Dispersive Lamb systems

Abstract

Under periodic boundary conditions, a one-dimensional dispersive medium driven by a Lamb oscillator exhibits a smooth response when the dispersion relation is asymptotically linear or superlinear at large wave numbers, but unusual fractal solution profiles emerge when the dispersion relation is asymptotically sublinear. Strikingly, this is exactly the opposite of the superlinear asymptotic regime required for fractalization and dispersive quantization, also known as the Talbot effect, of the unforced medium induced by discontinuous initial conditions. solution to the periodic initial-boundary value problem produced by rough initial data, e.g., a step function, is "quantized" as a discontinuous but piecewise constant profile or, more generally, piecewise smooth or even piecewise fractal, at times which are rational multiples of L n /π n−1 , where L is the length of the interval, but exhibits a continuous but non-differentiable fractal profile at all other times. (An interesting question is whether such fractal profiles enjoy selfsimilarity properties similar to the Riemann and Weierstrass non-differentiable functions, as explored in [6]. However, the Fourier series used to construct the latter are quite different in character, and so the problem remains open and challenging.)Dispersive quantization relies on the slow, conditional convergence of the Fourier series solutions, and requires the dispersion relation to be asymptotically polynomial at large wave number.The key references include the 1990's discovery of Michael Berry and collaborators, [1], in the context of optics and quantum mechanics, and the subsequent analytical work of Oskolkov, [18]. In particular, this effect underlies the experimentally observed phenomenon of quantum revival, in which an electron that is initially concentrated at a single location of its orbital shell is, at rational times, re-concentrated at a finite number of orbital locations. The subsequent rediscovery by the first author, [2,16], showed that the phenomenon appears in a range of linear dispersive partial differential equations, while other models arising in fluid mechanics, plasma dynamics, elasticity, DNA dynamics, etc. exhibit a fascinating range of as yet poorly understood behaviors, whose qualitative features are tied to the large wave number asymptotics of the dispersion relation. These studies were then extended, through careful numerical simulations, to nonlinear dispersive equations, including both integrable models, such as the cubic nonlinear Schrödinger, Korteweg-deVries, and modified Korteweg-deVries equations, as well as non-integrable generalizations with higher degree nonlinearities. Some of these numerical observations were subsequently rigorously confirmed in papers of Chousionis, Erdogan, and Tzirakis, [3,9,10], and, further, in the very recent paper by Erdogan and Shakan, [8], which extends the analysis to non-polynomial dispersion relations, but much more work remains to be completed, including extensions to other types of boundary conditions and dispersiv...

Fractal solutions of dispersive partial differential equations on the torus

“…In this paper, using exponential sum estimates, we study the fractal dimension of the graphs of solutions of a large class of dispersive PDE. Various aspects of this problem have been considered by many authors, e.g., [BeLe,Os1,Be,BeKl,KaRo,Ro,BMS,Os2,OsCh1,Ol,ChOl1,ErTz1,ErTz2,OsCh2,ChOl2,HoVe,CET,Ve,ErTz3,OlSh,OlTs]. One of our goals is to give a theoretical justification to some observations of Berry [Be], ChOl2] and Olver-Sheils [OlSh].…”

“…In Section 4, we obtain dimension estimates for the graphs of equations with nonpolynomial dispersion relations. Olver and his collaborators [Ol,ChOl1,ChOl2,OlSh] provided numerical simulations of the Talbot effect for a large class of dispersive equations. In the case of polynomial dispersion, they numerically confirmed the rational/irrational dichotomy discussed above.…”

The limiting amplitude principle for the nonlinear Lamb system

“…, u n (x, t)) ∈ R n , y(t) ∈ R n , M is a positivedefinite symmetric n×n matrix, S is skew, F (y) = −∇V (y) ∈ C 1 (R n ; R n ), where the potential energy V ∈ C 2 (R n ; R) satisfies the condition (6). In the linear case, when F (y) = Λy with some symmetric matrix Λ, the model (35) is called the gyroscopic Lamb system, see, e. g., [4,13]. We study the Cauchy problem for system (35) with the initial conditions…”

“…The system (2) -(4) was introduced first by Lamb [7] for the linear case, i. e., when F (y) = −ry with a positive constant r. This system can be considered as a simple model of radiation damping experienced by a vibrating body in an energy conducting medium, for example, vibrations of an elastic sphere in a gaseous medium, relativistic radiation of energy from a concentrated mass by gravity waves and so on. For details, see, e. g., [4,13]. For general nonlinear functions F (y), this model was studied by Komech [5,6] for finite-energy solutions.…”

Radiation-induced instability of a finite-chord Nemtsov membrane

“…Quite surprisingly, he had found that the emission of traveling waves in the continuum by the oscillating mass contributes an effective Rayleigh damping correction term to the oscillator equation yielding decay of its vertical motion [1]. In the course of time the radiating Lamb oscillator became paradigmatic for understanding the radiation damping in open and damped subsystems of closed conservative systems and gave rise to a number of abstract models of dispersion of energy from a 'small', usually finitedimensional, subsystem to a 'large', infinite-dimensional wave field [2,3,4,5,6,7,8,9,10,11,12,13].…”