## V5B4 Selected Topics in PDE and Mathematical Models - Dispersive Equations

#### Wintersemester 2017/2018

**Instructor:**
Dr. Xian Liao

** Lectures: **

**Topics:** We will focus on the mathematical theory of nonlinear Schrödinger equations (NLS)

- Well-posedness issue of (NLS)
- - Local & Global well-posedness in L2 / H1, by use of Strichartz estimates & Sobolev embedding & conservation laws

- Long time behaviour of the solutions of (NLS)
- - Blowup & Scattering, by use of Virial & Morawetz idenities

- Solitary waves of (NLS)
- - Orbital stability, by use of variational description & concentration-compactness

- Conserved energies for one dimensional cubic (NLS)
- - Conserved energies, by use of invariant transmission coefficient

**Prerequisites:**

**References:**

- T. Cazenave: Semilinear Schrödinger equations.
- F. Linares, G. Ponce: Introduction to nonlinear dispersive equations.
- T. Tao: Nonlinear dispersive equations - local and global analysis.
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T.Tao: The theory of nonlinear Schödinger equations.
- H. Koch, D. Tataru: Conserved energies for the cubic NLS in 1-d.

**Oral Exams:**
31.01.2018 & 21.03.2018

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