Undergraduate Course: Nonlinear Schrödinger equations (MATH11137)
Course Outline
| School | School of Mathematics |
College | College of Science and Engineering |
| Credit level (Normal year taken) | SCQF Level 11 (Year 5 Undergraduate) |
Availability | Available to all students |
| SCQF Credits | 10 |
ECTS Credits | 5 |
| Summary | This course is an introduction to analytical treatment of dispersive partial differential equations. In particular, the course focuses on the theoretical study of the nonlinear Schrödinger equations (NLS). The students will first learn Fourier transform, relevant function spaces and useful inequalities, and then use them to prove existence of unique solutions to NLS and further study their long time behaviour. The course aims to provide a glimpse of analysis in the theory of PDEs. |
| Course description |
- Review of the following topics: Lebesgue spaces, Hölder, Minkowski (integral) and interpolation inequalities. Fourier transform: Plancherel identity, Hausdorff-Young's inequality. Convolution: Young's inequality, duality of products and convolutions under Fourier transform
- (fractional) Sobolev spaces: Sobolev embedding theorem via Fourier transform, algebra property of Sobolev spaces
- space-time function spaces
- local well-posedness (I): via Sobolev embedding and Banach fixed point theorem
- virial identity, finite-time blowup solutions
- linear solutions: dispersive estimate, Strichartz estimate
- local well-posedness (II): via Strichartz estimate
- conservation laws, global existence
- a glimpse of scattering theory
The main focus is on how to use inequalities and establish estimates. Hence, some inequalities will be given without proofs and some operations such as switching limits and integrals will be performed without rigorous justifications.
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Information for Visiting Students
| Pre-requisites | None |
Course Delivery Information
| Not being delivered |
Learning Outcomes
Students should be able to:
- Explain the concept of well-posedness of an evolution PDE
- Comfortably work on Fourier transforms and relevant estimates
- Describe different function spaces such as Lebegue spaces and Sobolev spaces
- State, prove and use Sobolev embedding theorem
- State and use Strichartz estimates
- Prove conservation of mass, momentum and Hamiltonian
- Feel comfortable in applying inequalities to establish linear and nonlinear estimates
- Prove short time existence of unique solutions to NLS and discuss possible global-in-time behaviour
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Reading List
The following is suggested as references:
F. Linares, Felipe and G. Ponce. Introduction to nonlinear dispersive equations. Universitext. Springer, New York, 2009
Students might also find the following useful for reference:
T. Cazenave. Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
T. Tao. Dispersive PDE: Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006 |
Additional Information
| Graduate Attributes and Skills |
Not entered |
| Keywords | NSE |
Contacts
| Course organiser | Dr Thomas Leinster
Tel: (0131 6)50 5057
Email: |
Course secretary | Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: |
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