cubic Klein–Gordon equation. CRC 1173 Preprint 2022/64, Karlsruhe Institute of Technology, November . [bibtex]

  3. and . Eigenvalue analysis of the Lax operator for the one-dimensional cubic nonlinear defocusing Schrödinger equation. CRC 1173 Preprint 2022/30, Karlsruhe Institute of Technology, July . [bibtex]

  4. and . Conserved energies for the one dimensional Gross–Pitaevskii quation: low regularity case. CRC 1173 Preprint 2022/23, Karlsruhe Institute of Technology, April . [bibtex]

Former staff
Name Title Function
Dr. Postdoctoral researcher