4,127 research outputs found
Self-similar Radiation from Numerical Rosenau-Hyman Compactons
The numerical simulation of compactons, solitary waves with compact support,
is characterized by the presence of spurious phenomena, as numerically-induced
radiation, which is illustrated here using four numerical methods applied to
the Rosenau-Hyman K(p,p) equation. Both forward and backward radiations are
emitted from the compacton presenting a self-similar shape which has been
illustrated graphically by the proper scaling. A grid refinement study shows
that the amplitude of the radiations decreases as the grid size does,
confirming its numerical origin. The front velocity and the amplitude of both
radiations have been studied as a function of both the compacton and the
numerical parameters. The amplitude of the radiations decreases exponentially
in time, being characterized by a nearly constant scaling exponent. An ansatz
for both the backward and forward radiations corresponding to a self-similar
function characterized by the scaling exponent is suggested by the present
numerical results.Comment: To be published in Journal of Computational Physic
Numerical Solitons of Generalized Korteweg-de Vries Equations
We propose a numerical method for finding solitary wave solutions of
generalized Korteweg-de Vries equations by solving the nonlinear eigenvalue
problem on an unbounded domain. The artificial boundary conditions are obtained
to make the domain finite. We specially discuss the soliton solutions of the
K(m, n) equation and KdV-K(m,n) equation. Furthermore for the mixed models of
linear and nonlinear dispersion, the collision behaviors of soliton-soliton and
soliton-antisoliton are observed.Comment: 9 pages, 4 figure
Dissipative perturbations for the K(n,n) Rosenau-Hyman equation
Compactons are compactly supported solitary waves for nondissipative
evolution equations with nonlinear dispersion. In applications, these model
equations are accompanied by dissipative terms which can be treated as small
perturbations. We apply the method of adiabatic perturbations to compactons
governed by the K(n,n) Rosenau-Hyman equation in the presence of dissipative
terms preserving the "mass" of the compactons. The evolution equations for both
the velocity and the amplitude of the compactons are determined for some linear
and nonlinear dissipative terms: second-, fourth-, and sixth-order in the
former case, and second- and fourth-order in the latter one. The numerical
validation of the method is presented for a fourth-order, linear, dissipative
perturbation which corresponds to a singular perturbation term
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
Exact solutions of the generalized equations
Family of equations, which is the generalization of the equation, is
considered. Periodic wave solutions for the family of nonlinear equations are
constructed
The Camassa-Holm Equation: A Loop Group Approach
A map is presented that associates with each element of a loop group a
solution of an equation related by a simple change of coordinates to the
Camassa-Holm (CH) Equation. Certain simple automorphisms of the loop group give
rise to Backlund transformations of the equation. These are used to find
2-soliton solutions of the CH equation, as well as some novel singular
solutions.Comment: 19 pages, 7 figures; LaTeX with psfi
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
Exact solitary and periodic-wave solutions of the K(2,2) equation (defocusing branch)
An auxiliary elliptic equation method is presented for constructing exact solitary and periodic travelling-wave solutions of the K(2, 2) equation (defocusing branch). Some known results in the literature are recovered more efficiently, and some new exact travelling-wave solutions are obtained. Also, new stationary-wave solutions are obtained
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