## Full metadata

Title

Global behavior of finite energy solutions to the focusing nonlinear Schrödinger Equation in d dimension

Description

Nonlinear dispersive equations model nonlinear waves in a wide range of physical and mathematics contexts. They reinforce or dissipate effects of linear dispersion and nonlinear interactions, and thus, may be of a focusing or defocusing nature. The nonlinear Schrödinger equation or NLS is an example of such equations. It appears as a model in hydrodynamics, nonlinear optics, quantum condensates, heat pulses in solids and various other nonlinear instability phenomena. In mathematics, one of the interests is to look at the wave interaction: waves propagation with different speeds and/or different directions produces either small perturbations comparable with linear behavior, or creates solitary waves, or even leads to singular solutions. This dissertation studies the global behavior of finite energy solutions to the $d$-dimensional focusing NLS equation, $i partial _t u+Delta u+ |u|^{p-1}u=0, $ with initial data $u_0in H^1,; x in Rn$; the nonlinearity power $p$ and the dimension $d$ are chosen so that the scaling index $s=frac{d}{2}-frac{2}{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$ For solutions with $ME[u_0]<1$ ($ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Namely, if the renormalized gradient $g_u$ of a solution $u$ to NLS is initially less than 1, i.e., $g_u(0)<1,$ then the solution exists globally in time and scatters in $H^1$ (approaches some linear Schr"odinger evolution as $ttopminfty$); if the renormalized gradient $g_u(0)>1,$ then the solution exhibits a blowup behavior, that is, either a finite time blowup occurs, or there is a divergence of $H^1$ norm in infinite time. This work generalizes the results for the 3d cubic NLS obtained in a series of papers by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko with the key ingredients, the concentration compactness and localized variance, developed in the context of the energy-critical NLS and Nonlinear Wave equations by Kenig and Merle. One of the difficulties is fractional powers of nonlinearities which are overcome by considering Besov-Strichartz estimates and various fractional differentiation rules.

Date Created

2011

Contributors

- Guevara, Cristi Darley (Author)
- Roudenko, Svetlana (Thesis advisor)
- Castillo_Chavez, Carlos (Committee member)
- Jones, Donald (Committee member)
- Mahalov, Alex (Committee member)
- Suslov, Sergei (Committee member)
- Arizona State University (Publisher)

Topical Subject

Resource Type

Extent

viii, 108 p. : ill

Language

Copyright Statement

In Copyright

Primary Member of

Peer-reviewed

No

Open Access

No

Handle

https://hdl.handle.net/2286/R.I.9026

Statement of Responsibility

by Cristi Darley Guevara

Description Source

Retrieved Sept. 19, 2012

Level of coding

full

System Created

- 2011-08-12 03:51:53

System Modified

- 2021-08-30 01:54:19
- 2 years 3 months ago

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