Abstract. In the present paper we are concerned with the Novikov–Veselov equation at negative energy, i.e. with the (2+1)–dimensional analog of the KdV equation integrable by the method of inverse scattering for the two–dimensional Schrödinger equation at negative energy. We show that the solution of the Cauchy problem for this equation with non–singular scattering data behaves asymptotically a...

Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based on the elimination of the unknown boundary values by proper restrictions of the functional space and of the spectral variable complex domain. Illustrative exa...

We show that initial data near an unperturbed stable plane wave can evolve into a regime of spatiotemporal chaos in the slightly forced conservative periodic one-dimensional nonlinear Schrödinger equation. Statistical measures are employed to demonstrate that this spatiotemporal chaos is intermittent: there are windows in time for which the solution gains spatial coherence. The parameters and i...

The metric of some Lorentzian wormholes in the background of the FRW universe is obtained. It is shown that for a de Sitter space-time the new solution is supported by Phantom Energy. The wave equation for a scalar field in such backgrounds is separable. The form of the potential for the Schrödinger type one dimensional wave equation is found. PACS numbers: 04.20.-q, 04.20.Jb

We study the scattering theory for the nonlinear Schrödinger equations with cubic and quadratic nonlinearities in one and two space dimensions, respectively. For example, the nonlinearities are sum of gauge invariant term and non-gauge invariant terms such as λ0|u|2u + λ1u + λ2uū + λ3ū in one dimensional case, where λ0 ∈ R and λ1, λ2, λ3 ∈ C. The scattering theory for these equations belongs to...

We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic nonlinear Schrödinger equation in a suitable scaling limit. The result is extended to k-particle density matrices for all positive integer k. AMS Classification Number (2000): 35Q55, 81Q15, 81T18, 81V70. Runni...

As it has been proven, the determination of general one-dimensional Schrödinger Hamiltonians having third-order differential ladder operators requires to solve the Painlevé IV equation. In this work, it will be shown that some specific subsets of the higher-order supersymmetric partners of the harmonic oscillator possess third-order differential ladder operators. This allows us to introduce a s...

on the basis of a reproducing kernel space, an iterative algorithm for solving the one-dimensional linear and nonlinear schrödinger equations is presented. the analytical solution is shown in a series form in the reproducing kernel space and the approximate solution is constructed by truncating the series. the convergence of the approximate solution to the analytical solution is also proved. th...