This course is a problem-oriented course. It is designed for those students who are interested
in the interaction of mathematics and science,and plan to do research in this direction.
This course (Special Topics on Applied Mathematics) will focus on Nonlinear
Schr\"odinger Equations and Related Topics, such as Bose-Einstein condensates and Super-conductivities. Mathematical analysis and computational methods will be emphasized.
I will first give some introductory talks including introduction to quantum mechanics, introduction to Bose-Einstein condensates and Superductivity.
Then students will give talks on specific topics including
\I Exact solution of linear Schr\"odinger equations in one dimension.
\I Hydrogen atoms
\I Spins
\I Identical bosons
\I Derivation of Gross-Pitaevskii equation
\I Ground states of the Gross-Pitaevskii equations
\I Dynamics of the Gross-Pitaevskii equations
\I Dimension reduction issues
\I Normalized Gradient Flow method for computing ground states
\I Continuation method for computing ground states
\I Time splitting methods for compuing dynamics of Gross-Pitaevskii equation
\I Error estimates
\I WKB method and Hamilton-Jacobi theory
\I Semi-classical scaling and limits
\I Dipolar BECs
\I Multi-component BECs
\I Spinor BECs
\I Ginzburg-Landau theory for Superconductivity