33,669 research outputs found
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
The theory of optical dispersive shock waves in photorefractive media
The theory of optical dispersive shocks generated in propagation of light
beams through photorefractive media is developed. Full one-dimensional
analytical theory based on the Whitham modulation approach is given for the
simplest case of sharp step-like initial discontinuity in a beam with
one-dimensional strip-like geometry. This approach is confirmed by numerical
simulations which are extended also to beams with cylindrical symmetry. The
theory explains recent experiments where such dispersive shock waves have been
observed.Comment: 26 page
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
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
Complex singularities and PDEs
In this paper we give a review on the computational methods used to
characterize the complex singularities developed by some relevant PDEs. We
begin by reviewing the singularity tracking method based on the analysis of the
Fourier spectrum. We then introduce other methods generally used to detect the
hidden singularities. In particular we show some applications of the Pad\'e
approximation, of the Kida method, and of Borel-Polya method. We apply these
techniques to the study of the singularity formation of some nonlinear
dispersive and dissipative one dimensional PDE of the 2D Prandtl equation, of
the 2D KP equation, and to Navier-Stokes equation for high Reynolds number
incompressible flows in the case of interaction with rigid boundaries
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