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Title: Well-posedness of the one-dimensional derivative nonlinear Schrödinger equation
Author: Moşincat, Răzvan Octavian
ISNI:       0000 0004 7654 6974
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2018
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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.
Supervisor: Oh, Tadahiro ; Pocovnicu, Oana Sponsor: Engineering and Physical Sciences Research Council (EPSRC)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: nonlinear Schro¨dinger equations ; local well-posedness ; DNLS