Oh, Tadahiro

Pocovnicu, Oana

Moşincat, Răzvan Octavian

Engineering and Physical Sciences Research Council (EPSRC)

2018-11-19T13:05:17Z

2018-11-19T13:05:17Z

2018-11-29

This thesis is concerned with the well-posedness of the one-dimensional derivative non-linear Schrodinger equation (DNLS). In particular, we study the initial-value problem associated to DNLS with low-regularity initial data in two settings: (i) on the torus (namely with the periodic boundary condition) and (ii) on the real line. Our first main goal is to study the global-in-time behaviour of solutions to DNLS in the periodic setting, where global well-posedness is known to hold under a small mass assumption. In Chapter 2, we relax the smallness assumption on the mass and establish global well-posedness of DNLS for smooth initial data. In Chapter 3, we then extend this result for rougher initial data. In particular, we employ the I-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao and show the global well-posedness of the periodic DNLS at the end-point regularity. In the implementation of the I-method, we apply normal form reductions to construct higher order modified energy functionals. In Chapter 4, we turn our attention to the uniqueness of solutions to DNLS on the real line. By using an infinite iteration of normal form reductions introduced by Guo, Kwon, and Oh in the context of one-dimensional cubic NLS on the torus, we construct solutions to DNLS without using any auxiliary function space. As a result, we prove the unconditional uniqueness of solutions to DNLS on the real line in an almost end-point regularity.

http://hdl.handle.net/1842/33244

en

The University of Edinburgh

R. Mosincat and T. Oh, A remark on global well-posedness of the derivative nonlinear Schrodinger equation on the circle, C.R. Acad. Sci. Paris, Ser. I 353 (2015), pp. 837-841

R. Mosincat, Global well-posedness of the derivative nonlinear Schrodinger equation with periodic boundary condition in H 1 2 , J. Differential Equations 263 (2017), 4658-4722.

nonlinear Schrödinger equations

local well-posedness

DNLS

Well-posedness of the one-dimensional derivative nonlinear Schrödinger equation

Thesis or Dissertation

Doctoral

PhD Doctor of Philosophy