Justin Holmer

The Initial-Boundary Value Problem for the Korteweg–de Vries Equation

We prove local well-posedness of the initial-boundary value problem for the Korteweg–de Vries equation on right half-line, left half-line, and line segment, in the low regularity setting. This is accomplished by introducing an analytic family of boundary forcing operators.

Fast Soliton Scattering by Delta Impurities

We study the Gross-Pitaevskii equation with a repulsive delta function potential. We show that a high velocity incoming soliton is split into a transmitted component and a reflected component. The transmitted mass (L2 norm squared) is shown to be in good agreement with the quantum transmission rate of the delta function potential. We further show that the transmitted and reflected components resolve into solitons plus dispersive radiation, and quantify the mass and phase of these solitons.

Soliton Splitting by External Delta Potentials

We show that in the dynamics of the nonlinear Schrodinger equation a soliton scattered by an external delta potential splits into two solitons and a radiation term. Theoretical analysis gives the amplitudes and phases of the reflected and transmitted solitons with errors going to zero as the velocity of the incoming soliton tends to infinity. Numerical analysis shows that this asymptotic relation is valid for all but very slow solitons. We also show that the total transmitted mass, that is, the square of the L2 norm of the solution restricted on the transmitted side of the delta potential, is in good agreement with the quantum transmission rate of the delta potential.

On the 2D Zakharov system with L2 Schrödinger data

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L2 × H−1/2 × H−3/2. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schrodinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data—a result which is false for the cubic nonlinear Schrodinger equation in dimension two—and it is optimal because Glangetas–Merles solutions blow up at that time.

Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems

We prove low regularity global well-posedness for the Id Zakharov system and the 3d Klein-Gordon-Schrodinger system, which are systems in two variables u : R d x x R t → C and n: R d x x R t → R. The Zakharov system is known to be locally well-posed in (u,n) ∈ L 2 x H -½ and the Klein-Gordon-Schrodinger system is known to be locally well-posed in (u, n) ∈ L 2 x L 2 . Here, we show that the Zakharov and Klein-Gordon-Schrodinger systems are globally well-posed in these spaces, respectively, by using an available conservation law for the L 2 norm of u and controlling the growth of n via the estimates in the local theory.

Focusing quantum many-body dynamics, II : The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D

We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by

On the Rigorous Derivation of the 2D Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-Body Dynamics

a^{3\beta-1}V(a^{\beta}\cdot)

Divergence of Infinite-Variance Nonradial Solutions to the 3D NLS Equation

where

Fast Soliton Scattering by Attractive Delta Impurities

\int V\leqslant 0

Blow-up criteria for the 3D cubic nonlinear Schrödinger equation

and