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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

SYMMETRIES AND DROMION SOLUTION OF A (2+1)-DIMENSIONAL NONLINEAR SOHR?DINGER EQUATION

The Painlevé property, infinitely many symmetries and exact solutions of a (2+1)-dimensional nonlinear Schr?dinger equation, which are obtained from the constraints of the Kadomtsev-Petviashvili equation, are studied in this paper. The Painlevé property is proved by the Weiss-Kruskal approach, the infinitely many symmetries are obtained by the formal series symmetry method and the dromion-like solution which is localized exponentially in all directions is obtained by a variable separation method.     

SYMMETRIES AND DROMION SOLUTION OF A (2+1)-DIMENSIONAL NONLINEAR SOHR?DINGER EQUATION

The Painlevé property, infinitely many symmetries and exact solutions of a (2+1)-dimensional nonlinear Schr?dinger equation, which are obtained from the constraints of the Kadomtsev-Petviashvili equation, are studied in this paper. The Painlevé property is proved by the Weiss-Kruskal approach, the infinitely many symmetries are obtained by the formal series symmetry method and the dromion-like solution which is localized exponentially in all directions is obtained by a variable separation method.     

CONSERVATION LAWS FOR THE SCHRÖDINGER–NEWTON EQUATIONS

In this Letter a first-order Lagrangian for the Schrödinger–Newton equations is derived by modifying a second-order Lagrangian proposed by Christian [Exactly soluble sector of quantum gravity, Phys. Rev. D 56(8) (1997) 4844–4877]. Then Noether's theorem is applied to the Lie point symmetries determined by Robertshaw and Tod [Lie point symmetries and an approximate solution for the Schrödinger–Newton equations, Nonlinearity 19(7) (2006) 1507–1514] in order to find conservation laws of the Schrödinger–Newton equations.     

NONLINEAR STABILITY ANALYSIS OF THE EMDEN–FOWLER EQUATION

In this paper, we qualitatively study radial solutions of the semilinear elliptic equation Δu+uⁿ = 0 with u(0) = 1 and u′(0) = 0 on the positive real line, called the Emden–Fowler or Lane–Emden equation. This equation is of great importance in Newtonian astrophysics and the constant n is called the polytropic index.

By introducing a set of new variables, the Emden–Fowler equation can be written as an autonomous system of two ordinary differential equations which can be analyzed using linear and nonlinear stability analysis. We perform the study of stability by using linear stability analysis, the Jacobi stability analysis (Kosambi–Cartan–Chern-theory) and the Lyapunov function method. Depending on the values of n these different methods yield different results. We identify a parameter range for n where all three methods imply stability.     

GEOMETRIC PHASES AND SCHR?DINGER'S CAT STATE

A matrix method is presented for treating the dynamical phases, adiabatic phases and nonadiabatic phases of quantum superposition states. It is effective for any parameter-varying Hamiltonian system. As two examples, the evolution of mass-varying harmonic oscillator and the evolution of coherent states under parameter-varying displaced operator have been studied, Some new phenomena are obtained in the first case and the possible producing of so-called Schr?dinger's cat state by geometric phases is pointed out. The quantum state useful for the quantum optical verification of Berry's phase is introduced.     

HAMILTON'S PRINCIPLE AND SCHRÖDINGER'S EQUATION DERIVED FROM GAUSS' PRINCIPLE OF LEAST SQUARES

It is shown that the Hamilton's principle in classical mechanics and the Schrödinger equation in quantum mechanics can both be derived from an application of Gauss' principle of least squares.     

FM-EVOKED RESPONSES AND FM-TONOTOPIC CHARACTERISTICS IN THE AUDITORY CORTEX OF THE CAT

Averaged FM-evoked responses were recorded from different points on the surface of the AI areain cats and response thresholds(△F_m)for different carrier frequencies(F)were determined.△F_m variedgreatly with Fand with the site of recording.The majority of the △F_m-versus-F curves were markedwith the presence of one or more deep valleys.High,middle and low characteristic-F points of FM-responses intermingled in the AI area,a distribution entirely different from the tonotopic mapping ofthe auditory cortex in tuning responses.This was discussed in view of fine neuro-structural differentia-tion at the cortex level for auditory perception and discrimination.The smallest △F_m obtained incats with the FM-response method was lower than 1%F for F>1000Hz and lower than 10Hz forF<1000Hz.     

PARTIAL RESTORATION OF THE CHIRAL SYMMETRY AND THE π CONDENSATION IN NUCLEAR MATTER

From the relativistic semiclassieal theory of π condensation[1], we deduced that, in the σ model, because of the variation of the radius of the magic circle with nuclear density, the π condensation in nuclear matter is possible. The calculated critical density is slightly higher than that of the normal ground state nuclear matter.     

SLOW CORTICAL RESPONSE EVOKED BY AMPLITUDE MODULATION OF SOUND AND RESPONSE THRESHOLDS IN GUINEA PIGS

AM-evoked slow cortical responses(SCR)were recorded in awake guinea pigs and the response thresholdin terms of amplitude increament in dB was determined.A quasi-rectangular pulse of 200 ms durationand with a repetition rate of 1/sec served as the modulating wave.The typical AM-evoked SCR assumes apositive-negative-positive triphasic waveform appearing 40—200ms after the on-set or off-set of modulation.For white noise,for repetitive clicks(1000pps)and for tones(125Hz—16kHz)in a very wide carrier levelrange(30—90 dB L_p),the values are only around 0.5 dB,quite close to those for human obtained bypsychophysical methods,suggesting that the SCR values can represent the intensity difference limens forthe animals.A guinea pig curve,i.e.,the functional curve of versus I,is constructed.This curve issupposed to be the first complete and convincing curve for animal.