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We study the Gross-Pitaevskii equation with a slowly varying smooth potential, V (x) = W (hx). We show that up to time log(1/h)/h and errors of size h in H, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian, (ξ + sech ∗ V (x))/2. This provides an improvement (h→ h) compared to previous works, and is(More)
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 (L norm squared) is shown to be in good agreement with the quantum transmission rate of the delta function potential. We further show that the(More)
We show that 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(More)
Scattering of radial H solutions to the 3D focusing cubic nonlinear Schrödinger equation below a mass-energy threshold M [u]E[u] < M [Q]E[Q] and satisfying an initial mass-gradient bound ‖u0‖L2‖∇u0‖L2 < ‖Q‖L2‖∇Q‖L2 , where Q is the ground state, was established in Holmer-Roudenko [8]. In this note, we extend the result in [8] to non-radial H data. For this,(More)
We consider the problem of identifying sharp criteria under which radial H (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂tu + ∆u + |u|2u = 0 scatter, i.e. approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the scale-invariant quantities ‖u0‖L2‖∇u0‖L2 and M(More)
We study the Gross-Pitaevskii equation with a delta function potential, qδ0, where |q| is small and analyze the solutions for which the initial condition is a soliton with initial velocity v0. We show that up to time (|q|+ v 0)− 1 2 log(1/|q|) the bulk of the solution is a soliton evolving according the classical dynamics of a natural effective Hamiltonian,(More)
Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system    i∂tu +∆u = nu ∂ t n −∆n = ∆|u| u(x, 0) = u0(x) n(x, 0) = n0(x), ∂tn(x, 0) = n1(x) u = u(x, t) ∈ C n = n(x, t) ∈ R x ∈ R, t ∈ R for any dimension d, in the inhomogeneous Sobolev spaces (u, n) ∈ Hk(Rd)×Hs(Rd) for a range of exponents k, s depending on d. Here we(More)
In the 2-d setting, given an H solution v(t) to the linear Schrödinger equation i∂tv + ∆v = 0, we prove the existence (but not uniqueness) of an H solution u(t) to the defocusing nonlinear Schrödinger (NLS) equation i∂tu+∆u− |u|p−1u = 0 for nonlinear powers 2 < p < 3 and the existence of an H solution u(t) to the defocusing Hartree equation i∂tu +∆u −(More)
We consider a 2D time-dependent quantum system of N -bosons with harmonic external confining and attractive interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of matter type estimates showing that the k-th power of the energy controls the H Sobolev norm of the solution over k-particles. This estimate is new and more diffi cult(More)