1,223 research outputs found
Behavior of a Model Dynamical System with Applications to Weak Turbulence
We experimentally explore solutions to a model Hamiltonian dynamical system
derived in Colliander et al., 2012, to study frequency cascades in the cubic
defocusing nonlinear Schr\"odinger equation on the torus. Our results include a
statistical analysis of the evolution of data with localized amplitudes and
random phases, which supports the conjecture that energy cascades are a generic
phenomenon. We also identify stationary solutions, periodic solutions in an
associated problem and find experimental evidence of hyperbolic behavior. Many
of our results rely upon reframing the dynamical system using a hydrodynamic
formulation.Comment: 22 pages, 14 figure
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: where the usual fields of the
generalized KdV equation are defined by . For an
arbitrary continuous parameter 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 . 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
Weierstrass's criterion and compact solitary waves
Weierstrass's theory is a standard qualitative tool for single degree of
freedom equations, used in classical mechanics and in many textbooks. In this
Brief Report we show how a simple generalization of this tool makes it possible
to identify some differential equations for which compact and even semicompact
traveling solitary waves exist. In the framework of continuum mechanics, these
differential equations correspond to bulk shear waves for a special class of
constitutive laws.Comment: 4 page
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)}} + \alpha(\vp_{x})^{p} (\vp_{xx})^{2} \right)
dx, where the usual fields 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 , 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 we
find soliton-like solutions for all , moving with fixed shape and constant
velocity, . We show that the velocity, mass, and energy of the variational
travelling wave solutions are related by , where , independent of .\newline \newline PACS numbers: 03.40.Kf,
47.20.Ky, Nb, 52.35.SbComment: 16 pages. LaTeX. Figures available upon request (Postscript or hard
copy
A Continuum Description of Rarefied Gas Dynamics (I)--- Derivation From Kinetic Theory
We describe an asymptotic procedure for deriving continuum equations from the
kinetic theory of a simple gas. As in the works of Hilbert, of Chapman and of
Enskog, we expand in the mean flight time of the constituent particles of the
gas, but we do not adopt the Chapman-Enskog device of simplifying the formulae
at each order by using results from previous orders. In this way, we are able
to derive a new set of fluid dynamical equations from kinetic theory, as we
illustrate here for the relaxation model for monatomic gases. We obtain a
stress tensor that contains a dynamical pressure term (or bulk viscosity) that
is process-dependent and our heat current depends on the gradients of both
temperature and density. On account of these features, the equations apply to a
greater range of Knudsen number (the ratio of mean free path to macroscopic
scale) than do the Navier-Stokes equations, as we see in the accompanying
paper. In the limit of vanishing Knudsen number, our equations reduce to the
usual Navier-Stokes equations with no bulk viscosity.Comment: 16 page
Logarithmically Slow Expansion of Hot Bubbles in Gases
We report logarithmically slow expansion of hot bubbles in gases in the
process of cooling. A model problem first solved, when the temperature has
compact support. Then temperature profile decaying exponentially at large
distances is considered. The periphery of the bubble is shown to remain
essentially static ("glassy") in the process of cooling until it is taken over
by a logarithmically slowly expanding "core". An analytical solution to the
problem is obtained by matched asymptotic expansion. This problem gives an
example of how logarithmic corrections enter dynamic scaling.Comment: 4 pages, 1 figur
Symmetries of a class of nonlinear fourth order partial differential equations
In this paper we study symmetry reductions of a class of nonlinear fourth
order partial differential equations \be u_{tt} = \left(\kappa u + \gamma
u^2\right)_{xx} + u u_{xxxx} +\mu u_{xxtt}+\alpha u_x u_{xxx} + \beta u_{xx}^2,
\ee where , , , and are constants. This
equation may be thought of as a fourth order analogue of a generalization of
the Camassa-Holm equation, about which there has been considerable recent
interest. Further equation (1) is a ``Boussinesq-type'' equation which arises
as a model of vibrations of an anharmonic mass-spring chain and admits both
``compacton'' and conventional solitons. A catalogue of symmetry reductions for
equation (1) is obtained using the classical Lie method and the nonclassical
method due to Bluman and Cole. In particular we obtain several reductions using
the nonclassical method which are no} obtainable through the classical method
Singularites in the Bousseneq equation and in the generalized KdV equation
In this paper, two kinds of the exact singular solutions are obtained by the
improved homogeneous balance (HB) method and a nonlinear transformation. The
two exact solutions show that special singular wave patterns exists in the
classical model of some nonlinear wave problems
- …