29 research outputs found
Dispersive shock waves and modulation theory
There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G. B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg–de Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs
Hydrodynamic optical soliton tunneling
A conceptually new notion of hydrodynamic optical soliton tunneling is introduced in which
a dark soliton is incident upon an evolving, broad potential barrier that arises from an appropriate variation of the input signal. The barriers considered include smooth rarefaction waves and highly oscillatory dispersive shock waves. Both the soliton and the barrier satisfy the same one-dimensional defocusing nonlinear Schrodinger (NLS) equation, which admits a convenient dispersive hydrodynamic interpretation. Under the scale separation assumption of nonlinear wave
(Whitham) modulation theory, the highly nontrivial nonlinear interaction between the soliton and the evolving hydrodynamic barrier is described in terms of self-similar, simple wave solutions to an asymptotic reduction of the Whitham-NLS partial differential equations. One of the Riemann invariants of the reduced modulation system determines the characteristics of a soliton interacting
with a mean flow that results in soliton tunneling or trapping. Another Riemann invariant yields the tunneled soliton's phase shift due to hydrodynamic interaction. Under certain conditions, soliton interaction with hydrodynamic barriers gives rise to new effects that include reversal of the soliton propagation direction and spontaneous soliton cavitation, which further suggest possible methods of dark soliton control in optical fibers
Oblique spatial dispersive shock waves in nonlinear Schrodinger flows
In dispersive media, hydrodynamic singularities are resolved by coherent wavetrains known as dispersive shock waves (DSWs). Only dynamically expanding, temporal DSWs are possible in one-dimensional media. The additional degree of freedom inherent in two-dimensional media allows for the generation of time-independent DSWs that exhibit spatial expansion. Spatial oblique DSWs, dispersive analogs of oblique shocks in classical media, are constructed utilizing Whitham modulation theory for a class of nonlinear Schrodinger boundary value problems. Self-similar, simple wave solutions of the modulation equations yield relations between the DSW’s orientation and the
upstream/downstream flow fields. Time dependent numerical simulations demonstrate a convective or absolute instability of oblique DSWs in supersonic flow over obstacles. The convective instability results in an effective stabilization of the DSW
Stationary expansion shocks for a regularized Boussinesq system
Stationary expansion shocks have been recently identified as a new type of solution to hyperbolic conservation laws regularized by non-local dispersive terms that naturally arise in shallow-water theory. These expansion shocks were studied in [1] for the Benjamin-Bona-Mahony equation using matched asymptotic expansions. In this paper, we extend the analysis of [1] to the regularized Boussinesq system by using Riemann invariants of the underlying dispersionless shallow water equations. The extension for a system is non-trivial, requiring a combination of small amplitude, long-wave expansions with high order matched asymptotics. The constructed asymptotic solution is shown to be in excellent agreement with
accurate numerical simulations of the Boussinesq system for a range of appropriately smoothed Riemann data
Expansion shock waves in regularised shallow water theory
We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularised shallow water equations that include the Benjamin-Bona-Mahoney (BBM) and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock’s existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock
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
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
Propagating two-dimensional magnetic droplets
Propagating, solitary magnetic wave solutions of the Landau-Lifshitz equation
with uniaxial, easy-axis anisotropy in thin (two-dimensional) magnetic films
are investigated. These localized, nontopological wave structures, parametrized
by their precessional frequency and propagation speed, extend the stationary,
coherently precessing "magnon droplet" to the moving frame, a non-trivial
generalization due to the lack of Galilean invariance. Propagating droplets
move on a spin wave background with a nonlinear droplet dispersion relation
that yields a limited range of allowable droplet speeds and frequencies. An
iterative numerical technique is used to compute the propagating droplet's
structure and properties. The results agree with previous asymptotic
calculations in the weakly nonlinear regime. Furthermore, an analytical
criterion for the droplet's orbital stability is confirmed. Time-dependent
numerical simulations further verify the propagating droplet's robustness to
perturbation when its frequency and speed lie within the allowable range.Comment: 16 pages, 11 figure
Toll-Like Receptor induced CD11b and L-selectin response in patients with coronary artery disease
Toll-Like Receptor (TLR) -2 and -4 expression and TLR-induced cytokine response of inflammatory cells are related to atherogenesis and atherosclerotic plaque progression. We examined whether immediate TLR induced changes in CD11b and L-selectin (CD62L) expression are able to discriminate the presence and severity of atherosclerotic disease by exploring single dose whole blood TLR stimulation and detailed dose-response curves. Blood samples were obtained from 125 coronary artery disease (CAD) patients and 28 controls. CD11b and L-selectin expression on CD14+ monocytes was measured after whole blood stimulation with multiple concentrations of the TLR4 ligand LPS (0.01-10 ng/ml) and the TLR2 ligand P3C (0.5-500 ng/ml). Subsequently, dose-response curves were created and the following parameters were calculated: hillslope, EC50, area under the curve (AUC) and delta. These parameters provide information about the maximum response following activation, as well as the minimum trigger required to induce activation and the intensity of the response. CAD patients showed a significantly higher L-selectin, but not CD11b response to TLR ligation than controls after single dose stimulations as well as significant differences in the hillslope and EC50 of the dose-response curves. Within the CAD patient group, dose-response curves of L-selectin showed significant differences in the presence of hypertension, dyslipidemia, coronary occlusion and degree of stenosis, whereas CD11b expression had the strongest discriminating power after single dose stimulation. In conclusion, single dose stimulations and dose-response curves of CD11b and L-selectin expression after TLR stimulation provide diverse but limited information about atherosclerotic disease severity in stable angina patients. However, both single dose stimulation and dose-response curves of LPS-induced L-selectin expression can discriminate between controls and CAD patients.Biopharmaceutic
Dispersive hydrodynamics: Preface
This paper was accepted for publication in the journal Physica D: Nonlinear Phenomena and the definitive published version is available at http://dx.doi.org/10.1016/j.physd.2016.07.002.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