Related papers: WKB analysis of generalized derivative nonlinear S…
We consider the large time behavior of solutions to defocusing nonlinear Schrodinger equation in the presence of a time dependent external potential. The main assumption on the potential is that it grows at most quadratically in space,…
In this paper we investigate the global well-posedness and long-term behavior of solutions to the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on the real line. The equation incorporates both local cubic nonlinearities with…
Complex Wadati-type potentials of the form $V(x)=-w^2(x) + iw_x(x)$, where $w(x)$ is a real-valued function, are known to possess a number of intriguing features, unusual for generic non-Hermitian potentials. In the present work, we…
This is the first part of a two-paper series studying nonlinear Schr\"odinger equations with quasi-periodic initial data. In this paper, we consider the standard nonlinear Schr\"odinger equation. Under the assumption that the Fourier…
We study the initial-boundary value problem for the derivative nonlinear Schr\"odinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost…
We consider Darboux transformations for the derivative nonlinear Schr\"odinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant…
We prove a general dispersive estimate for a Schroedinger type equation on a product manifold, under the assumption that the equation restricted to each factor satisfies suitable dispersive estimates. Among the applications are the…
A new family of solvable potentials related to the Schroedinger-Riccati equation has been investigated. This one-dimensional potential family depends on parameters and is restricted to the real interval. It is shown that this potential…
Two important cases, where boundary conditions and solutions of the well-known integrable equations on a semi-strip are uniquely determined by the initial conditions, are rigorously studied in detail. First, the case of rectangular matrix…
We consider a nonlinear Klein-Gordon equation with a quasilinear quadratic term. The Nonlinear Schr\"odinger (NLS) equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated…
In the WKB approximation the $\nabla^2S$ term in Schrodinger's equation is subordinate to the |\nabla S|^2 term. Here we study an anti-WKB approximation in which the $\nabla^2 S$ term dominates (after a guess for S is supplied). Our…
We consider non-gauge-invariant cubic nonlinear Schr\"odinger equations in one space dimension. We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^\infty$ decay up to time…
Stationary solutions asymptoting to nonlinear plane waves of the nonlinear Schr\"odinger equation with a PT-symmetric, complex linear potential are characterized. The potential includes both a spatially varying gain-loss profile and a…
We consider a number of linear and non-linear boundary value problems involving generalized Schr\"odinger equations. The model case is $-\Delta u=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R^n}$. We use the Sobolev…
The necessary and sufficient conditions for the exactness of the semiclassical approximation for the solution of the Schr\"odinger and Klein-Gordon equations are obtained. It is shown that the existence of an exact semiclassical solution of…
We consider the time-dependent non linear Schrodinger equations with a double well potential in dimensions d =1 and d=2. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest two eigenvalues…
We prove global existence of small solutions to the initial value problem for a class of cubic derivative nonlinear Schr\"odinger systems with the masses satisfying suitable non-resonance relations. The large-time asymptotics of the…
The aim of this work is to study the numerical solution of the nonlinear Schrodinger problem using a combination between Witt basis and finite difference approximations. We construct a discrete fundamental solution for the non-stationary…
In this paper, we consider the final state problem for the nonlinear Schr\"odinger equation with a homogeneous nonlinearity which is of the long range critical order and is not necessarily a polynomial, in one and two space dimensions. As…
We study the reducibility of a Linear Schr\"odinger equation subject to a small unbounded almost-periodic perturbation which is analytic in time and space. Under appropriate assumptions on the smallness, analiticity and on the frequency of…