New Integrable Equations of Nonlinear Schrödinger Type

New integrable matrix nonlinear evolution partial differential equations in (1 + 1)-dimensions are derived, via a treatment which starts from an appropriate matrix generalization of the Zakharov–Shabat spectral problem. Via appropriate parametrizations, multi-vector versions of these equations are also exhibited. Generally these equations feature solitons that do not move with constant velocities: they rather behave as boomerons or as trappons, namely, up to a Galileian transformation, they typically boomerang back to where they came from, or they are trapped to oscillate around some fixed position determined by their initial data. In this paper, meant to be the first of a series, we focus on the derivation and exhibition of new coupled evolution equations of nonlinear Schrödinger type and on the behavior of their single-soliton solutions.     

Upper Estimates for a Moving Boundary Problem for Resonant Nonlinear Schrödinger Equations

Pointwise bounds are obtained for the solution of an initial boundary value problem for the resonant nonlinear Schrödinger equations. The context is that of a straight-line region with prescribed moving boundaries, expanding or noncontracting, upon which zero (Dirichlet) conditions are imposed.     

高阶Schrödinger方程的加权估计

本文主要研究的是相函数为齐次椭圆多项式的自由高阶Schrodinger方程.通过相函数等值面的几何性质,得到了解算子的Strichartz加权估计和极大算子加权估计.     

Discretization of Coupled Nonlinear Schrödinger Equations

The soliton solutions for discrete coupled nonlinear Schrödinger equations are investigated by using bilinear formalism. Pfaffian expressions of the N -soliton solutions of dark–dark and bright–bright types are explicitly given for the defocusing–defocusing and focusing–focusing cases, respectively.     

Well-Posedness of the Cauchy Problem for Stochastic Evolution Functional Equations

We prove the existence, uniqueness, and continuous dependence on the initial data of the solutions of the Cauchy problem for stochastic evolution functional equations with random coefficients in Hilbert spaces. We propose a method for constructing an approximating sequence for the solution of the Cauchy problem and obtain an estimate for the rate of convergence to the exact solution.     

An Inverse Problem for the Matrix Schrödinger Equation

By modifying and generalizing some old techniques of N. Levinson, a uniqueness theorem is established for an inverse problem related to periodic and Sturm-Liouville boundary value problems for the matrix Schrödinger equation.     

Cauchy Problem for a System of Equations of Ultraparabolic Type

This paper is devoted to the proof of the existence of a solution of the Cauchy problem for a system of equations of ultraparabolic type.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 768–774.Original Russian Text Copyright ©2005 by S. A. Tersenov.     

Evolution of solitary waves for a perturbed nonlinear Schrödinger equation

Soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear Schrödinger equation. Perturbation terms, which model wide classes of physically relevant perturbations, are considered. An analytical solution is found for the first-order correction of the evolving solitary wave. This solution for the solitary wave tail is in integral form and an explicit expression is found, for large time. Singularity theory, usually used for combustion problems, is applied to the large time expression for the solitary wave tail. Analytical results are obtained, such as the parameter regions in which qualitatively different types of solitary wave tails occur, the location of zeros and the location and amplitude of peaks, in the solitary wave tail. Two examples, the near-continuum limit of a discrete NLS equation and an explicit numerical scheme for the NLS equation, are considered in detail. For the discrete NLS equation it is found that three qualitatively different types of solitary wave tail can occur, while for the explicit finite-difference scheme, only one type of solitary wave tail occurs. An excellent comparison between the perturbation solution and numerical simulations, for the solitary wave tail, is found for both examples.     

On the Cauchy problem for a coupled system of third-order nonlinear Schrödinger equations

We investigate some well-posedness issues for the initial value problem (IVP) associated with the system

     

Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces

We study the well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations

     

The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament

The Cauchy problem of one-dimensional fourth-order nonlinear Schrödinger equation related to the vortex filament is studied. Local well-posedness for initial data in is obtained by the Fourier restriction norm method under certain coefficient condition.     

The Schrödinger operator

In this paper we study the maximum dissipative extension of the Schrödinger operator, introduce the generalized indefinite metric space, obtain the representation of the maximum dissipative extension of the Schrödinger operator in the natural boundary space and make preparation for the further study of the longtime chaotic behavior of the infinite-dimensional dynamics system in the Schrödinger equation.

     

Parabolic Schrödinger operators

We characterize the domain of the parabolic Schrödinger operator t_∂−Δ+V in Lp(Rn+1), 1<p<∞, where the potential V is nonnegative and belongs to the Parabolic Reverse Hölder class p(PB).     

On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations

We investigate the global well-posedness, scattering and blow up phenomena when the 3-D quintic nonlinear Schrödinger equation, which is energy-critical, is perturbed by a subcritical nonlinearity λ₁p|u|u. We find when the quintic term is defocussing, then the solution is always global no matter what the sign of λ₁ is. Scattering will occur either when the perturbation is defocussing and or when the mass of the solution is small enough and . When the quintic term is focusing, we show the blow up for certain solutions.     

The Cauchy Problem for Shilov Parabolic Equations

We establish well-posed solvability of the Cauchy problem for the Shilov parabolic equations with time-dependent coefficients whose initial data are tempered distributions. For a certain class of equations we state necessary and sufficient conditions for unique solvability of the Cauchy problem whose properties with respect to the spatial variable are typical for a fundamental solution. The results are characterized only by the order and the parabolicity exponent.     

The Cauchy Problem for Partial Difference Equations

We want to discuss partial difference equations, first of all with respect to the existence and uniqueness of their solution. These equations are considered with solutions on arbitrary subsets of the n-dimensional grid Zn. The basic theorem enables one to formulate the Cauchy problem for such equations. The solution is proven to be recursively computable for partial difference equations under very mild restrictions. (Variable coefficients for linear equations, systems of equations as well as nonlinear equations are not excluded.) The construction of solutions presented here also allows for some qualitative conclusions, such as boundedness of solutions.