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Thesis
Peer Reviewed

Perturbations of tides and traveling waves for the Korteweg–de Vries equation

UCLA Electronic Theses and Dissertations (2023)

This work explores the existence and behavior of solutions to the Korteweg–de Vries equation on the line for large perturbations of certain classical solutions. First, we show that given a suitable solution $V(t,x)$, KdV is globally well-posed for initial data $u(0,x) \in V (0,x) + H^{-1}(\mathbb{R})$. Our conditions on $V$ do include regularity but do not impose any assumptions on spatial asymptotics. In particular, we show that smooth periodic and step-like profiles $V(0,x)$ satisfy our hypotheses.

Our second main objective is to prove a variational characterization of KdV multisolitons. Maddocks and Sachs used that $n$-solitons are local constrained minimizers of the polynomial conserved quantities in order to prove that $n$-solitons are orbitally stable in $H^n(\mathbb{R})$. We show that multisolitons are the unique global constrained minimizers for this problem. We then use this characterization to provide a new proof of the Maddocks–Sachs orbital stability result via concentration compactness.

Cover page: Perturbations of tides and traveling waves for the Korteweg–de Vries equation

Thesis
Peer Reviewed

Benjamin–Ono at Low Regularity: An Integrability Approach

UCLA Electronic Theses and Dissertations (2021)

We employ the integrable structure of the Benjamin--Ono equation in order to study its rough solutions. For rough data, our most useful tools are the Lax pair formalism and, as in the inverse scattering transform, the structural information embedded in solutions to the scattering equation. Using these, we prove that Sobolev norms are conserved and locally smoothed for rough initial data. Using the integrable structure, we construct a Hamiltonian that usefully approximates the Benjamin--Ono Hamiltonian. With this we may provide a short new proof that the Cauchy problem is well-posed in L2.

Cover page: Benjamin–Ono at Low Regularity: An Integrability Approach

Thesis
Peer Reviewed

Strichartz estimates for the Schrödinger flow on compact symmetric spaces

UCLA Electronic Theses and Dissertations (2018)

This thesis studies scaling critical Strichartz estimates for the Schrödinger flow on compact symmetric spaces. A general scaling critical Strichartz estimate (with an ε-loss, respectively) is given conditional on a conjectured dispersive estimate (with an ε-loss, respectively) on general compact symmetric spaces. The dispersive estimate is then proved for the special case of connected compact Lie groups. Slightly more generally, for products of connected compact Lie groups and spheres of odd dimension, the dispersive estimate is proved with an ε-loss.

Cover page: Strichartz estimates for the Schrödinger flow on compact symmetric spaces

Thesis
Peer Reviewed

Symplectic non-squeezing for the Korteweg--de Vries flow on the line

UCLA Electronic Theses and Dissertations (2020)

The main goal of this work is to prove a symplectic non-squeezing result for the Korteweg--de Vries (KdV) equation on the line $\R$. This is achieved via a finite-dimensional approximation argument. Our choice of finite-dimensional Hamiltonian system that effectively approximates the KdV flow is inspired by the recent breakthrough in the well-posedness theory of KdV in low regularity spaces \cite{KV18}, relying on its completely integrable structure. The employment of our methods also provides us with a new concise proof of symplectic non-squeezing for the same equation on the circle $\T$, recovering the result of \cite{CKSTT}.

Cover page: Symplectic non-squeezing for the Korteweg--de Vries flow on the line

Thesis
Peer Reviewed

Doubly Critical Semilinear Schrödinger Equations.

UCLA Electronic Theses and Dissertations (2017)

We present several new results regarding the two and three dimensional energy-critical non- linear Schro ̈dinger equation in the presence of a second critical nonlinearity.

Cover page: Doubly Critical Semilinear Schrödinger Equations.

Thesis
Peer Reviewed

Local existence and breakdown of scattering behavior for semilinear Schrödinger equations

UCLA Electronic Theses and Dissertations (2021)

In this thesis, we study the behavior of solutions to some semilinear Schr\"odinger equations at short and long time scales. We first consider the nonlinear Schr\"odinger equations with power-type nonlinearity in three dimensions with periodic boundary conditions. We show that this equation is locally well-posed in critically scaling Sobolev spaces $H^s(\bb{T}^3)$. We then investigate the long-time asymptotic behavior of solutions to NLS in Euclidean space with defocusing, mass-subcritical power-type and Hartree nonlinearities. We discuss the divide between the wealth of results on the scattering theory for these equations in weighted $L^2$ spaces and the paucity of analogous results in $L^2(\bb{R}^d)$. To explain this, we show that the scattering problems for these equations are well-posed in weighted $L^2$ spaces in the sense that the scattering operators attain their natural and maximal regularity. Furthermore, we show that these scattering problems are ill-posed in $L^2$ in the sense that the scattering operators cannot be extended to all of $L^2$ without losing a positive (and, in the case of Hartree, infinite) amount of regularity.

Cover page: Local existence and breakdown of scattering behavior for semilinear Schrödinger equations