1,124 research outputs found
Asymptotic dynamics of short-waves in nonlinear dispersive models
The multiple-scale perturbation theory, well known for long-waves, is
extended to the study of the far-field behaviour of short-waves, commonly
called ripples. It is proved that the Benjamin-Bona-Mahony- Peregrine equation
can propagates short-waves. This result contradict the Benjamin hypothesis that
short-waves tends not to propagate in this model and close a part of the old
controversy between Korteweg-de Vries and Benjamin-Bona-Mahony-Peregrine
equations. We shown that a nonlinear (quadratic) Klein-Gordon type equation
substitutes in a short-wave analysis the ubiquitous Korteweg-de Vries equation
of long-wave approach. Moreover the kink solutions of phi-4 and sine-Gordon
equations are understood as an all orders asymptotic behaviour of short-waves.
It is proved that the antikink solution of phi-4 model which was never obtained
perturbatively can be obtained by perturbation expansion in the wave-number k
in the short-wave limit.Comment: to appears in Physical Review E. 4 pages, revtex file
Vector Nonlinear Klein-Gordon Lattices: General Derivation of Small Amplitude Envelope Soliton Solutions
Group velocity and group velocity dispersion for a wave packet in vectorial
discrete Klein-Gordon models are obtained by an expansion, based on
perturbation theory, of the linear system giving the dispersion relation and
the normal modes.
We show how to map this expansion on the Multiple Scale Expansion in the real
space and how to find Non Linear Schr\"odinger small amplitude solutions when a
nonlinear one site potential balances the group velocity dispersion effect
On the evolution of scattering data under perturbations of the Toda lattice
We present the results of an analytical and numerical study of the long-time
behavior for certain Fermi-Pasta-Ulam (FPU) lattices viewed as perturbations of
the completely integrable Toda lattice. Our main tools are the direct and
inverse scattering transforms for doubly-infinite Jacobi matrices, which are
well-known to linearize the Toda flow. We focus in particular on the evolution
of the associated scattering data under the perturbed vs. the unperturbed
equations. We find that the eigenvalues present initially in the scattering
data converge to new, slightly perturbed eigenvalues under the perturbed
dynamics of the lattice equation. To these eigenvalues correspond solitary
waves that emerge from the solitons in the initial data. We also find that new
eigenvalues emerge from the continuous spectrum as the lattice system is let to
evolve under the perturbed dynamics.Comment: 27 pages, 17 figures. Revised Introduction and Discussion section
A system of ODEs for a Perturbation of a Minimal Mass Soliton
We study soliton solutions to a nonlinear Schrodinger equation with a
saturated nonlinearity. Such nonlinearities are known to possess minimal mass
soliton solutions. We consider a small perturbation of a minimal mass soliton,
and identify a system of ODEs similar to those from Comech and Pelinovsky
(2003), which model the behavior of the perturbation for short times. We then
provide numerical evidence that under this system of ODEs there are two
possible dynamical outcomes, which is in accord with the conclusions of
Pelinovsky, Afanasjev, and Kivshar (1996). For initial data which supports a
soliton structure, a generic initial perturbation oscillates around the stable
family of solitons. For initial data which is expected to disperse, the finite
dimensional dynamics follow the unstable portion of the soliton curve.Comment: Minor edit
Solving linear and nonlinear klein-gordon equations by new perturbation iteration transform method
We present an effective algorithm to solve the Linear and Nonlinear KleinGordon equation, which is based on the Perturbation Iteration Transform Method (PITM). The Klein-Gordon equation is the name given to the equation of motion of a quantum scalar or pseudo scalar field, a field whose quanta are spin-less particles. It describes the quantum amplitude for finding a point particle in various places, the relativistic wave function, but the particle propagates both forwards and backwards in time. The Perturbation Iteration Transform Method (PITM) is a combined form of the Laplace Transform Method and Perturbation Iteration Algorithm. The method provides the solution in the form of a rapidly convergent series. Some numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. The results show that the PITM is very efficient, simple and can be applied to other nonlinear problems.Publisher's Versio
Hylomorphic solitons
This paper is devoted to the study of solitary waves and solitons whose
existence is related to the ratio energy/charge. These solitary waves are
called hylomorphic. This class includes the Q-balls, which are spherically
symmetric solutions of the nonlinear Klein-Gordon equation (NKG), as well as
solitary waves and vortices which occur, by the same mechanism, in the
nonlinear Schroedinger equation and in gauge theories. This paper is devoted to
the study of hylomorphic soliton. Mainly we will be interested in the very
general principles which are at the base of their existence such as the
Variational Principle, the Invariance Principle, the Noether theorem, the
Hamilton-Jacobi theory etc.
We give a general definition of hylomorphic solitons and an interpretation of
their nature (swarm interpretation) which is very helpful in understanding
their behavior.
We apply these ideas to the Nonlinear Schroedinger Equation (NS) and to the
Nonlinear Klein-Gordon Equation (NKG) repectively
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