Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances

In this joint theoretical, numerical and experimental study, we investigate the phenomenon of forced generation of nonlinear waves by disturbances moving steadily with a transcritical velocity through a layer of shallow water. The plane motion considered here is modelled by the generalized Boussinesq equations and the forced Korteweg-de Vries (fKdV) equation, both of which admit two types of forcing agencies in the form of an external surface pressure and a bottom topography. Numerical results are obtained using both theoretical models for the two types of forcings. These results illustrate that within a transcritical speed range, a succession of solitary waves are generated, periodically and indefinitely, to form a procession advancing upstream of the disturbance, while a train of weakly nonlinear and weakly dispersive waves develops downstream of an ever elongating stretch of a uniformly depressed water surface immediately behind the disturbance. This is a beautiful example showing that the response of a dynamic system to steady forcing need not asymptotically tend to a steady state, but can be conspicuously periodic, after an impulsive start, when the system is being forced at resonance. A series of laboratory experiments was conducted with a cambered bottom topography impulsively started from rest to a constant transcritical velocity U, the corresponding depth Froude number F = U/(gh[sub]0)^1/2 (g being the gravitational constant and h[sub]0 the original uniform water depth) being nearly the critical value of unity. For the two types of forcing, the generalized Boussinesq model indicates that the surface pressure can be more effective in generating the precursor solitary waves than the submerged topography of the same normalized spatial distribution. However, according to the fKdV model, these two types of forcing are entirely equivalent. Besides these and some other rather refined differences, a broad agreement is found between theory and experiment, both in respect of the amplitudes and phases of the waves generated, when the speed is nearly critical (0.9 F > 0.2, finally disappear at F ~= 0.2. In the other direction, as the Froude number is increased beyond F ~= 1.2, the precursor soliton phenomenon was found also to evanesce as no finite-amplitude solitary waves can outrun, nor can any two-dimensional waves continue to follow, the rapidly moving disturbance. In this supercritical range and for asymptotically large times, all the effects remain only local to the disturbance. Thus, the criterion of the fascinating phenomenon of the generation of precursor solitons is ascertained

Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field.Comment: To appear in Inventiones Mathematica

Contributions of plasma physics to chaos and nonlinear dynamics

This topical review focusses on the contributions of plasma physics to chaos and nonlinear dynamics bringing new methods which are or can be used in other scientific domains. It starts with the development of the theory of Hamiltonian chaos, and then deals with order or quasi order, for instance adiabatic and soliton theories. It ends with a shorter account of dissipative and high dimensional Hamiltonian dynamics, and of quantum chaos. Most of these contributions are a spin-off of the research on thermonuclear fusion by magnetic confinement, which started in the fifties. Their presentation is both exhaustive and compact. [15 April 2016

