61,424 research outputs found

### One parameter family of Compacton Solutions in a class of Generalized Korteweg-DeVries Equations

We study the generalized Korteweg-DeVries equations derivable from the
Lagrangian: $L(l,p) = \int \left( \frac{1}{2} \varphi_{x} \varphi_{t} - {
{(\varphi_{x})^{l}} \over {l(l-1)}} + \alpha(\varphi_{x})^{p}
(\varphi_{xx})^{2} \right) dx,$ where the usual fields $u(x,t)$ of the
generalized KdV equation are defined by $u(x,t) = \varphi_{x}(x,t)$. For $p$ an
arbitrary continuous parameter $0< p \leq 2 ,l=p+2$ we find compacton solutions
to these equations which have the feature that their width is independent of
the amplitude. This generalizes previous results which considered $p=1,2$. For
the exact compactons we find a relation between the energy, mass and velocity
of the solitons. We show that this relationship can also be obtained using a
variational method based on the principle of least action.Comment: Latex 4 pages and one figure available on reques

### 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

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