Undergraduate Course: Nonlinear Schrodinger 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  NB. This course is delivered *biennially* with the next instance being in 202122. It is anticipated that it would then be delivered every other session thereafter.
In academic year 202223 and later, MATH10101 Metric spaces is recommended for this course.
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, HausdorffYoung'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
 spacetime function spaces
 local wellposedness (I): via Sobolev embedding and Banach fixed point theorem
 virial identity, finitetime blowup solutions
 linear solutions: dispersive estimate, Strichartz estimate
 local wellposedness (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.

Information for Visiting Students
Prerequisites  None 
High Demand Course? 
Yes 
Course Delivery Information
Not being delivered 
Learning Outcomes
Students should be able to:
 Explain the concept of wellposedness 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 globalintime behaviour

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 Tadahiro Oh
Tel: (0131 6)50 5844
Email: hiro.oh@ed.ac.uk 
Course secretary  Mr Martin Delaney
Tel: (0131 6)50 6427
Email: Martin.Delaney@ed.ac.uk 

