4,831 research outputs found
Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations
We consider the stability and instability of periodic travling waves for
Korteweg-de Vries type equations with fractional dispersion and other nonlinear
dispersive equations. We establish that a constrained minimizer for the related
variational problem is nonlinearly stable to period preserving perturbations.
We also discuss when the associated linearized equation admits exponentially
growing solutions. The proof utilizes the variational structure of the
equation.Comment: 27 pages. Missing assumption added to stability theorem. arXiv admin
note: text overlap with arXiv:1303.260
Numerical Study of Nonlinear Dispersive Wave Models with SpecTraVVave
In nonlinear dispersive evolution equations, the competing effects of
nonlinearity and dispersion make a number of interesting phenomena possible. In
the current work, the focus is on the numerical approximation of traveling-wave
solutions of such equations. We describe our efforts to write a dedicated
Python code which is able to compute traveling-wave solutions of nonlinear
dispersive equations of the general form \begin{equation*} u_t + [f(u)]_{x} +
\mathcal{L} u_x = 0, \end{equation*} where is a self-adjoint
operator, and is a real-valued function with .
The SpectraVVave code uses a continuation method coupled with a spectral
projection to compute approximations of steady symmetric solutions of this
equation. The code is used in a number of situations to gain an understanding
of traveling-wave solutions. The first case is the Whitham equation, where
numerical evidence points to the conclusion that the main bifurcation branch
features three distinct points of interest, namely a turning point, a point of
stability inversion, and a terminal point which corresponds to a cusped wave.
The second case is the so-called modified Benjamin-Ono equation where the
interaction of two solitary waves is investigated. It is found that is possible
for two solitary waves to interact in such a way that the smaller wave is
annihilated. The third case concerns the Benjamin equation which features two
competing dispersive operators. In this case, it is found that bifurcation
curves of periodic traveling-wave solutions may cross and connect high up on
the branch in the nonlinear regime
The Nikolaevskiy equation with dispersion
The Nikolaevskiy equation was originally proposed as a model for seismic
waves and is also a model for a wide variety of systems incorporating a
neutral, Goldstone mode, including electroconvection and reaction-diffusion
systems. It is known to exhibit chaotic dynamics at the onset of pattern
formation, at least when the dispersive terms in the equation are suppressed,
as is commonly the practice in previous analyses. In this paper, the effects of
reinstating the dispersive terms are examined. It is shown that such terms can
stabilise some of the spatially periodic traveling waves; this allows us to
study the loss of stability and transition to chaos of the waves. The secondary
stability diagram (Busse balloon) for the traveling waves can be remarkably
complicated.Comment: 24 pages; accepted for publication in Phys. Rev.
On the nonlinear dynamics of the traveling-wave solutions of the Serre system
We numerically study nonlinear phenomena related to the dynamics of traveling
wave solutions of the Serre equations including the stability, the persistence,
the interactions and the breaking of solitary waves. The numerical method
utilizes a high-order finite-element method with smooth, periodic splines in
space and explicit Runge-Kutta methods in time. Other forms of solutions such
as cnoidal waves and dispersive shock waves are also considered. The
differences between solutions of the Serre equations and the Euler equations
are also studied.Comment: 28 pages, 20 figures, 3 tables, 33 references. Other author's papers
can be downloaded at http://www.denys-dutykh.com
Space-modulated Stability and Averaged Dynamics
In this brief note we give a brief overview of the comprehensive theory,
recently obtained by the author jointly with Johnson, Noble and Zumbrun, that
describes the nonlinear dynamics about spectrally stable periodic waves of
parabolic systems and announce parallel results for the linearized dynamics
near cnoidal waves of the Korteweg-de Vries equation. The latter are expected
to contribute to the development of a dispersive theory, still to come.Comment: Proceedings of the "Journ\'ees \'Equations aux d\'eriv\'ees
partielles", Roscoff 201
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
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