Convergence to Scattering States in the Nonlinear Schr\"odinger Equation

In this paper, we consider global solutions of the following nonlinear Schr\"odinger equation $iu_t+\Delta u+\lambda|u|^\alpha u = 0,$ in $\R^N,$ with $\lambda\in\R,$ $\alpha\in(0,\frac{4}{N-2})$ $(\alpha\in(0,\infty)$ if $N=1)$ and \linebreak $u(0)\in X\equiv H^1(\R^N)\cap L^2(|x|^2;dx).$ We show that, under suitable conditions, if the solution $u$ satisfies $e^{-it\Delta}u(t)-u_ \pm\to0$ in $X$ as $t\to\pm\infty$ then $u(t)-e^{it\Delta}u_\pm\to0$ in $X$ as $t\to\pm\infty.$ We also study the converse. Finally, we estimate $|\:\|u(t)\|_X-\|e^{it\Delta}u_\pm\|_X\:|$ under some less restrictive assumptions

LECTURES ON NONLINEAR DISPERSIVE EQUATIONS II

Program N. Tzvetkov Ill-posedness issues for nonlinear dispersive equations H. Koch Dispersive estimates and application

LECTURES ON NONLINEAR DISPERSIVE EQUATIONS II

Program N. Tzvetkov Ill-posedness issues for nonlinear dispersive equations H. Koch Dispersive estimates and application

Edge Current due to Majorana Fermions in Superfluid 3^3He A- and B-Phases

We propose a method utilizing edge current to observe Majorana fermions in the surface Andreev bound state for the superfluid $^3$He A- and B-phases. The proposal is based on self-consistent analytic solutions of quasi-classical Green's function with an edge. The local density of states and edge mass current in the A-phase or edge spin current in the B-phase can be obtained from these solutions. The edge current carried by the Majorana fermions is partially cancelled by quasiparticles (QPs) in the continuum state outside the superfluid gap. QPs contributing to the edge current in the continuum state are distributed in energy even away from the superfluid gap. The effect of Majorana fermions emerges in the depletion of the edge current by temperature within a low-temperature range. The observations that the reduction in the mass current is changed by $T^2$-power in the A-phase and the reduction in the spin current is changed by $T^3$-power in the B-phase establish the existence of Majorana fermions. We also point out another possibility for observing Majorana fermions by controlling surface roughness.Comment: 13 pages, 4 figures, published versio

Majorana bound state in rotating superfluid 3He-A between parallel plates

A concrete and experimentally feasible example for testing the putative Majorana zero energy state bound in a vortex is theoretically proposed for a parallel plate geometry of superfluid $^3$He-A phase. We examine the experimental setup in connection with ongoing rotating cryostat experiments. The theoretical analysis is based on the well-established Ginzburg--Landau functional, supplemented by microscopic calculations of the Bogoliubov--de Gennes equation, both of which allow the precise location of the parameter regions of the Majorana state to be found in realistic situations.Comment: 5 pages, 4 figure

Spontaneous mass current and textures of p-wave superfluids of trapped Fermionic atom gases at rest and under rotation

It is found theoretically based on the Ginzburg-Landau framework that p-wave superfluids of neutral atom gases in three dimension harmonic traps exhibit spontaneous mass current at rest, whose direction depends on trap geometry. Under rotation various types of the order parameter textures are stabilized, including Mermin-Ho and Anderson-Toulouse-Chechetkin vortices. In a cigar shape trap spontaneous current flows longitudial to the rotation axis and thus perpendicular to the ordinary rotational current. These features, spontaneous mass current at rest and texture formation, can be used as diagnoses for p-wave superfluidity.Comment: 5 pages, 5 figure

LECTURES ON NONLINEAR DISPERSIVE EQUATIONS I

CONTENTS J. Bona Derivation and some fundamental properties of nonlinear dispersive waves equations F. Planchon Schr\"odinger equations with variable coecients P. Rapha\"el On the blow up phenomenon for the L^2 critical non linear Schrodinger Equatio