arXiv subjects

Harvey Shepard

Publications and source records attributed to Harvey Shepard.

5 recordsLinked to original sources

arXiv1994

Semiquantum Chaos and the Large N Expansion

We consider the dynamical system consisting of a quantum degree of freedom $A$ interacting with $N$ quantum oscillators described by the Lagrangian \bq L = {1\over 2}\dot{A}^2 + \sum_{i=1}^{N} \left\{{1\over 2}\dot{x}_i^2 - {1\over 2}( m^2 + e^2 A^2)x_i^2 \right\}. \eq In the limit $N \rightarrow \infty$, with $e^2 N$ fixed, the quantum fluctuations in $A$ are of order $1/N$. In this limit, the $x$ oscillators behave as harmonic oscillators with a time dependent mass determined by the solution of a semiclassical equation for the expectation value $\VEV{A(t)}$. This system can be described, when $\VEV{x(t)}= 0$, by a classical Hamiltonian for the variables $G(t) = \VEV{x^2(t)}$, $\dot{G}(t)$, $A_c(t) = \VEV{A(t)}$, and $\dot{A_c}(t)$. The dynamics of this latter system turns out to be chaotic. We propose to study the nature of this large-$N$ limit by considering both the exact quantum system as well as by studying an expansion in powers of $1/N$ for the equations of motion using the closed time path formalism of quantum dynamics.

chao-dyn↗

arXiv1994

SUSY-Based Variational Method for the Anharmonic Oscillator

Using a newly suggested algorithm of Gozzi, Reuter, and Thacker for calculating the excited states of one dimensional systems, we determine approximately the eigenvalues and eigenfunctions of the anharmonic oscillator, described by the Hamiltonian $H= p^2/2 + g x^4$. We use ground state post-gaussian trial wave functions of the form $Ψ(x) = N{\rm{exp}}[-b |x|^{2n}]$, where $n$ and $b$ are continuous variational parameters. This algorithm is based on the hierarchy of Hamiltonians related by supersymmetry (SUSY) and the factorization method. We find that our two parameter family of trial wave functions yields excellent energy eigenvalues and wave functions for the first few levels of the anharmonic oscillator.

patt-sol↗

arXiv1994

Semi-Quantum Chaos

We consider a system in which a classical oscillator is interacting with a purely quantum mechanical oscillator, described by the Lagrangian $ L = \frac{1}{2} \dot{x}^2 + \frac{1}{2} \dot{A}^2 - \frac{1}{2} ( m^2 + e^2 A^2) x^2 \>, $ where $A$ is a classical variable and $x$ is a quantum operator. With $\langle x(t) \rangle = 0$, the relevant variable for the quantum oscillator is $\langle x(t) x(t) \rangle = G(t)$. The classical Hamiltonian dynamics governing the variables $A(t)$, $Π_A(t)$, $G(t)$ and $Π_G(t)$ is chaotic so that the results of making measurements on the quantum system at later times are sensitive to initial conditions. This system arises as the zero momentum part of the problem of pair production of charged scalar particles by a strong external electric field.

chao-dyn↗

arXiv1993

Solitons in the Camassa-Holm Shallow Water Equation

We study the class of shallow water equations of Camassa and Holm derived from the Lagrangian: $ L= \int \left( \frac{1}{2} (φ_{xxx}-φ_{x} )φ_{t} - {1 \over 2} {(φ_{x})^{3}} - {1 \over 2}φ_{x}(φ_{xx})^{2} - {1 \over 2} κφ_{x}^{2} \right) dx, $

patt-sol↗

arXiv1993

Solitary Waves and Compactons in a class of Generalized Korteweg-DeVries Equations

We study the class of generalized Korteweg-DeVries equations derivable from the Lagrangian: $ L(l,p) = \int \left( \frac{1}{2} \vp_{x} \vp_{t} - { {(\vp_{x})^{l}} \over {l(l-1)}} + α(\vp_{x})^{p} (\vp_{xx})^{2} \right) dx, $ where the usual fields $u(x,t)$ of the generalized KdV equation are defined by $u(x,t) = \vp_{x}(x,t)$. This class contains compactons, which are solitary waves with compact support, and when $l=p+2$, these solutions have the feature that their width is independent of the amplitude. We consider the Hamiltonian structure and integrability properties of this class of KdV equations. We show that many of the properties of the solitary waves and compactons are easily obtained using a variational method based on the principle of least action. Using a class of trial variational functions of the form $u(x,t) = A(t) \exp \left[-β(t) \left|x-q(t) \right|^{2n} \right]$ we find soliton-like solutions for all $n$, moving with fixed shape and constant velocity, $c$. We show that the velocity, mass, and energy of the variational travelling wave solutions are related by $ c = 2 r E M^{-1}$, where $ r = (p+l+2)/(p+6-l)$, independent of $n$.\newline \newline PACS numbers: 03.40.Kf, 47.20.Ky, Nb, 52.35.Sb

patt-sol↗