245 research outputs found
Nonlinear wave propagation through cold plasma
Electromagnetic wave propagation through cold collision free plasma is
studied using the nonlinear perturbation method. It is found that the equations
can be reduced to the modified Kortweg-de Vries equation
Discrete Reductive Perturbation Technique
We expand a partial difference equation (PE) on multiple lattices and
obtain the PE which governs its far field behaviour. The
perturbative--reductive approach is here performed on well known nonlinear
PEs, both integrable and non integrable. We study the cases of the
lattice modified Korteweg--de Vries (mKdV) equation, the Hietarinta equation,
the lattice Volterra--Kac--Van Moerbeke (VKVM) equation and a non integrable
lattice KdV equation. Such reductions allow us to obtain many new PEs
of the nonlinear Schr\"odinger (NLS) type.Comment: 18 pages, 1 figure. submitted to Journal of Mathematical Physic
Multiscale expansion and integrability properties of the lattice potential KdV equation
We apply the discrete multiscale expansion to the Lax pair and to the first
few symmetries of the lattice potential Korteweg-de Vries equation. From these
calculations we show that, like the lowest order secularity conditions give a
nonlinear Schroedinger equation, the Lax pair gives at the same order the
Zakharov and Shabat spectral problem and the symmetries the hierarchy of point
and generalized symmetries of the nonlinear Schroedinger equation.Comment: 10 pages, contribution to the proceedings of the NEEDS 2007
Conferenc
Renormalization Group Method and Reductive Perturbation Method
It is shown that the renormalization group method does not necessarily
eliminate all secular terms in perturbation series to partial differential
equations and a functional subspace of renormalizable secular solutions
corresponds to a choice of scales of independent variables in the reductive
perturbation method.Comment: 5 pages, late
Supersonic Discrete Kink-Solitons and Sinusoidal Patterns with "Magic" wavenumber in Anharmonic Lattices
The sharp pulse method is applied to Fermi-Pasta-Ulam (FPU) and Lennard-Jones
(LJ) anharmonic lattices. Numerical simulations reveal the presence of high
energy strongly localized ``discrete'' kink-solitons (DK), which move with
supersonic velocities that are proportional to kink amplitudes. For small
amplitudes, the DK's of the FPU lattice reduce to the well-known ``continuous''
kink-soliton solutions of the modified Korteweg-de Vries equation. For high
amplitudes, we obtain a consistent description of these DK's in terms of
approximate solutions of the lattice equations that are obtained by restricting
to a bounded support in space exact solutions with sinusoidal pattern
characterized by the ``magic'' wavenumber . Relative displacement
patterns, velocity versus amplitude, dispersion relation and exponential tails
found in numerical simulations are shown to agree very well with analytical
predictions, for both FPU and LJ lattices.Comment: Europhysics Letters (in print
Kink Solution in a Fluid Model of Traffic Flows
Traffic jam in a fluid model of traffic flows proposed by Kerner and
Konh\"auser (B. S. Kerner and P. Konh\"auser, Phys. Rev. E 52 (1995), 5574.) is
analyzed. An analytic scaling solution is presented near the critical point of
the hetero-clinic bifurcation. The validity of the solution has been confirmed
from the comparison with the simulation of the model.Comment: RevTeX v3.1, 6 pages, and 2 figure
Multiscale reduction of discrete nonlinear Schroedinger equations
We use a discrete multiscale analysis to study the asymptotic integrability
of differential-difference equations. In particular, we show that multiscale
perturbation techniques provide an analytic tool to derive necessary
integrability conditions for two well-known discretizations of the nonlinear
Schroedinger equation.Comment: 12 page
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
Multiscale Analysis of Discrete Nonlinear Evolution Equations
The method of multiscale analysis is constructed for dicrete systems of
evolution equations for which the problem is that of the far behavior of an
input boundary datum. Discrete slow space variables are introduced in a general
setting and the related finite differences are constructed. The method is
applied to a series of representative examples: the Toda lattice, the nonlinear
Klein-Gordon chain, the Takeno system and a discrete version of the
Benjamin-Bona-Mahoney equation. Among the resulting limit models we find a
discrete nonlinear Schroedinger equation (with reversed space-time), a 3-wave
resonant interaction system and a discrete modified Volterra model.Comment: published in J. Phys. A : Math. Gen. 32 (1999) 927-94
Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV
We consider multiple lattices and functions defined on them. We introduce
slow varying conditions for functions defined on the lattice and express the
variation of a function in terms of an asymptotic expansion with respect to the
slow varying lattices.
We use these results to perform the multiple--scale reduction of the lattice
potential Korteweg--de Vries equation.Comment: 17 pages. 1 figur
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