543 research outputs found
An energy-based stability criterion for solitary traveling waves in Hamiltonian lattices
In this work, we revisit a criterion, originally proposed in [Nonlinearity
{\bf 17}, 207 (2004)], for the stability of solitary traveling waves in
Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the
implications of this criterion from the point of view of stability theory, both
at the level of the spectral analysis of the advance-delay differential
equations in the co-traveling frame, as well as at that of the Floquet problem
arising when considering the traveling wave as a periodic orbit modulo a shift.
We establish the correspondence of these perspectives for the pertinent
eigenvalue and Floquet multiplier and provide explicit expressions for their
dependence on the velocity of the traveling wave in the vicinity of the
critical point. Numerical results are used to corroborate the relevant
predictions in two different models, where the stability may change twice. Some
extensions, generalizations and future directions of this investigation are
also discussed
On the Stability of Periodic Solutions of the Generalized Benjamin-Bona-Mahony Equation
We study the stability of a four parameter family of spatially periodic
traveling wave solutions of the generalized Benjamin-Bona-Mahony equation to
two classes of perturbations: periodic perturbations with the same periodic
structure as the underlying wave, and long-wavelength localized perturbations.
In particular, we derive necessary conditions for spectral instability to
perturbations to both classes of perturbations by deriving appropriate
asymptotic expansions of the periodic Evans function, and we outline a
nonlinear stability theory to periodic perturbations based on variational
methods which effectively extends our periodic spectral stability results.Comment: 27 pages, 3 figure
Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation
In this paper, we investigate the spectral instability of periodic traveling
wave solutions of the generalized Korteweg-de Vries equation to long wavelength
transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By
analyzing high and low frequency limits of the appropriate periodic Evans
function, we derive an orientation index which yields sufficient conditions for
such an instability to occur. This index is geometric in nature and applies to
arbitrary periodic traveling waves with minor smoothness and convexity
assumptions on the nonlinearity. Using the integrable structure of the ordinary
differential equation governing the traveling wave profiles, we are then able
to calculate the resulting orientation index for the elliptic function
solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations.Comment: 26 pages. Sign error corrected in Lemma 3. Statement of main theorem
corrected. Exposition updated and references added
Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg-de Vries Equation
In this paper, we study the orbital stability for a four-parameter family of
periodic stationary traveling wave solutions to the generalized Korteweg-de
Vries equation. In particular, we derive sufficient conditions for such a
solution to be orbitally stable in terms of the Hessian of the classical action
of the corresponding traveling wave ordinary differential equation restricted
to the manifold of periodic traveling wave solution. We show this condition is
equivalent to the solution being spectrally stable with respect to
perturbations of the same period in the case of the Korteweg-de Vries equation,
and in neighborhoods of the homoclinic and equilibrium solutions in the case of
a power-law nonlinearity.Comment: 24 page
Instability and stability properties of traveling waves for the double dispersion equation
In this article we are concerned with the instability and stability
properties of traveling wave solutions of the double dispersion equation
for ,
. The main characteristic of this equation is the existence of two
sources of dispersion, characterized by the terms and . We
obtain an explicit condition in terms of , and on wave velocities
ensuring that traveling wave solutions of the double dispersion equation are
strongly unstable by blow up. In the special case of the Boussinesq equation
(), our condition reduces to the one given in the literature. For the
double dispersion equation, we also investigate orbital stability of traveling
waves by considering the convexity of a scalar function. We provide both
analytical and numerical results on the variation of the stability region of
wave velocities with , and and then state explicitly the conditions
under which the traveling waves are orbitally stable.Comment: 16 pages, 4 figure
Instability and stability properties of traveling waves for the double dispersion equation
In this article we are concerned with the instability and stability
properties of traveling wave solutions of the double dispersion equation
for ,
. The main characteristic of this equation is the existence of two
sources of dispersion, characterized by the terms and . We
obtain an explicit condition in terms of , and on wave velocities
ensuring that traveling wave solutions of the double dispersion equation are
strongly unstable by blow up. In the special case of the Boussinesq equation
(), our condition reduces to the one given in the literature. For the
double dispersion equation, we also investigate orbital stability of traveling
waves by considering the convexity of a scalar function. We provide both
analytical and numerical results on the variation of the stability region of
wave velocities with , and and then state explicitly the conditions
under which the traveling waves are orbitally stable.Comment: 16 pages, 4 figure
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