Related papers: Time-like boundary conditions in the NLS model
Sequences of bivariate orthogonal polynomials written as vector polynomials of increasing size satisfy a couple of three term relations with matrix coefficients. In this work, introducing a time-dependent parameter, we analyse a Lax-type…
A two component nonlocal vector nonlinear Schr\"odinger equation (VNLSE) is considered with a self-induced $ {\cal PT}$ symmetric potential. It is shown that the system possess a Lax pair and an infinite number of conserved quantities and…
In this paper, we present a microscopic derivation of the two-dimensional focusing cubic nonlinear Schr\"odinger equation starting from an interacting $N$-particle system of Bosons. The interaction potential we consider is given by…
The problem of classifying boundary points of space-time, for example singularities, regular points and points at infinity, is an unexpectedly subtle one. Due to the fact that whether or not two boundary points are identified or even…
We consider nonlinear Schrodinger equations with either local or nonlocal nonlinearities. In addition, we include periodic potentials as used, for example, in matter wave experiments in optical lattices. By considering the corresponding…
We discuss the concept of Lax-Darboux scheme and illustrate it on well known examples associated with the Nonlinear Schrodinger (NLS) equation. We explore the Darboux links of the NLS hierarchy with the hierarchy of Heisenberg model,…
A well-posedness and maximal regularity result for the time-periodic Cahn-Hilliard-Gurtin system in the half space is proved. For this purpose, we introduce a novel class of complementing boundary conditions, extending the classical…
We investigate the initial-boundary value problem for the general three-component nonlinear Schrodinger (gtc-NLS) equation with a 4x4 Lax pair on a finite interval by extending the Fokas unified approach. The solutions of the gtc-NLS…
Globally hyperbolic spacetimes with timelike boundary $(\overline{M} = M \cup \partial M, g)$ are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if $\overline{M}$ is obtained by means of a…
The results on the initial boundary value problem for Einstein's vacuum field equation obtained in \cite{friedrich:nagy} rely on an unusual gauge. One of the defining gauge source functions represents the mean extrinsic curvature of the…
We consider the cubic nonlinear Schr\"odinger (NLS) equation set on a two dimensional box of size $L$ with periodic boundary conditions. By taking the large box limit $L \to \infty$ in the weakly nonlinear regime (characterized by smallness…
We consider the one-dimensional linear free space Schr\"odinger equation on a bounded interval subject to homogeneous linear boundary conditions. We prove that, in the case of pseudoperiodic boundary conditions, the solution of the…
The Schr\"odinger-Robertson inequality for relativistic position and momentum operators X^\mu, P_\nu, \mu, \nu = 0,1,2,3, is interpreted in terms of Born reciprocity and `non-commutative' relativistic phase space geometry. For states which…
A method of solving the time-dependent Schr\"odinger equation is presented, in which a finite region of space is treated explicitly, with the boundary conditions for matching the wave-functions on to the rest of the system replaced by an…
If $L^x$ is the total occupation local time of $d$-dimensional super-Brownian motion, $X$, for $d=2$ and $d=3$, we construct a random measure $\mathcal{L}$, called the boundary local time measure, as a rescaling of $L^x e^{-\lambda L^x} dx$…
Spacetime is modelled by binary relations - by the classes of the automorphisms $\GL(\C^2)$ of a complex 2-dimensional vector space with respect to the definite unitary subgroup $\U(2)$. In extension of Feynman propagators for particle…
In this paper, we investigate the space-time shifted nonlocal derivative nonlinear Schr\"{o}dinger (DNLS) equation under nonzero boundary conditions using the Riemann--Hilbert (RH) approach for the first time. To begin with, in the direct…
In this paper, we study the initial boundary value problem for nonlinear Schr\"odinger equations on the half-line with nonlinear boundary conditions of type $u_x(0,t)+\lambda|u(0,t)|^ru(0,t)=0,$ $\lambda\in\mathbb{R}-\{0\}$, $r> 0$. We…
We consider an inverse problem of recovering all spatial dependent coefficients in the time dependent Schr\"odinger equation defined on an open bounded domain in $\mathbb{R}^n$, $n\geq 2$, with smooth enough boundary. We show that by…
We make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary…