22 research outputs found
Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions
The small dispersion limit of solutions to the Camassa-Holm (CH) equation is
characterized by the appearance of a zone of rapid modulated oscillations. An
asymptotic description of these oscillations is given, for short times, by the
one-phase solution to the CH equation, where the branch points of the
corresponding elliptic curve depend on the physical coordinates via the Whitham
equations. We present a conjecture for the phase of the asymptotic solution. A
numerical study of this limit for smooth hump-like initial data provides strong
evidence for the validity of this conjecture. We present a quantitative
numerical comparison between the CH and the asymptotic solution. The dependence
on the small dispersion parameter is studied in the interior and at
the boundaries of the Whitham zone. In the interior of the zone, the difference
between CH and asymptotic solution is of the order , at the trailing
edge of the order and at the leading edge of the order
. For the latter we present a multiscale expansion which
describes the amplitude of the oscillations in terms of the Hastings-McLeod
solution of the Painlev\'e II equation. We show numerically that this
multiscale solution provides an enhanced asymptotic description near the
leading edge.Comment: 25 pages, 15 figure
Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions
We study numerically the small dispersion limit for the Korteweg-de Vries
(KdV) equation for and give a
quantitative comparison of the numerical solution with various asymptotic
formulae for small in the whole -plane. The matching of the
asymptotic solutions is studied numerically
Numerical study of fractional Camassa-Holm equations
A numerical study of fractional Camassa-Holm equations is presented. Smooth
solitary waves are constructed numerically. Their stability is studied as well
as the long time behavior of solutions for general localised initial data from
the Schwartz class of rapidly decreasing functions. The appearence of
dispersive shock waves is explored
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