Optical solitons with spatio-temporal dispersion by first integral approach and functional variable method

This paper applies the first integral method and functional variable technique in order to obtain optical solitons from the governing nonlinear Schrödinger's equation with spatio-temporal dispersion. There are four types of nonlinear media that are taken into account. These are Kerr law, power law, parabolic law as well as the dual-power law nonlinearity. Several constraint conditions naturally emerge from the results obtained and these conditions are also listed.     

Optical solitons for the resonant nonlinear Schrödinger's equation with time-dependent coefficients by the first integral method

In this paper, the resonant nonlinear Schrödinger's equation is studied with three forms of nonlinearity. This equation is also considered with time-dependent coefficients. The first integral method is used to carry out the integration. Exact soliton solutions of this equation are found. These solutions are constructed through the established first integrals. The power of this manageable method is confirmed.     

Dispersive optical solitons by Kudryashov's method

This paper studies the dynamics of dispersive optical solitions that is modeled by the fourth order nonlinear Schrödinger's equation and Schrödinger–Hirota equation, the latter of which is considered with power law nonlinearity. Kudryashov's method is applied to obtain soliton solutions to the model equations. These results and the solution methodology makes a profound impact in the study of optical solitons.     

Bright and dark solitons in a cascaded system

This paper studies coupled nonlinear Schrödinger's equation (NLSE) that appears in a cascaded system. Both Kerr law and power law nonlinearities are considered. Bright and dark soliton solutions are retrieved for these nonlinearities. The corresponding constraint conditions naturally fall out that from the mathematical expressions that must remain valid for solitons to exist.     

Monte carlo simulation of a quantized universe

A Monte Carlo simulation method which yields groundstate wave functions for multielectron atoms is applied to quantized cosmological models. In quantum mechanics, the propagator for the Schrödinger equation reduces to the absolute value squared of the groundstate wave function in the limit of infinite Euclidean time. The wave function of the universe as the solution to the Wheeler-DeWitt equation may be regarded as the zero energy mode of a Schrödinger equation in coordinate time. The simulation evaluates the path integral formulation of the propagator by constructing a large number of paths and computing their contribution to the path integral using the Metropolis algorithm to drive the paths toward a global minimum in the path energy. The result agrees with a solution to the Wheeler-DeWitt equation which has the characteristics of a nodeless groundstate wave function. Oscillatory behavior cannot be reproduced although the simulation results may be physically reasonable. The primary advantage of the simulations is that they may easily be extended to cosmologies with many degrees of freedom. Examples with one, two, and three degrees of freedom (d.f.) are presented.This essay was awarded Honorable Mention by the Gravity Research Foundation.Part of this work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract number W-7405-ENG-48 and part was supported by National Science Foundation Grant PHY82-13411 to Oakland University.     

Path integral solution of linear second order partial differential equations I: the general construction

A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schrödinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette.     

Dark optical solitons with power law nonlinearity using G′/G-expansion

This paper studies the dynamics of dark optical solitons. The G′/G-expansion approach is utilized. The byproduct of this approach is the singular periodic solution of the governing nonlinear Schrödinger's equation for its corresponding parameter regime. The constraint conditions are also in place for the existence of dark solitons.     

Qualitative analysis and traveling wave solutions for the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity

In this Letter, we investigate the perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law nonlinearity. All explicit expressions of the bounded traveling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded traveling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Dark optical solitons and conservation laws for parabolic and dual-power law nonlinearities in (2 + 1)-dimensions

This paper studies the dynamics of optical solitons with parabolic and dual-power law nonlinearities. The dark 1-soliton solution is first obtained by the ansatz method along with the necessary constraint conditions, for both of these nonlinearities. Subsequently, the invariance, conservation laws and double reductions of the governing nonlinear Schrödinger's equation are studied and the conserved densities are thus revealed.     

Electromagnetic Field Theory Without Divergence Problems 2. A Least Invasively Quantized Theory

Classical electrodynamics based on the Maxwell–Born–Infeld field equations coupled with a Hamilton–Jacobi law of point charge motion is partially quantized. The Hamilton–Jacobi phase function is supplemented by a dynamical amplitude field on configuration space. Both together combine into a single complex wave function satisfying a relativistic Klein–Gordon equation that is self-consistently coupled to the evolution equations for the point charges and the electromagnetic fields. Radiation-free stationary states exist. The hydrogen spectrum is discussed in some detail. Upper bounds for Born's “aether constant” are obtained. In the limit of small velocities of and negligible radiation from the point charges, the model reduces to Schrödinger's equation with Coulomb Hamiltonian, coupled with the de Broglie–Bohm guiding equation.     

Nonspreading wave packets in quantum mechanics

In this paper a nonspreading, unnormalizable wave packet satisfying the Schrödinger equation is constructed. A modification of the Schrödinger equation is considered which allows the normalization of the wave packet. The case is generalized for relativistic mechanics.     

Perturbation of chiral solitons

This paper studies the perturbation of soliton due to the chiral nonlinear Schrödinger's equation by the aid of soliton perturbation theory. The perturbation term that is studied is the quantum potential perturbation of the chiral soliton that is known as Bohm potential. The stable fixed point of the chiral soliton parameters is obtained.     

