Stability of Solitary Waves for the nonlinear Schrödinger Equation
Stability of Solitary Waves for the nonlinear Schrödinger Equation
dc.contributor.advisor | Koch, Herbert | |
dc.contributor.author | Ritschl, Tillman | |
dc.date.accessioned | 2024-07-22T09:16:51Z | |
dc.date.available | 2024-07-22T09:16:51Z | |
dc.date.issued | 22.07.2024 | |
dc.identifier.uri | https://hdl.handle.net/20.500.11811/11701 | |
dc.description.abstract | Consider the one-dimensional focusing nonlinear Schrödinger equation with subcritical/critical exponent.
This thesis examines a question derived from the so-called soliton resolution conjecture. The NLS admits regular solutions called solitons. The soliton resolution conjecture claims that every global solution of the NLS will eventually resolve into a sum of soliton-like solutions and a radiation component which disperses like a linear solution. We consider the related question of 'asymptotic stability'. For initial data close to a soliton, does the solution resolve into a soliton-like solution and radiation? Specifically, we examine the linearisation of the NLS around the soliton. Let L denote the Hamiltonian of the resulting linear equation. We show the following in this thesis. Firstly, we fully characterise the spectrum of L. Apart from several well-known eigenvalues in 0, iL admits a resonance in ±1 for p = 3, a symmetrical pair of eigenvalues ±E in (-1, 1){0} for 3 < p < 5, as well as two additional generalised eigenvalues in 0 for p = 5. Secondly, based on the above characterisation of the spectrum of L, we show the existence of a wave operator for 3 < p < 5, mapping L onto the free Schrödinger operator. This is accomplished by constructing a distorted Fourier transform mapping L onto a multiplication operator. Thirdly, we show that the wave operator acts as a bounded operator from L^q to L^q for every 1 = q = 8. As a consequence, for 3 < p < 5, the linearised equation allows for the same dispersive estimates as the free equation. Lastly, for 3 < p < 5, we show a local smoothing estimate for the linearised equation. Due to the absence of resonances, this local smoothing estimate allows for significantly stronger local decay than the case of the free equation. | en |
dc.language.iso | eng | |
dc.rights | In Copyright | |
dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | |
dc.subject | Soliton | |
dc.subject | Schrödinger | |
dc.subject | asymptotische Stabilität | |
dc.subject | Resonanz | |
dc.subject | Spektrum | |
dc.subject | solitary wave | |
dc.subject | Schrödinger equation | |
dc.subject | asymptotic stability | |
dc.subject | soliton resolution conjecture | |
dc.subject | distorted Fourier transform | |
dc.subject | hypergeometric equation | |
dc.subject | Riemann equation | |
dc.subject | unstable eigenvalue | |
dc.subject | local smoothing | |
dc.subject | resonance | |
dc.subject | scattering | |
dc.subject | spectrum | |
dc.subject.ddc | 510 Mathematik | |
dc.title | Stability of Solitary Waves for the nonlinear Schrödinger Equation | |
dc.type | Dissertation oder Habilitation | |
dc.publisher.name | Universitäts- und Landesbibliothek Bonn | |
dc.publisher.location | Bonn | |
dc.rights.accessRights | openAccess | |
dc.identifier.urn | https://nbn-resolving.org/urn:nbn:de:hbz:5-77234 | |
ulbbn.pubtype | Erstveröffentlichung | |
ulbbnediss.affiliation.name | Rheinische Friedrich-Wilhelms-Universität Bonn | |
ulbbnediss.affiliation.location | Bonn | |
ulbbnediss.thesis.level | Dissertation | |
ulbbnediss.dissID | 7723 | |
ulbbnediss.date.accepted | 16.07.2024 | |
ulbbnediss.institute | Mathematisch-Naturwissenschaftliche Fakultät : Fachgruppe Mathematik / Mathematisches Institut | |
ulbbnediss.fakultaet | Mathematisch-Naturwissenschaftliche Fakultät | |
dc.contributor.coReferee | Donninger, Roland |
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