129 research outputs found
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
Refraction of dispersive shock waves
We study a dispersive counterpart of the classical gas dynamics problem of
the interaction of a shock wave with a counter-propagating simple rarefaction
wave often referred to as the shock wave refraction. The refraction of a
one-dimensional dispersive shock wave (DSW) due to its head-on collision with
the centred rarefaction wave (RW) is considered in the framework of defocusing
nonlinear Schr\"odinger (NLS) equation. For the integrable cubic nonlinearity
case we present a full asymptotic description of the DSW refraction by
constructing appropriate exact solutions of the Whitham modulation equations in
Riemann invariants. For the NLS equation with saturable nonlinearity, whose
modulation system does not possess Riemann invariants, we take advantage of the
recently developed method for the DSW description in non-integrable dispersive
systems to obtain main physical parameters of the DSW refraction. The key
features of the DSW-RW interaction predicted by our modulation theory analysis
are confirmed by direct numerical solutions of the full dispersive problem.Comment: 45 pages, 23 figures, minor revisio
Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
We consider two physically and mathematically distinct regularization
mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the
combination of diffusion and dispersion are known to give rise to monotonic and
oscillatory traveling waves that approximate shock waves. The zero-diffusion
limits of these traveling waves are dynamically expanding dispersive shock
waves (DSWs). A richer set of wave solutions can be found when the flux is
non-convex. This review compares the structure of solutions of Riemann problems
for a conservation law with non-convex, cubic flux regularized by two different
mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation;
and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation.
In the first case, the possible dynamics involve two qualitatively different
types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the
second case, in addition to RWs, there are traveling wave solutions
approximating both classical (Lax) and non-classical (undercompressive) shock
waves. Despite the singular nature of the zero-diffusion limit and rather
differing analytical approaches employed in the descriptions of dispersive and
diffusive-dispersive regularization, the resulting comparison of the two cases
reveals a number of striking parallels. In contrast to the case of convex flux,
the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is
identified as an undercompressive DSW. Other prominent features, such as
shock-rarefactions, also find their purely dispersive counterparts involving
special contact DSWs, which exhibit features analogous to contact
discontinuities. This review describes an important link between two major
areas of applied mathematics, hyperbolic conservation laws and nonlinear
dispersive waves.Comment: Revision from v2; 57 pages, 19 figure
Dam break problem for the focusing nonlinear Schr\"odinger equation and the generation of rogue waves
We propose a novel, analytically tractable, scenario of the rogue wave
formation in the framework of the small-dispersion focusing nonlinear
Schr\"odinger (NLS) equation with the initial condition in the form of a
rectangular barrier (a "box"). We use the Whitham modulation theory combined
with the nonlinear steepest descent for the semi-classical inverse scattering
transform, to describe the evolution and interaction of two counter-propagating
nonlinear wave trains --- the dispersive dam break flows --- generated in the
NLS box problem. We show that the interaction dynamics results in the emergence
of modulated large-amplitude quasi-periodic breather lattices whose amplitude
profiles are closely approximated by the Akhmediev and Peregrine breathers
within certain space-time domain. Our semi-classical analytical results are
shown to be in excellent agreement with the results of direct numerical
simulations of the small-dispersion focusing NLS equation.Comment: 29 pages, 15 figures, major revisio
Unsteady undular bores in fully nonlinear shallow-water theory
We consider unsteady undular bores for a pair of coupled equations of
Boussinesq-type which contain the familiar fully nonlinear dissipationless
shallow-water dynamics and the leading-order fully nonlinear dispersive terms.
This system contains one horizontal space dimension and time and can be
systematically derived from the full Euler equations for irrotational flows
with a free surface using a standard long-wave asymptotic expansion.
In this context the system was first derived by Su and Gardner. It coincides
with the one-dimensional flat-bottom reduction of the Green-Naghdi system and,
additionally, has recently found a number of fluid dynamics applications other
than the present context of shallow-water gravity waves. We then use the
Whitham modulation theory for a one-phase periodic travelling wave to obtain an
asymptotic analytical description of an undular bore in the Su-Gardner system
for a full range of "depth" ratios across the bore. The positions of the
leading and trailing edges of the undular bore and the amplitude of the leading
solitary wave of the bore are found as functions of this "depth ratio". The
formation of a partial undular bore with a rapidly-varying finite-amplitude
trailing wave front is predicted for ``depth ratios'' across the bore exceeding
1.43. The analytical results from the modulation theory are shown to be in
excellent agreement with full numerical solutions for the development of an
undular bore in the Su-Gardner system.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9
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Dispersive hydrodynamics: Preface
This Special Issue on Dispersive Hydrodynamics is dedicated to the memory and work of G. B. Whitham who was one of
the pioneers in this field of physical applied mathematics. Some of the papers appearing here are related to work reported
on at the workshop \Dispersive Hydrodynamics: The Mathematics of Dispersive Shock Waves and Applications" held in
May 2015 at the Banff International Research Station. This Preface provides a broad overview of the field and summaries
of the various contributions to the Special Issue, placing them in a unified context
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