The dynamic natures of microscopic particles described by nonlinear Schrödinger equation

There are a lot of difficulties and troubles in quantum mechanics, when the linear Schrödinger equation is used to describe microscopic particles. Thus, we here replace it by a nonlinear Schrödinger equation to investigate the properties and rule of microscopic particles. In such a case we find that the motion of microscopic particle satisfies classical rule and obeys the Hamiltonian principle, Lagrangian and Hamilton equations. We verify further the correctness of these conclusions by the results of nonlinear Schrödinger equation under actions of different externally applied potential. From these studies, we see clearly that rules and features of motion of microscopic particle described by nonlinear Schrödinger equation are greatly different from those in the linear Schrödinger equation, they have many classical properties, which are consistent with concept of corpuscles. Thus, we should use the nonlinear Schrödinger equation to describe microscopic particles.     

Exact X-wave solutions of the hyperbolic nonlinear Schrödinger equation with a supporting potential

We find exact solutions of the two- and three-dimensional nonlinear Schrödinger equation with a supporting potential. We focus in the case where the diffraction operator is of the hyperbolic type and both the potential and the solution have the form of an X-wave. Following similar arguments, several additional families of exact solutions can also can be found irrespectively of the type of the diffraction operator (hyperbolic or elliptic) or the dimensionality of the problem. In particular we present two such examples: The one-dimensional nonlinear Schrödinger equation with a stationary and a “breathing” potential and the two-dimensional nonlinear Schrödinger with a Bessel potential.     

Emulation of the evolution of a Bose–Einstein condensate in a time-dependent harmonic trap

We emulate the ground state of a Bose–Einstein condensate in a time-dependent isotropic harmonic trap by constructing analytic simulacra of a transformed wavefunction in the regions around the origin and far from the origin. This transformed wavefunction is obtained through a pseudoconformal transformation and is a function of new spatial and temporal variables, while the simulacra are generalisations of asymptotic solutions of the nonlinear Schrödinger equation and they are matched by requiring continuity not only of the wavefunction and of its slope, but of its curvature as well. The resulting piecewise analytic simulacra coincide almost perfectly with the numerically obtained solutions of the time-dependent nonlinear Schrödinger equation and constitute an easy and accurate analytic method for describing fully the condensate ground state.     

Exact solutions of the nonlinear Schrödinger equation in time-dependent parabolic density profiles

By using covariance properties of an extended Schrödinger formalism, exact soliton-like solutions of the nonlinear Schrödinger equation in time-dependent inhomogeneous media (parabolic density profiles) are constructed.     

Periodic Wave Solutions of Generalized Derivative Nonlinear Schrödinger Equation

A Darboux transformation of the generalized derivative nonlinear Schrodinger equation is derived. As an application, some new periodic wave solutions of the generalized derivative nonlinear Schrodinger equation are explicitly given.     

A general method for the solution of nonlinear soliton and kink Schrödinger equations

We present a method by which one-dimensional nonlinear soliton and kink Schrödinger equations can be solved in closed form. The hermitean nonlinear soliton operator may contain up to second derivatives of the wave function and the vanishing condition must hold. The method is applied to solve known nonlinear Schrödinger equations for one-soliton and one-kink solutions and, by inverting the procedure, to derive new operators with wave packet solutions of algebraic and arbitrary shapes. One of them is equivalent to the Derivative Nonlinear Schrödinger equation.     

Analytical study of the nonlinear Schrödinger equation with an arbitrary linear time-dependent potential in quasi-one-dimensional Bose-Einstein condensates

Under investigation in this paper is a nonlinear Schrödinger equation with an arbitrary linear time-dependent potential, which governs the soliton dynamics in quasi-one-dimensional Bose-Einstein condensates (quasi-1DBECs). With Painlevé analysis method performed to this model, its integrability is firstly examined. Then, the distinct treatments based on the truncated Painlevé expansion, respectively, give the bilinear form and the Painlevé-Bäcklund transformation with a family of new exact solutions. Furthermore, via the computerized symbolic computation, a direct method is employed to easily and directly derive the exact analytical dark- and bright-solitonic solutions. At last, of physical and experimental interests, these solutions are graphically discussed so as to better understand the soliton dynamics in quasi-1DBECs.     

