931 research outputs found
On the inelastic 2-soliton collision for gKdV equations with general nonlinearity
We study the problem of 2-soliton collision for the generalized Korteweg-de
Vries equations, completing some recent works of Y. Martel and F. Merle. We
classify the nonlinearities for which collisions are elastic or inelastic. Our
main result states that in the case of small solitons, with one soliton smaller
than the other one, the unique nonlinearities allowing a perfectly elastic
collision are precisely the integrable cases, namely the quadratic (KdV), cubic
(mKdV) and Gardner nonlinearities.Comment: 54 pages, submitted; corrected a gap in a previous computatio
On the soliton dynamics under a slowly varying medium for generalized KdV equations
We consider the problem of the soliton propagation, in a slowly varying
medium, for a generalized Korteweg - de Vries equations (gKdV). We study the
effects of inhomogeneities on the dynamics of a standard soliton. We prove that
slowly varying media induce on the soliton solution large dispersive effects at
large time. Moreover, unlike gKdV equations, we prove that there is no
pure-soliton solution in this regime.Comment: Submitted. Corrected some typos, updated reference
Nonlinear stability of mKdV breathers
Breather solutions of the modified Korteweg-de Vries equation are shown to be
globally stable in a natural H^2 topology. Our proof introduces a new Lyapunov
functional, at the H^2 level, which allows to describe the dynamics of small
perturbations, including oscillations induced by the periodicity of the
solution, as well as a direct control of the corresponding instability modes.
In particular, degenerate directions are controlled using low-regularity
conservation laws.Comment: 24 pp., submitte
Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers
We study the long-time dynamics of complex-valued modified Korteweg-de Vries
(mKdV) solitons, which are recognized because they blow-up in finite time. We
establish stability properties at the H^1 level of regularity, uniformly away
from each blow-up point. These new properties are used to prove that mKdV
breathers are H^1 stable, improving our previous result, where we only proved
H^2 stability. The main new ingredient of the proof is the use of a B\"acklund
transformation which links the behavior of breathers, complex-valued solitons
and small real-valued solutions of the mKdV equation. We also prove that
negative energy breathers are asymptotically stable. Since we do not use any
method relying on the Inverse Scattering Transformation, our proof works even
under rough perturbations, provided a corresponding local well-posedness theory
is available.Comment: 45 pages, we thank Yvan Martel for pointing us a gap in the previous
version of this pape
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