Effective Field Theories for Disordered Systems from a Generalized Phase Formulation

We consider a spinless particle moving in a d-dimensional box, subject to periodic boundary conditions, and in the presence of a random potential. Introducing the logarithm of the wave function transforms the time-independent Schrödinger equation into a stochastic differential equation with the random potential acting as the source. Using this as our starting point we write functional integral representations for the disorder averaged density of states, the two point correlator of the absolute value of the wave function, and inverse participation ratios. We also show how a deterministic or random magnetic field can be included in the formalism.     

Classical variational derivation and physical interpretation of Dirac's equation

A simple random walk model has been shown by Gaveauet al. to give rise to the Klein-Gordon equation under analytic continuation. This absolutely most probable path implies that the components of the Dirac wave function have a common phase; the influence of spin on the motion is neglected. There is a nonclassical path of relative maximum likelihood which satisfies the constraint that the probability density coincide with the quantum mechanical definition. In three space dimensions, and in the presence of electromagnetic interaction, the Lagrangian for this optimal, nonclassical path coincides with the Lagrangian of the Dirac particle. The nonrelativistic, or diffusion, limit is shown to be a formal consequence of Einstein's dynamical equilibrium condition; the continuity equation reduces to the same diffusion equation derived from Schrödinger's equation. The relativistic, massless limit, which would describe a neutrino, is comparable (in the sense of analytic continuation) to a nonviscous liquid whose molecules possess internal degrees of freedom.Dedicated to Professor Alfonso Maria Liquori on the occasion of his 60th birthday.     

Topological solitons of resonant nonlinear Schödinger'sequation with dual-power law nonlinearity by G′/G-expansion technique

This paper considers the resonant nonlinear Schrödinger's equation with dual-power law nonlinearity. The G′/G-expansion method is applied to integrate this equation. The soliton solutions are thus obtained. Both constant coefficients as well as time-dependent coefficients are considered. The results for parabolic law nonlinearity fall out as a special case.     

Unitarity Schrödinger Equation and Ground State Properties of the Finite Trapped Superfluid Fermi Gases in a BCS-BEC Crossover

On the basis of quantum hydrodynamical equations we derive a unitarity Schrödinger equation of a finite trapped superfluid Fermi gas valid in the whole interaction regime from BCS superfluid to BEC. This equation is just the Ginzburg-Laudau-type equation for the fermionic Cooper pairs in the BCS side, the Gross-Pitaevskii-type equation for the bosonic dimers in the BEC side, and a unitarity equation for a strongly interacting Fermi superfluid in the unitarity limit. By taking a modified Gauss-like trial wave function, we solve the unitarity Schrödinger equation, calculate the energy, chemical potential, sizes and profiles of the ground-state condensate, and discuss the properties of the ground state in the entire BCS-BEC crossover regimes.     

Where and why the generalized Hamilton-Jacobi representation describes microstates of the Schrödinger wave function

A generalized Hamilton-Jacobi representation describes microstates of the Schrödinger wave function for bound states. At the very points that boundary values are applied to the bound state Schrödinger wave function, the generalized Hamilton-Jacobi equation for quantum mechanics exhibits a nodal singularity. For initial value problems, the two representations are equivalent.     

具有运动边界与含时频率的一类环状非球谐振子模型势的精确解(英文)

在球坐标系中研究了一类具有运动边界与含时频率的环状非球谐振子模型势的Schrdinger方程.应用坐标变换将运动边界转化为固定边界,从而获得了系统的精确波函数.研究表明,系统的角向波函数是一个推广的缔合勒让德多项式,径向波函数可以表示为贝赛耳函数.最后我们简单讨论了指数运动边界和指数含时频率这一特殊情况.     

Fresnel Type Path Integral for the Stochastic Schrödinger Equation

A path integral representation is obtained for the stochastic partial differential equation of Schrödinger type arising in the theory of open quantum systems subject to continuous nondemolition measurement and filtering, known as the a posteriori or Belavkin equation. The result is established by means of Fresnel-type integrals over paths in configuration space. This is achieved by modifying the classical action functional in the expression for the amplitude along each path by means of a stochastic Itô integral. This modification can be regarded as an extension of Menski's path integral formula for a quantum system subject to continuous measurement to the case of the stochastic Schrödinger equation.     

A general solution to the Schrödinger–Poisson equation for a charged hard wall: Application to potential profile of an AlN/GaN barrier structure

A general, system-independent, formulation of the parabolic Schrödinger–Poisson equation is presented for a charged hard wall in the limit of complete screening by the ground state. It is solved numerically using iteration and asymptotic boundary conditions. The solution gives a simple relation between the band bending and sheet charge density at an interface. Approximative analytical expressions for the potential profile and wave function are developed based on properties of the exact solution. Specific tests of the validity of the assumptions leading to the general solution are made. The assumption of complete screening by the ground state is found be a limitation; however, the general solution provides a fair approximate account of the potential profile when the bulk is doped. The general solution is further used in a simple model for the potential profile of an AlN/GaN barrier structure. The result compares well with the solution of the full Schrödinger–Poisson equation.