Long Waves in Ocean and Coastal Waters

Water waves occurring in the ocean have a wide spectrum of wavelength and period, ranging from capillary waves of 1 cm or shorter wavelength to long waves with wavelength being large compared to ocean depth, anywhere from tens to thousands of kilometers. Of the various long-wavegenic sources, distant body forces can act as the continuous ponderomotive force for the tides. Hurricanes and storms in the sea can develop a sea state, with the waves being worked on by winds and eventually cascading down to swells after a long distance of travel away from their birthplace. Large tsunamis can be ascribed to a rapidly occurring tectonic displacement of the ocean floor (usually near the coast of the Pacific Ocean) over a large horizontal dimension (of hundreds to over a thousand square kilometers) during strong earthquakes, causing vertical displacements to ocean floor of tens of meters. Other generation mechanisms include underwater subsidence or land avalanche in the ocean and submarine volcanic eruption. Gigantic rockfalls and long-period seismic waves can also produce gravity waves in lakes, reservoirs, and rivers. Generation, propagation, and evolution of such long waves in the ocean and their effects in coastal waters and harbors is a subject of increasing importance in civil, coastal, and environmental engineering and science. Of the various long wave phenomena, tsunami appears to stand out in possessing a broad variation of wave characteristics and scaling parameters on the one hand, and, on the other, in having the capacity of inflicting a disastrous effect on the target area. In taking tsunamis as a representative case for the study of long waves in the ocean, it can be said that large tsunamis are generated with a great source of potential energy (as high as 10^15-10^16J ), though the detailed source motion of a specific tsunami is generally difficult to determine. The large size of source region implies that the "new born" waves would be initially long and the energy contained in the large wave-number part (k, nondimensionalized with respect to the local ocean depth, h) would be unimportant. Soon after leaving the source region, the low wave-number components of the source spectrum are further dispersed effectively by the factor sech kh into the even lower wave-number parts. Tsunamis thus evolve into a train of long waves, with wavelength continually increasing from about 50 km to as high as 250 km, but with a quite small amplitude, typically of 1/2 m or smaller, as they travel across the Pacific Ocean at a speed of 650 km/h-760 km/h. There is experimental evidence indicating that tsunamis continually, though slowly, evolve due to dispersion while propagating in the open ocean; this property has been observed by Van Dorn (16) from the data taken at Wake Island of the March 9, 1957 Aleutian tsunami. One of our primary interests is, of course, the evolution of tsumanis in coastal waters and their terminal effects. Large tsunamis can have their wave height amplified many fold in climbing up the continental slope and propagating into shallower water, producing devastating waves (up to 20 m or higher on record) upon arriving at a beach. The terminal amplification can be crucially affected by three-dimensional configurations of the coastal environment enroute to beach. These factors dictate the transmission, reflection, rate of growth, and trapping of tsunamis in their terminal stage. After the first hit on target, a tsunami is partly reflected to travel once over across the Pacific Ocean, with some degree of attenuation -- a process which is still unclear, but is generally known to be small. Based on observations, Munk (13) suggests the figure of the "decay time" (intensity reducing to 1/e) being about 112 day, and the "reverberation time" (intensity falling off to 10^-6) about a week, while the reflection frequency (across the Pacific) is around 1.7/day. To fix idea, the pertinent physical characteristics and their scaling parameters of a tsunami through its life span of evolution can be described qualitatively in Table I. From the aforementioned estimate we note that the dispersion parameter, h/[lambda], and the amplitude parameter, a/h, are both small in general. However, their competitive roles as rated by the Ursell number Ur, can increase from some small values in the deep ocean, typically of order 10^-2 for large tsunamis, by a factor of 10^3 upon arriving in near-shore waters. This indicates that the effects of nonlinearity (amplitude dispersion) are practically nonexistent in the deep ocean, but gradually become more important and can no longer be neglected when the Ursell number increases to order unity or greater during the terminal stage in which the coastal effects manifest. The small values of the dimensionless wave number, kh = 2[pi]h/[lamda] being in the range of 0.6-0.03 during travel in open ocean, suggests that a slight dispersive effect is still present and this can lead to an accumulated effect in predicting the phase position over very large distances of travel. The overall evolution of tsunamis, as only crudely characterized in Table 1, depends in fact on many factors such as the features of source motion, nonlinear and dispersive effects on propagation in one and two dimensions, the three-dimensional configuration of the coastal region, the direction of incidence, converging or diverging passage of the waves, local reflection and adsorption, density stratification in water, etc. While these aspects of physical behavior are akin to tsunamis, they are also relevant to the consideration of other long wave phenomena. With an intent to provide a sound basis for general applications to long wave phenomena in nature, this paper presents (in the section on three-dimensional long-wave models) a basic long-wave equation which is of the Boussinesq class with special reference to tsunami propagation in two horizontal dimensions through water having spatial and temporal variations in depth. Under certain particular conditions (such as the propagation in one space dimension, or primarily one space dimensional of long waves in water of constant depth) this equation reduces to the Korteweg-de Vries equation or the nonlinear Schrodinger equation. In these special cases we have seen the impressive developments in recent studies of the "soliton-bearing" nonlinear partial differential equations by means of such methods as the variational modulation, the inverse scattering analysis, and modern differential geometry (12,14,17). While extensions of these methods to more general cases will require further major developments, the present analysis and survey will concentrate on the three-dimensional (with propagation in two horizontal dimensions) effects under various conditions by examining the validity of different wave models (based on neglecting the effects of nonlinearity, dispersion, or reflection) in different circumstances. From the example of self focusing of weakly-nonlinear waves (given in the section on converging cylindrical long waves), the effects of nonlinearity, dispersion, and reflection will be seen all to play such a major role that the present basic equation cannot be further modified without suffering from a significant loss of accuracy

Extreme Events in Nonlinear Lattices

The spatiotemporal complexity induced by perturbed initial excitations through the development of modulational instability in nonlinear lattices with or without disorder, may lead to the formation of very high amplitude, localized transient structures that can be named as extreme events. We analyze the statistics of the appearance of these collective events in two different universal lattice models; a one-dimensional nonlinear model that interpolates between the integrable Ablowitz-Ladik (AL) equation and the nonintegrable discrete nonlinear Schr\"odinger (DNLS) equation, and a two-dimensional disordered DNLS equation. In both cases, extreme events arise in the form of discrete rogue waves as a result of nonlinear interaction and rapid coalescence between mobile discrete breathers. In the former model, we find power-law dependence of the wave amplitude distribution and significant probability for the appearance of extreme events close to the integrable limit. In the latter model, more importantly, we find a transition in the the return time probability of extreme events from exponential to power-law regime. Weak nonlinearity and moderate levels of disorder, corresponding to weak chaos regime, favour the appearance of extreme events in that case.Comment: Invited Chapter in a Special Volume, World Scientific. 19 pages, 9 figure