Interactions of large amplitude solitary waves in viscous fluid conduits

The free interface separating an exterior, viscous fluid from an intrusive conduit of buoyant, less viscous fluid is known to support strongly nonlinear solitary waves due to a balance between viscosity-induced dispersion and buoyancy-induced nonlinearity. The overtaking, pairwise interaction of weakly nonlinear solitary waves has been classified theoretically for the Korteweg-de Vries equation and experimentally in the context of shallow water waves, but a theoretical and experimental classification of strongly nonlinear solitary wave interactions is lacking. The interactions of large amplitude solitary waves in viscous fluid conduits, a model physical system for the study of one-dimensional, truly dissipationless, dispersive nonlinear waves, are classified. Using a combined numerical and experimental approach, three classes of nonlinear interaction behavior are identified: purely bimodal, purely unimodal, and a mixed type. The magnitude of the dispersive radiation due to solitary wave interactions is quantified numerically and observed to be beyond the sensitivity of our experiments, suggesting that conduit solitary waves behave as "physical solitons." Experimental data are shown to be in excellent agreement with numerical simulations of the reduced model. Experimental movies are available with the online version of the paper.Comment: 13 pages, 4 figure

Burgers type equations, Gelfand's problem and Schumpeterian dynamics

Burgers equations have been introduced to study different models of fluids Bateman, 1915, Burgers, 1939, Hopf, 1950, Cole, 1951, Lighthill, Whitham, 1955.... The difference-differential analogs of these equations have been proposed for Schumpeterian models of economic development Iwai, 1984, Polterovich, Henkin, 1988, Belenky, 1990, 1996, Henkin, Polterovich, 1999, Shananin, Tashlitskaya, 2000.... This paper is a survey of recent results and conjectures on Burgers type equations, motivated both by fluid mechanics and by Schumpeterian dynamics. Abriged proofs of new results are given. This paper is an extended version of the paper [Henkin, 2011] prepared for the talk at the conference "General Equilibrium Analysis" at Higher School of Economics, June, 2011

Dispersive Riemann problems for the Benjamin-Bona-Mahony equation

Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin‐Bona‐Mahony (BBM) equation u t + u u x = u xxt are studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Korteweg‐de Vries equation u t + u u x + u xxx = 0 . The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompanied by the generation of two‐phase linear wavetrains, solitary wave shedding, and expansion shocks. Both narrow and broad initial widths give rise to two‐phase nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock for symmetric data. The dispersive Lax shock is described by an approximate self‐similar solution of the BBM equation whose limit as t → ∞ is a stationary, discontinuous weak solution. By introducing a slight asymmetry in the data for the dispersive Lax shock, the generation of an incoherent solitary wavetrain is observed. Further asymmetry leads to the DSW implosion regime that is effectively described by a pair of coupled nonlinear Schrödinger equations. The complex interplay between nonlocality, nonlinearity, and dispersion in the BBM equation underlies the rich variety of nonclassical dispersive hydrodynamic solutions to the dispersive Riemann problem

Whitham modulation theory and two-phase instabilities for generalized nonlinear Schr\"{o}dinger equations with full dispersion

The generalized nonlinear Schr\"odinger equation with full dispersion (FDNLS) is considered in the semiclassical regime. The Whitham modulation equations are obtained for the FDNLS equation with general linear dispersion and a generalized, local nonlinearity. Assuming the existence of a four-parameter family of two-phase solutions, a multiple-scales approach yields a system of four independent, first order, quasi-linear conservation laws of hydrodynamic type that correspond to the slow evolution of the two wavenumbers, mass, and momentum of modulated periodic traveling waves. The modulation equations are further analyzed in the dispersionless and weakly nonlinear regimes. The ill-posedness of the dispersionless equations corresponds to the classical criterion for modulational instability (MI). For modulations of linear waves, ill-posedness coincides with the generalized MI criterion, recently identified by Amiranashvili and Tobisch (New J. Phys. 21 (2019)). A new instability index is identified by the transition from real to complex characteristics for the weakly nonlinear modulation equations. This instability is associated with long-wavelength modulations of nonlinear two-phase wavetrains and can exist even when the corresponding one-phase wavetrain is stable according to the generalized MI criterion. Another interpretation is that, while infinitesimal perturbations of a periodic wave may not grow, small but finite amplitude perturbations may grow, hence this index identifies a nonlinear instability mechanism for one-phase waves. Classifications of instability indices for multiple FDNLS equations with higher order dispersion, including applications to finite depth water waves and the discrete NLS equation are presented and compared with direct numerical simulations.Comment: 26 pages, 7 figure

