Dispersive shock waves and modulation theory

Abstract

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 Schr\"{o}dinger 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.Comment: review article, 68 pages, 52 figure

Experimental Generation of Riemann Waves in Optics: A Route to Shock Wave Control

“…In shock-bearing systems, the Riemann simple waves which allow describing the dynamics in terms of a single evolving Riemann invariant are crucial for understanding shock dynamics in different areas [5][6][7] and for the formulation of a modulation theory [8,9]. Yet Riemann waves (RWs) and their importance in terms of applications have been so far overlooked.…”

“…In essence, until the shock point, the profiles obtained from the NLSE or the IBE are indistinguishable, illustrating how the Riemann pulse maintains its proportionality between chirp and amplitude. For z ≥ 500 m, however, the IBE approximation loses validity due to the increasing effect of dispersive regularization: Although unnoticeable in the power profile of the pulse (which lays on a null background [36,37]), chirp oscillations develop on its trailing edge as a typical signature of dispersive shock wave formation [9,14,32,33,[38][39][40].…”

Riemann problem for the photon fluid: Self-steepening effects

“…Now these two pulses have different profiles and propagate with different group velocities. This is manifestation of lack of the time inversion invariance mentioned in the introduction, which is caused by the last term in the mNLS equation (1). It should be noted that the asymptotic solution (11) describes well the wave packet even for not very large x.…”

“…[1]). In the fiber optics applications, the dynamics of pulses is described usually by the nonlinear Schrödinger (NLS) equation that accounts for two main effects-quadratic normal dispersion and Kerr nonlinearity.…”

“…The weakly nonlinear waves are discussed in section III where we show that in weakly nonlinear case these two modes obey either to KdV or mKdV equation what results in very different their behavior. In section IV we obtain the periodic solutions to equation (1) by the finite-gap integration method which yields these solutions in the form convenient for applications in the Whitham theory of modulations and the Whitham equations are also derived in section IV. In section V we describe the elementary wave structures that appear as building blocks in the general wave patterns.…”