Related papers: The Nonlinear Schr\"odinger Equation and Conservat…
A method to find exact solutions to nonlinear Schr\"odinger equation, defined on a line and on a plane, is found by connecting it with second order linear ordinary differential equation. The connection is essentially made using Riccati…
In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schr\"odinger equation. After presenting the basic properties of the equation, we give a self contained proof of…
We give a complete description of nontrivial local conservation laws of all orders for a natural generalization of the nonlinear progressive wave equation and, in particular, show that there is an infinite number of such conservation laws.
We show that a nonlinear Schr\"odinger wave equation can reproduce all the features of linear quantum mechanics. This nonlinear wave equation is obtained by exploring, in a uniform language, the transition from fully classical theory…
We prove global well-posedness for the cubic nonlinear Schr\"odinger equation with nonlinearity concentrated on a homogeneous Poisson process.
We prove the existence of non-trivial solutions to a system of coupled, nonlinear, Schroedinger equations with general nonlinearity.
In this article, we consider the nonlinear Schr\"odinger equation on the cylinder $\mathbb{R}^d\times \mathbb{T}$. In the long range case, we show there is no linear scattering state of the nonlinear Schr\"odinger equation on $\mathbb{R}^d…
We consider the 3-dimensional nonlinear Schr\"{o}dinger equation (NLS) with average nonlinearity. This is a limiting model of NLS with strong magnetic confinement and a generalized model of the resonant system of NLS with a partial harmonic…
For the first time, the general nonlinear Schr\"odinger equation is investigated, in which the chromatic dispersion and potential are specified by two arbitrary functions. The equation in question is a natural generalization of a wide class…
The initial-boundary value problem in a bounded domain with moving boundaries and nonhomogeneous boundary conditions for a higher order nonlinear Schr\"odinger (HNLS) equation is considered. Existence and uniqueness of global weak solutions…
We review some recent results concerning the Initial Value Problem of 1d-cubic non-linear Schr\"odinger equation (NLS) and other related systems as the Schr\"odinger Map. For the latter we prove the existence of a cascade of energy.…
In this paper, we study the schrodinger equation and wave equation with the Dirichlet boundary condition on a connected finite graph. The explicit expressions for solutions are given and the energy conservations are derived. Applications to…
We consider construction of ansatzes for nonlinear Schrodinger equations in three space dimensions and arbitrary nonlinearity, and conditions of their reduction to ordinary differential equations. Complete description of ansatzes of certain…
General features of nonlinear quantum mechanics are discussed in the context of applications to two-level atoms.
We study a system of nonlinear Schr\"odinger equations with cubic interactions in one space dimension. The orbital stability and instability of semitrivial standing wave solutions are studied for both non-degenerate and degenerate cases.
On the basis of analytical results, we present a numerical example that indicates inconsistency of a widely used ansatz with cubically nonlinear Schr\"odinger equation.
Conservation laws are computed for various nonlinear partial differential equations that arise in elasticity and acoustics. Using a scaling homogeneity approach, conservation laws are established for two models describing shear wave…
We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential…
In this paper we study a mixed problem for the nonlinear Schr\"odinger equation globally that have a nonlinear adding, in which the coefficient is a generalized function. Here is proved a global solvability theorem of the considered problem…
This paper is devoted to the well-posedness of stochastic nonlinear Schr\"odinger equations in the energy space H1(Rd), which is a natural continuation of our recent work [1]. We consider both focusing and defocusing nonlinearities and…