On the Dynamics of Noncircular Accretion Discs

The classical picture of an accretion disc is of a geometrically thin, nearly axisymmetric, fluid flow undergoing supersonic circular motion and slowly accreting due to angular momentum transport by the disc turbulence. There are, however, strong theoretical and observational grounds for considering non-circular motion in accretion discs. In this thesis I study the dynamics of such non-circular accretion discs using a mix of analytical and semi-analytical methods. One such example of a non-circular accretion disc is an eccentric disc, where the dominant fluid motion consists of slowly evolving, nested, confocal, Keplerian ellipses. I focus on the dynamics of these eccentric discs, along with the dynamical non-axisymmetric vertical structure that they set up. In the first part of this thesis I consider eccentric waves in ideal fluid discs. I present a secular Hamiltonian theory describing the evolution of the disc orbits due to pressure gradients and show that it can be used to calculate the eccentric standing wave patterns in the disc (the eccentric modes). I derive a ``short wavelength'' theory for nearly circular orbits that nevertheless can have substantially non-axisymmetric surface density/pressure due to nonlinear eccentricity gradients and disc twist. I use this to show that the pressure profile and precessional forces (such as general relativistic apsidal precession) can focus eccentric waves, causing them to become highly nonlinear in the inner disc. In the second part of this thesis I consider the action of non-ideal terms on the eccentric disc, such as excitation and damping by viscosity. I derive the ordinary differential equations describing simple horizontally invariant ``laminar flows'' in a local model of an eccentric disc, which can be used to study the disc's dynamical vertical structure. I also move beyond the purely hydrodynamic models of the preceding chapters and consider the action of magnetic fields in a local model. Finally I apply the theory presented in this thesis to the highly eccentric discs expected from the tidal disruption of a star by a supermassive black hole. It is currently an open question how, and indeed if, the discs in tidal disruption events circularise. As a step towards understanding the evolution of the disc orbits, I calculate the dynamical vertical structure of highly eccentric discs, emphasising the role of radiation pressure and thermal stability, and showing that magnetic fields may be important where the disc is highly compressed near the periapsis

The long-wave potential-vorticity dynamics of coastal fronts

This paper studies the propagation of free, long waves on a potential vorticity front in the presence of a vertical coast, using a -layer, quasi-geostrophic model with piecewise-constant potential vorticity. The coastal boundary induces flow through image vorticity and a Kelvin wave, either of which can reinforce or oppose the Rossby wave dynamics at the front. The behaviour of the front depends strongly on the relative strengths of these three mechanisms, which are explicit in our model. The richest behaviour, which includes kink solitons (under-compressive shocks) and compound-wave structures, occurs in the regime where vortical effects are dominant. The evolution of the front is described by a fully nonlinear finite-amplitude equation including first-order dispersive effects, which is related to the modified Korteweg–de Vries equation. The different behaviours are classified using the canonical example of the Riemann problem, which we analyse using El’s technique of ‘dispersive shock-fitting’. Contour-dynamic simulations show that the dispersive long-wave theory captures the behaviour of the full quasi-geostrophic system to a high degree of accuracy

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

Analysis and control of rogue waves in fibre lasers and in hydrodynamics: integrable turbulence framework

Understanding mechanisms underlying the formation of extreme events is the problem of primary importance in various domains of study including hydrodynamics, optics, forecasting natural disasters etc. In these domains, extreme events are known as RogueWaves (RWs). RWs are localised coherent structures of unusually large amplitude spontaneously emerging in nonlinear random wave fields, and as such, can have damaging effect on the environment (e.g. offshore engineering structures) or on the medium they propagate through (e.g. optical fibres). Within this PhD project several problems related to the emergence, control and manipulation of RWs in fibre optics and in hydrodynamics have been investigated. The particular emphasis is on the study of RWs emerging in the propagation of the so-called partially coherent waves described by the focusing nonlinear Schr¨odinger equation (fNLSE), the universal model for the propagation of modulationally unstable quasi-monochromatic wavepackets in a broad range of physical media. fNLSE belongs to the class of the completely integrable equations possessing deep mathematical structure and amenable to analytical methods such as Inverse Scattering Transform and Finite-Gap Integration. We use recent mathematical discoveries related to the semi classical, or small-dispersion, limit of fNLSE to investigate analytically, numerically and experimentally the formation of RWs within the framework of integrable turbulence—the emerging theory of random waves in integrable systems. The study of the RW formation in this project has also prompted the investigation of a closely related problem concerned with dynamics of soliton and breather gases as special types of integrable turbulence. The project’s findings fall in five categories: (i) the analytical description of the emergence of the so-called “heavy tails” in the probability distribution for the field intensity at the early stage of the development of integrable turbulence; (ii) the development and experimental realisation in a water tank of nonlinear spectral engineering, the IST-based method of control and manipulation of RWs; (iii) the development of the spectral theory of bidirectional soliton gases; (iv) numerical synthesis of breather gases and the verification of the recently developed spectral kinetic theory for such gases; (v) the investigation of the RWformation in the compression of broad optical pulses in the highly nonlinear propagation regimes, when the higher order effects such as self steepening, third order dispersion and Raman scattering need to be taken into account