Dispersive equations with multiple potentials

Abstract

In this thesis, we study dispersive equations with several moving,potentials, a.k.a. charge transfer Hamiltonians. We mainly focus on,two models: the Sch\odinger equation and the wave equation. We prove,linear estimates and analyze nonlinear models based on them.,In Chapter 1, we briefly survey the historical backgrounds and motivation,for the main results in this thesis. ,Then, in Chapter 2, we prove Strichartz estimates for scattering states,of the scalar charge transfer models in $\mathbb{R}^{3}$. More precisely,,we study the time-dependent charge transfer Hamiltonian ,\begin{equation},H(t)=-\frac{1}{2}\Delta+\sum_{j=1}^{m}V_{j}(x-\vec{v}_{j}t),\end{equation},with rapidly decaying smooth potentials $V_{j}(x)$, say, exponentially,decaying and a set of mutually non-parallel constant velocities $\vec{v}_{j}$.,We prove Strichartz estimates for the evolution ,\begin{equation},\frac{1}{i}\partial_{t}\psi+H(t)\psi=0.,\end{equation},Based on the idea of the proof of Strichartz estimates which follows,\cite{CM,RSS}, we also show the energy of the whole evolution is,bounded independent of time without using the phase space method,,for example, in \cite{Graf}. One can easily generalize our argument,to $\mathbb{R}^{n}$ for $n\geq3$. We also discuss the extension,of above results to matrix charge transfer models in $\mathbb{R}^{3}$. ,Next, in Chapter 3, we prove Strichartz estimates (both regular and,reversed) for a scattering state to the wave equation with a charge,transfer Hamiltonian in $\mathbb{R}^{3}$: ,\begin{equation},\partial_{tt}u-\Delta u+\sum_{j=1}^{m}V_{j}\left(x-\vec{v}_{j}t\right)u=0,\end{equation},where $V_{j}(x)$'s are rapidly decaying smooth potentials and $\left\{ \vec{v}_{j}\right\} $,is a set of distinct constant velocities such that ,\begin{equation},\left|\vec{v}_{i}\right|<1,\,1\leq i\leq m.,\end{equation},The energy estimate and the local energy decay of a scattering state,are also established. In order to study nonlinear multisoltion systems,,we will present the inhomogeneous generalizations of Strichartz estimates.,As an application of our results, we show that scattering states indeed,scatter to solutions to the free wave equation. ,In Chapter 4, we study the endpoint reversed Strichartz estimates,along general time-like trajectories for wave equations in $\mathbb{R}^{3}$.,We also discuss some applications of the reversed Strichartz estimates,and the structure formula of wave operators to the wave equation with one,potential. These techniques are useful to analyze the stability problem,of traveling solitons.,In Chapter 5, lastly, we construct multisoliton solutions to the defocusing energy critical,wave equation with potentials in $\mathbb{R}^{3}$,\begin{equation},\partial_{tt}u-\Delta u+\sum_{j=1}^{m}V_{j}\left(x-\vec{v}_{j}t\right)u+u^{5}=0,\end{equation},based on regular,and reversed Strichartz estimates developed in Chapter 3 for wave,equations with charge transfer Hamiltonians. We also show the,asymptotic stability of multisoliton solutions. The multisoliton structures with both stable and unstable solitons are covered. Since each soliton,decays slowly with rate $\frac{1}{\left\langle x\right\rangle }$, the interactions among the solitons are strong. Some reversed Strichartz estimates and local decay estimates for the charge transfer model are established to handle strong interactions.