1,653 research outputs found
Chiral Solitons in Generalized Korteweg-de Vries Equations
Generalizations of the Korteweg-de Vries equation are considered, and some
explicit solutions are presented. There are situations where solutions engender
the interesting property of being chiral, that is, of having velocity
determined in terms of the parameters that define the generalized equation,
with a definite sign.Comment: 9 pages, latex, no figures. References added, typos correcte
Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials
We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices,
consisting of a chain of particles coupled by fractional power nonlinearities
of order . This class of systems incorporates a classical Hertzian
model describing acoustic wave propagation in chains of touching beads in the
absence of precompression. We analyze the propagation of localized waves when
is close to unity. Solutions varying slowly in space and time are
searched with an appropriate scaling, and two asymptotic models of the chain of
particles are derived consistently. The first one is a logarithmic KdV
equation, and possesses linearly orbitally stable Gaussian solitary wave
solutions. The second model consists of a generalized KdV equation with
H\"older-continuous fractional power nonlinearity and admits compacton
solutions, i.e. solitary waves with compact support. When , we numerically establish the asymptotically Gaussian shape of exact FPU
solitary waves with near-sonic speed, and analytically check the pointwise
convergence of compactons towards the limiting Gaussian profile
Long-Time Dynamics of Variable Coefficient mKdV Solitary Waves
We study the Korteweg-de Vries-type equation dt u=-dx(dx^2 u+f(u)-B(t,x)u),
where B is a small and bounded, slowly varying function and f is a
nonlinearity. Many variable coefficient KdV-type equations can be rescaled into
this equation. We study the long time behaviour of solutions with initial
conditions close to a stable, B=0 solitary wave. We prove that for long time
intervals, such solutions have the form of the solitary wave, whose centre and
scale evolve according to a certain dynamical law involving the function
B(t,x), plus an H^1-small fluctuation.Comment: 19 page
Modulational Instability in Equations of KdV Type
It is a matter of experience that nonlinear waves in dispersive media,
propagating primarily in one direction, may appear periodic in small space and
time scales, but their characteristics --- amplitude, phase, wave number, etc.
--- slowly vary in large space and time scales. In the 1970's, Whitham
developed an asymptotic (WKB) method to study the effects of small
"modulations" on nonlinear periodic wave trains. Since then, there has been a
great deal of work aiming at rigorously justifying the predictions from
Whitham's formal theory. We discuss recent advances in the mathematical
understanding of the dynamics, in particular, the instability of slowly
modulated wave trains for nonlinear dispersive equations of KdV type.Comment: 40 pages. To appear in upcoming title in Lecture Notes in Physic
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations
In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of
the focusing nonlinear Schr\"odinger (NLS), focusing modified Korteweg-de Vries
(mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and
breather solutions, they demonstrate the lack of local well-posedness for these
equations below their respective endpoint regularities. In this paper, we study
the defocusing analogues of these equations, namely defocusing NLS, defocusing
mKdV, and real KdV, all in one spatial dimension, for which suitable soliton
and breather solutions are unavailable. We construct for each of these
equations classes of modified scattering solutions, which exist globally in
time, and are asymptotic to solutions of the corresponding linear equations up
to explicit phase shifts. These solutions are used to demonstrate lack of local
well-posedness in certain Sobolev spaces,in the sense that the dependence of
solutions upon initial data fails to be uniformly continuous. In particular, we
show that the mKdV flow is not uniformly continuous in the topology,
despite the existence of global weak solutions at this regularity.
Finally, we investigate the KdV equation at the endpoint regularity
, and construct solutions for both the real and complex KdV
equations. The construction provides a nontrivial time interval and a
locally Lipschitz continuous map taking the initial data in to a
distributional solution H^{-3/4})$ which is uniquely
defined for all smooth data. The proof uses a generalized Miura transform to
transfer the existing endpoint regularity theory for mKdV to KdV.Comment: minor edit
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