42,570 research outputs found
Stability of Compacton Solutions of Fifth-Order Nonlinear Dispersive Equations
We consider fifth-order nonlinear dispersive type equations to
study the effect of nonlinear dispersion. Using simple scaling arguments we
show, how, instead of the conventional solitary waves like solitons, the
interaction of the nonlinear dispersion with nonlinear convection generates
compactons - the compact solitary waves free of exponential tails. This
interaction also generates many other solitary wave structures like cuspons,
peakons, tipons etc. which are otherwise unattainable with linear dispersion.
Various self similar solutions of these higher order nonlinear dispersive
equations are also obtained using similarity transformations. Further, it is
shown that, like the third-order nonlinear equations, the fifth-order
nonlinear dispersive equations also have the same four conserved quantities and
further even any arbitrary odd order nonlinear dispersive type
equations also have the same three (and most likely the four) conserved
quantities. Finally, the stability of the compacton solutions for the
fifth-order nonlinear dispersive equations are studied using linear stability
analysis. From the results of the linear stability analysis it follows that,
unlike solitons, all the allowed compacton solutions are stable, since the
stability conditions are satisfied for arbitrary values of the nonlinear
parameters.Comment: 20 pages, To Appear in J.Phys.A (2000), several modification
Exact and approximate dynamics of the quantum mechanical O(N) model
We study a quantum dynamical system of N, O(N) symmetric, nonlinear
oscillators as a toy model to investigate the systematics of a 1/N expansion.
The closed time path (CTP) formalism melded with an expansion in 1/N is used to
derive time evolution equations valid to order 1/N (next-to-leading order). The
effective potential is also obtained to this order and its properties
areelucidated. In order to compare theoretical predictions against numerical
solutions of the time-dependent Schrodinger equation, we consider two initial
conditions consistent with O(N) symmetry, one of them a quantum roll, the other
a wave packet initially to one side of the potential minimum, whose center has
all coordinates equal. For the case of the quantum roll we map out the domain
of validity of the large-N expansion. We discuss unitarity violation in the 1/N
expansion; a well-known problem faced by moment truncation techniques. The 1/N
results, both static and dynamic, are also compared to those given by the
Hartree variational ansatz at given values of N. We conclude that late-time
behavior, where nonlinear effects are significant, is not well-described by
either approximation.Comment: 16 pages, 12 figrures, revte
Chaos in Time Dependent Variational Approximations to Quantum Dynamics
Dynamical chaos has recently been shown to exist in the Gaussian
approximation in quantum mechanics and in the self-consistent mean field
approach to studying the dynamics of quantum fields. In this study, we first
show that any variational approximation to the dynamics of a quantum system
based on the Dirac action principle leads to a classical Hamiltonian dynamics
for the variational parameters. Since this Hamiltonian is generically nonlinear
and nonintegrable, the dynamics thus generated can be chaotic, in distinction
to the exact quantum evolution. We then restrict attention to a system of two
biquadratically coupled quantum oscillators and study two variational schemes,
the leading order large N (four canonical variables) and Hartree (six canonical
variables) approximations. The chaos seen in the approximate dynamics is an
artifact of the approximations: this is demonstrated by the fact that its onset
occurs on the same characteristic time scale as the breakdown of the
approximations when compared to numerical solutions of the time-dependent
Schrodinger equation.Comment: 10 pages (12 figures), RevTeX (plus macro), uses epsf, minor typos
correcte
Dynamics of broken symmetry lambda phi^4 field theory
We study the domain of validity of a Schwinger-Dyson (SD) approach to
non-equilibrium dynamics when there is broken symmetry. We perform exact
numerical simulations of the one- and two-point functions of lambda phi^4 field
theory in 1+1 dimensions in the classical domain for initial conditions where <
phi(x) > not equal to 0. We compare these results to two self-consistent
truncations of the SD equations which ignore three-point vertex function
corrections. The first approximation, which sets the three-point function to
one (the bare vertex approximation (BVA)) gives an excellent description for <
phi(x) > = phi(t). The second approximation which ignores higher in 1/N
corrections to the 2-PI generating functional (2PI -1/N expansion) is not as
accurate for phi(t). Both approximations have serious deficiencies in
describing the two-point function when phi(0) > .4.Comment: 10 pages, 6 figure
An O(N) symmetric extension of the Sine-Gordon Equation
We discuss an O(N) exension of the Sine-Gordon (S-G)equation which allows us
to perform an expansion around the leading order in large-N result using
Path-Integral methods. In leading order we show our methods agree with the
results of a variational calculation at large-N. We discuss the striking
differences for a non-polynomial interaction between the form for the effective
potential in the Gaussian approximation that one obtains at large-N when
compared to the N=1 case. This is in contrast to the case when the classical
potential is a polynomial in the field and no such drastic differences occur.
We find for our large-N extension of the Sine-Gordon model that the unbroken
ground state is unstable as one increases the coupling constant (as it is for
the original S-G equation) and we determine the stability criteria.Comment: 21 pages, Latex (Revtex4) v3:minor grammatical changes and addition
Non-perturbative approach for the time-dependent symmetry breaking
We present a variational method which uses a quartic exponential function as
a trial wave-function to describe time-dependent quantum mechanical systems. We
introduce a new physical variable which is appropriate to describe the
shape of wave-packet, and calculate the effective action as a function of both
the dispersion and . The effective potential
successfully describes the transition of the system from the false vacuum to
the true vacuum. The present method well describes the long time evolution of
the wave-function of the system after the symmetry breaking, which is shown in
comparison with the direct numerical computations of wave-function.Comment: 8 pages, 3 figure
Isospectral Potentials from Modified Factorization
Factorization of quantum mechanical potentials has a long history extending
back to the earliest days of the subject. In the present paper, the
non-uniqueness of the factorization is exploited to derive new isospectral
non-singular potentials. Many one-parameter families of potentials can be
generated from known potentials using a factorization that involves
superpotentials defined in terms of excited states of a potential. For these
cases an operator representation is available. If ladder operators are known
for the original potential, then a straightforward procedure exists for
defining such operators for its isospectral partners. The generality of the
method is illustrated with a number of examples which may have many possible
applications in atomic and molecular physics.Comment: 8 pages, 4 figure
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