Dispersive estimates using scattering theory for matrix Hamiltonian equations
Jeremy L. Marzuola
Discrete & Continuous Dynamical Systems - A 2011, 30(4): 995-1035 doi: 10.3934/dcds.2011.30.995
We develop the techniques of [25] and [11] in order to derive dispersive estimates for a matrix Hamiltonian equation defined by linearizing about a minimal mass soliton solution of a saturated, focussing nonlinear Schrödinger equation

$\i u_t + \Delta u + \beta (|u|^2) u = 0$
$\u(0,x) = u_0 (x),$

in $\mathbb{R}^3$. These results have been seen before, though we present a new approach using scattering theory techniques. In further works, we will numerically and analytically study the existence of a minimal mass soliton, as well as the spectral assumptions made in the analysis presented here.
keywords: Dispersive estimates scattering theory matrix Hamiltonians.
Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations
Jeremy L. Marzuola Michael I. Weinstein
Discrete & Continuous Dynamical Systems - A 2010, 28(4): 1505-1554 doi: 10.3934/dcds.2010.28.1505
Schrödinger / Gross-Pitaevskii equations (NLS/GP) with a focusing (attractive) nonlinear potential and symmetric double well linear potential. NLS/GP plays a central role in the modeling of nonlinear optical and mean-field quantum many-body phenomena. It is known that there is a critical $L^2$ norm (optical power / particle number) at which there is a symmetry breaking bifurcation of the ground state. We study the rich dynamical behavior near the symmetry breaking point. The source of this behavior in the full Hamiltonian PDE is related to the dynamics of a finite-dimensional Hamiltonian reduction. We derive this reduction, analyze a part of its phase space and prove a shadowing theorem on the persistence of solutions, with oscillating mass-transport between wells, on very long, but finite, time scales within the full NLS/GP. The infinite time dynamics for NLS/GP are expected to depart, from the finite dimensional reduction, due to resonant coupling of discrete and continuum / radiation modes.
keywords: Nonlinear Schrödinger equation Double Well Potential.
Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation
Roy H. Goodman Jeremy L. Marzuola Michael I. Weinstein
Discrete & Continuous Dynamical Systems - A 2015, 35(1): 225-246 doi: 10.3934/dcds.2015.35.225
We study the long-time behavior of solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential. NLS/GP governs nearly-monochromatic guided optical beams in weakly coupled waveguides with both linear and nonlinear (Kerr) refractive indices and zero absorption, as well as the behavior of Bose-Einstein condensates. For small $L^2$ norm (low power), the solution executes beating oscillations between the two wells. There is a power threshold at which a symmetry breaking bifurcation occurs. The set of guided mode solutions splits into two families of solutions. One type of solution is concentrated in either well of the potential, but not both. Solutions in the second family undergo tunneling oscillations between the two wells. A finite dimensional reduction (system of ODEs) derived in [17] is expected to well-approximate the PDE dynamics on long time scales. In particular, we revisit this reduction, find a class of exact solutions and shadow them in the (NLS/GP) system by applying the approach of [17].
keywords: shadowing. tunneling Hamiltonian systems Nonlinear Schrödinger equation effective dynamics

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