EXACT SOLUTIONS TO THE N-BODY SCHRÖDINGER EQUATION FOR THE HARMONIC OSCILLATOR

The exact solutions to the N-body Schrödinger equation for the harmonic oscillator are presented analytically. The permutational symmetry of the solutions for the identical three-body system of the harmonic oscillator are discussed in some detail.     

EXPLICIT SOLITON ASYMPTOTICS FOR THE NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE

There exists a particular class of boundary value problems for integrable nonlinear evolution equations formulated on the half-line, called linearizable. For this class of boundary value problems, the Fokas method yields a formalism for the solution of the associated initial-boundary value problem, which is as efficient as the analogous formalism for the Cauchy problem. Here, we employ this formalism for the analysis of several concrete initial-boundary value problems for the nonlinear Schrödinger equation. This includes problems involving initial conditions of a hump type coupled with boundary conditions of Robin type.     

CLOSED FORM SOLUTIONS TO THE INTEGRABLE DISCRETE NONLINEAR SCHRÖDINGER EQUATION

In this article we derive explicit solutions of the matrix integrable discrete nonlinear Schrödinger equation by using the inverse scattering transform and the Marchenko method. The Marchenko equation is solved by separation of variables, where the Marchenko kernel is represented in separated form, using a matrix triplet (A, B, C). Here A has only eigenvalues of modulus larger than one. The class of solutions obtained contains the N-soliton and breather solutions as special cases. We also prove that these solutions reduce to known continuous matrix NLS solutions as the discretization step vanishes.     

ON FINITE NORMAL SERIES SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH IRREGULAR SINGULARITY

We compute all potentials with the following property: The one-dimensional nonrelativistic Schrödinger equation for these potentials has irregular singular points at infinity and/or zero and is solved by a finite normal series. We restrict to expansion order zero, discuss some properties of the potentials obtained and, as an application, calculate for some given potentials exact solutions and energies. The aim of this paper is to provide a tool for finding exact solutions of the Schrödinger equation for a large class of singular potentials.     

A POSSIBLE GENERALIZATION OF THE SCHRÖDINGER EQUATION BASED ON THE GAUSS PRINCIPLE OF LEAST SQUARES

The linear Schrödinger equation is generalized into non-linear equation based on the Gauss' principle of least squares. The weight function is assigned in such a way that it might be interpreted as occupation number density of hidden particles that obey the Fermi–Dirac stastistics. It is shown that the motion of a free particle, according to the so generalized non-linear equation, is described by a well behaved nondeforming wave packet moving with a constant velocity, in contrast to the always deforming wave packet according to the linear Schrödinger equation.     

A SOLITON PERTURBATION THEORY FOR THE UNSTABLE NONLINEAR SCHR?DINGER EQUATION WITH CORRECTIONS

Based on the Zakharov-Shabat equation of the inverse scattering transform for the unstable nonlinear Schr?dinger equation, for which a perturbation theory with corrections is developed in this paper. All necessary formulae for calculating the scattering data are derived. Based upon these formulae, the effect due to the corrections can be studied. As an example, the correction due to the damping is calculated.     

CONSTRUCTION OF THE SOLITON STATES OF THE QUANTUM NONLINEAR SCHR?DINGER EQUATION

The quantum nonlinear schr?dinger equation (QNSE) is exactly solved by Beth's ansatz method and we give a reasonable definition of the quantum soliton states. From the definition we construct the soliton states of the QNSE from its bound-state solutions. The dispersion effect of the quantum soliton is also exactly analysed.     

ON THE APPLICATION OF A GENERALIZED VERSION OF THE DRESSING METHOD TO THE INTEGRATION OF VARIABLE COEFFICIENT N-COUPLED NONLINEAR SCHRÖDINGER EQUATION

N-coupled nonlinear Schrödinger (NLS) equations have been proposed to describe N-pulse simultaneous propagation in optical fibers. When the fiber is nonuniform, N-coupled variable-coefficient NLS equations can arise. In this paper, a family of N-coupled integrable variable-coefficient NLS equations are studied by using a generalized version of the dressing method. We first extend the dressing method to the versions with (N + 1) × (N + 1) operators and (2N + 1) × (2N + 1) operators. Then, we obtain three types of N-coupled variable-coefficient equations (N-coupled NLS equations, N-coupled Hirota equations and N-coupled high-order NLS equations). Then, the compatibility conditions are given, which insure that these equations are integrable. Finally, the explicit solutions of the new integrable equations are obtained.     

CONSTRUCTION OF THE SOLITON STATES OF THE QUANTUM NONLINEAR SCHR?DINGER EQUATION

The quantum nonlinear schr?dinger equation (QNSE) is exactly solved by Beth's ansatz method and we give a reasonable definition of the quantum soliton states. From the definition we construct the soliton states of the QNSE from its bound-state solutions. The dispersion effect of the quantum soliton is also exactly analysed.     

THE NONLINEAR SCHR?DINGER EQUATION AND THE CROSS-PHASE MODULATION IN ERBIUM-DOPED FIBER AMPLIFIERS

The nonlinear Schr$\ddot{o}$dinger equation(NLSE) in erbium-doped fiber(EDF) was obtained. The cross-phase modulation (XPM) in the erbium-doped fiber amplifiers (EDFA) was studied based on this NLSE and the rate equations. A more generalized form of the propagation equation in EDFA was obtained which included the phase shifts the EDFA induced. An analytical expression was given to the XPM in the EDFA. It was found that the XPM in the EDFA dose not change very much with the wavelength except at the neighboring wavelengths (around 1531 nm) where the absorption and emission cross-sections of the erbium ions reach their maxima, and the XPMs have opposite signs on the two sides around 1531 nm. Furthermore, it was found that the XPM increases with the increase of the length of EDF.     

A DIRECT PERTURBATION THEORY OF THE NONLINEAR SCHR?DINGER EQUATION WITH CORRECTIONS

An exact direct perturbation theory of nonlinear Schrodinger equation with corrections is developed under the condition that the initial value of the perturbed solution is equal to the value of an exact multisoliton solution at a particular time. After showing the squared Jost functions are the eigenfunctions of the linearized operator with a vanishing eigenvalue,suitable definitions of adjoint functions and inner product are introduced. Orthogonal relations are derived and the expansion of the unity in terms of the squared Jost functions is naturally implied. The completeness of the squared Jost functions is shown by the generalized Marchenko equation. As an example,the evolution of a Raman loss compensated soliton in an optical fiber is treated.     

EULER–LAGRANGE EQUATIONS FOR FUNCTIONALS DEFINED ON FRÉCHET MANIFOLDS

We prove a version of the variational Euler–Lagrange equations valid for functionals defined on Fréchet manifolds, such as the spaces of sections of differentiable vector bundles appearing in various physical theories.     

GLOBAL EXISTENCE AND BLOW-UP PHENOMENA FOR THE PERIODIC HUNTER–SAXTON EQUATION WITH WEAK DISSIPATION

In this paper, we study the periodic Hunter–Saxton equation with weak dissipation. We first establish the local existence of strong solutions, blow-up scenario and blow-up criteria of the equation. Then, we investigate the blow-up rate for the blowing-up solutions to the equation. Finally, we prove that the equation has global solutions.