Rotating Nonlinear States in Gross-Pitaevskii Equations

In this thesis, we consider a class of nonlinear Schr\"odinger equations with angular momentum rotation term, which is also known as the Gross-Pitaevskii equation in the physics literature. This equation is a well-known model for ultra-cold quantum gases in rotating traps. Our results include two main directions: (1) Approximate solutions known as semiclassical wave packets to weakly nonlinear Schr\"odinger equations with rotation. (2) Existence and dynamical properties of rotating bound state solutions in both mass-subcritical and mass-supercritical regimes. The organization of the thesis is as follows: In Chapter 1, we give an overview of the Gross-Pitaevskii equation and briefly outline our motivations as well as our main results. In Chapter 2, we present some background definitions and lemmas used through the thesis. In Chapter 3, we consider semiclassically scaled, weakly nonlinear Schr\"odinger equations with external confining potentials and angular momentum rotation term. We construct asymptotic solutions in the form of semiclassical wave packets, which are concentrated in both space and momentum around the associated classical Hamiltonian flow. Our results include both the linear and the weakly nonlinear case. In Chapter 4, we prove the existence and orbital stability of a class of bound state solutions to nonlinear Schr\"odinger equations with super-quadratic confining potentials in the mass-subcritical regime. These solutions are given by time-dependent rotation of non radially symmetric solutions, which are obtained via doubly constrained energy minimization problem. One of these constraints is the total mass, while the other is given by the expectation value of the angular momentum around the z-axis. Our approach also yields a new description of the set of ground state solutions subject to a single mass constraint. Finally, Chapter 5 can be viewed as a complementary work of Chapter 4, where we extend our study to the focusing mass-supercritical case. We prove that there exist two non-radially symmetric solutions, one of which is a local minimizer and the other is a mountain pass type critical point of the associated energy, both with prescribed mass and angular momentum. We also establish that the local minimizer is orbitally stable, while the mountain pass type solution is strongly unstable against finite time blow-up.

Author dc:creator

• Xiaoan Shen (17720007)

• Partial Differential Equations
• Mathematical Physics

Statement dc:rights

• In Copyright
• Open Access after 2028-05-01

DOI dc:identifier

https://doi.org/10.25417/uic.32995181.v1

OAI identifier oai:identifier

oai:figshare.com:article/32995181