Mathematical and physical aspects of controlling the exact solutions of the 3D Gross-Pitaevskii equation

The possibility of the decomposition of the three-dimensional (3D) Gross-Pitaevskii equation (GPE) into a pair of coupled Schrödinger-type equations, is investigated. It is shown that, under suitable mathematical conditions, it is possible to construct the exact controlled solutions of the 3D GPE from the solutions of a linear 2D Schrödinger equation coupled with a 1D nonlinear Schrödinger equation (the transverse and longitudinal components of the GPE, respectively). The coupling between these two equations is the functional of the transverse and the longitudinal profiles. The applied method of nonlinear decomposition, called the controlling potential method (CPM), yields the full 3D solution in the form of the product of the solutions of the transverse and longitudinal components of the GPE. It is shown that the CPM constitutes a variational principle and sets up a condition on the controlling potential well. Its physical interpretation is given in terms of the minimization of the (energy) effects introduced by the control. The method is applied to the case of a parabolic external potential to construct analytically an exact BEC state in the form of a bright soliton, for which the quantitative comparison between the external and controlling potentials is presented.     

Exact and explicit solutions of higher-order nonlinear equations of Schrödinger type

New exact solutions for the higher-order nonlinear Schrödinger equation and coupled higher-order nonlinear Schrödinger equations are obtained by using the generalized Jacobi elliptic function method. Solutions in the limiting cases are also studied.     

Decay estimates for Schrödinger equations

We prove global existence and optimal decay estimates for classical solutions with small initial data for nonlinear nonlocal Schrödinger equations. The Laplacian in the Schrödinger equation can be replaced by an operator corresponding to a non-degenerate quadratic form of arbitrary signature. In particular, the Davey-Stewartson system is included in the the class of equations we discuss.Partially supported by NSF grant DMS-860-2031. Sloan Research Fellow     

具有色散系数的(2+1)维非线性Schrödinger方程的有理解和空间孤子

非线性Schrödinger方程是物理学中具有广泛应用的非线性模型之一. 本文采用相似变换, 将具有色散系数的(2+1)维非线性Schrödinger方程简化成熟知的Schrödinger方程, 进而得到原方程的有理解和一些空间孤子.     

Solitons for a generalized variable-coefficient nonlinear Schrdinger equation

In this paper,we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear Schrdinger equation (NLS) with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions,one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results,some previous one-and two-soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one-and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.     

Time-dependent quantum systems and chronoprojective geometry

Chronoprojective transformations in the framework of five-dimensional Schrödinger formalism are used to construct the solution of the Schrödinger equation with a time-dependent harmonic potential from the solution of a free Schrödinger equation.     

Analytic solutions for time-dependent Schrödinger equations with linear of nonlinear Hamiltonians

The decomposition method is applied to the time-dependent Schrödinger equation for linear or nonlinear Hamiltonian operators, without linearization, perturbation, or numerical methods, to obtain a rapidly converging analytic solution     

The modified propagation equation for TM polarized subwavelength spatial solitons in a nonlinear planar waveguide

The wave equation of TM polarized subwavelength beam propagations in a nonlinear planar waveguide is derived beyond the paraxial approximation. This modified equation contains more higher-order linear and nonlinear terms than the nonlinear Schrödinger equation. The propagation of fundamental subwavelength spatial solitons is numerically studied. It is shown that the effect of the higher nonlinear terms is significant. That is, for the propagation of narrower beam the modified nonlinear Schrödinger equation is more suitable than the nonlinear Schrödinger equation.     

An alternative derivation of the pulse solutions of the KdV hierarchy

An alternative approach issues from the Appelle transformation of the Schrödinger equation. One solves the inverse problem for the transformed equation, a general solution of which is a quadratic form of two independent solutions of the primary Schrödinger equation. If the potential in the Schrödinger equation obeys one equation of the KdV hierarchy, the time derivative of this form is a linear combination of the form and its space derivative. The coefficients in the combination depend on the potential and the energy parameter of the Schrödinger equation only. This relation also determines the time dependence of the spectral data which along with the solution of the inverse problem gives the solution of the KdV equations as usual.     

非线性与立方非线性Schrodinger方程的多级包络周期解

本文基于Jacobi椭圆函数和Lamé方程,应用摄动法研究了非线性与立方非线性Schrodinger方程,获得了其新的多级包络周期解。这些解在极限条件下可以退化为各种形式的包络孤波解。     

非线性与立方非线性Schrodinger方程的多级包络周期解

本文基于Jacobi椭圆函数和Lamé方程,应用摄动法研究了非线性与立方非线性Schrodinger方程,获得了其新的多级包络周期解。这些解在极限条件下可以退化为各种形式的包络孤波解。     

Linear and nonlinear Schrödinger equations

The Schrödinger equation for a point particle in a quartic potential and a nonlinear Schrödinger equation are solved by the decomposition method yielding convergent series for the solutions which converge quite rapidly in physical problems involving bounded inputs and analytic functions. Several examples are given to demonstrate use of the method.