A generalized approach to the quantization of a large class of maps on a torus, i.e. quantization via the von Neumann Equation, is described and a number of issues related to the quantization of model systems are discussed. The approach yields well behaved mixed quantum states for tori for which the corresponding Schrodinger equation has no solutions, as well as an extended spectrum for tori wh...

Gaussons: Solitons of the logarithmic Schrodinger equation. Iwo Bialynicki-Birula (Institute of Theoretical Physics, Warsaw University, Warsaw, Poland and Department of Physics, University of Pittsburgh, Pittsburgh, U.S.A.); and J erzy Mycielski (Institute of Theoretical Physics, Warsaw University, Warsaw, Poland). Physica Scripta (Sweden) 20,539-544,1979. Properties of the Schrodinger equation...

We derive the amplitude equation, in the weakly nonlinear regime, for an optical wave packet that propagates in an initially undistorted nematic liquid crystal. By using the dyad representation Q i j , we find the retarded and nonlocal equation for the nematic configuration and solve it in Fourier space. This allows us to calculate the amplitude dependent dispersion relation for a nematic liqui...

In this paper, we primarily explore numerical solutions to the Quantum 1D Infinite Square Well problem, and the 1D Quantum Scattering problem. We use different finite difference schemes to approximate the second derivative in the 1D Schrodinger’s Equation and linearize the problem. By doing so, we convert the Infinite Well problem to a simple Eigenvalue problem and the Scattering problem to a s...

Four-dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time. Moreover, as an application of four-dimensional differential transform, exact solutions of nonlinear system of partial differential equations have been investigated. The results of the present method are compared very well with analytical solution of the system. Diff...

We propose an approach that permits to avoid instability phenomena for the nonlinear Schrödinger equations. We show that by approximating the solution in a suitable way, relying on a frequency cut-off, global well-posedness is obtained in any Sobolev space with nonnegative regularity. The error between the exact solution and its approximation can be measured according to the regularity of the e...

Abstract. 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 the corresponding nonlinear problems are indicated. Mathematics Subject Classification 2000: 53Cxx,35Jxx