35 research outputs found
Inverse problem for an inhomogeneous Schr\"odinger equation
An inverse problem is considered for an inhomogeneous Schr\"odinger equation.
Assuming that the potential vanishes outside a finite interval and satisfies
some other technical assumptions, one proves the uniqueness of the recovery of
this potential from the knowledge of the wave function at the ends of the above
interval for all energies. An algorithm is given for the recovery of the
potential from the above data
Global well-posedness and scattering for the defocusing energy-critical nonlinear Schr\"odinger equation in
We obtain global well-posedness, scattering, uniform regularity, and global
spacetime bounds for energy-space solutions to the defocusing
energy-critical nonlinear Schr\"odinger equation in . Our
arguments closely follow those of Colliander-Keel-Staffilani-Takaoka-Tao,
though our derivation of the frequency-localized interaction Morawetz estimate
is somewhat simpler. As a consequence, our method yields a better bound on the
-norm
Complete integrability of derivative nonlinear Schr\"{o}dinger-type equations
We study matrix generalizations of derivative nonlinear Schr\"{o}dinger-type
equations, which were shown by Olver and Sokolov to possess a higher symmetry.
We prove that two of them are `C-integrable' and the rest of them are
`S-integrable' in Calogero's terminology.Comment: 14 pages, LaTeX2e (IOP style), to appear in Inverse Problem
ftp 147.26.103.110 or 129.120.3.113 (login: ftp) On multi-lump solutions to the non-linear
We present a new approach to proving the existence of semi-classical bound states of the non-linear Schrödinger equation which are concentrated near a finite set of non-degenerate critical points of the potential function. The method is based on considering a system of non-linear elliptic equations. The positivity of the solutions is considered. It is shown how the same method yields “multi-bump ” solutions “homoclinic” to an equilibrium point for non-autonomous Hamiltonian equations. The method provides a calculable asymptotic form for the solutions in terms of a small parameter.