Rogue waves in superfluid helium

Rogue waves have been observed in superfluid helium. The experimental system consists of high intensity second sound (temperature-entropy) waves within a resonant cavity. Under steady state conditions, with a constant oscillatory driving force at the resonant frequency, the waves are turbulent and there are fluxes of energy towards both high and low frequencies. Rogue waves appear under the nonequilibrium conditions that prevail shortly after the drive has been switched on, prior to establishment of the steady state. The experiment is described briefly, relevant results are presented and discussed theoretically in terms of nonlinear wave interactions, and possible connections to rogue waves on the ocean are considered.     

Rogue waves in shallow water

Most of the processes resulting in the formation of unexpectedly high surface waves in deep water (such as dispersive and geometrical focusing, interactions with currents and internal waves, reflection from caustic areas, etc.) are active also in shallow areas. Only the mechanism of modulational instability is not active in finite depth conditions. Instead, wave amplification along certain coastal profiles and the drastic dependence of the run-up height on the incident wave shape may substantially contribute to the formation of rogue waves in the nearshore. A unique source of long-living rogue waves (that has no analogues in the deep ocean) is the nonlinear interaction of obliquely propagating solitary shallow-water waves and an equivalent mechanism of Mach reflection of waves from the coast. The characteristic features of these processes are (i) extreme amplification of the steepness of the wave fronts, (ii) change in the orientation of the largest wave crests compared with that of the counterparts and (iii) rapid displacement of the location of the extreme wave humps along the crests of the interacting waves. The presence of coasts raises a number of related questions such as the possibility of conversion of rogue waves into sneaker waves with extremely high run-up. Also, the reaction of bottom sediments and the entire coastal zone to the rogue waves may be drastic.     

Rogue edge waves in the ocean

Theoretically possible rogue edge wave are studied over cylindrical bottom in the framework of nonlinear shallow water equations in a weakly nonlinear limit. The nonlinear mechanisms (nonlinear dispersion enhancement, modulation instability and multimodal interactions) of possible anomalous edge wave appearance are analyzed.     

Rogue waves and solitons on a cnoidal background

Solutions of the nonlinear Schrödinger equation, appearing as rogue waves on a spatially-periodic background envelope, are obtained using the Darboux transformation scheme. Several particular examples are illustrated numerically. These include soliton and breather solutions on a periodic background as well as higher-order structures. The results enrich our knowledge of possible analytic solutions that describe the appearance of rogue waves in a variety of situations. This work is prepared on the occasion of Prof. Helmut Brand's 60th birthday. He has made significant contributions to the science of solitons and his ideas have inspired our research into localised formations in various physical contexts.     

Rogue waves in Alfvénic turbulence

Rogue waves, in the form of giant breathers, are shown to develop in the Alfvén wave (AW) turbulence regime described by the randomly driven derivative nonlinear Schrödinger equation in the presence of a weak dissipation. The distribution of the instantaneous global maxima of the AW intensity fluctuations is seen to be accurately fitted by power laws, which contrasts with the integrable regime (absence of dissipation and forcing) where the behavior is rather exponential. As the dissipation is reduced, freak waves form less frequently but reach larger amplitudes.     

Rogue waves management by external potentials

We study the effect of time-dependent linear and quadratic potentials on the profile and dynamics of rogue waves represented by a Peregrine soliton. The Akhmediev breather, Ma breather, bright soliton, Peregrine soliton, and constant wave (CW) are all obtained by changing the value of one parameter in the general solution corresponding to the amplitude of the input CW. The corresponding solutions for the case with linear and quadratic potentials were derived by the similarity transformation method. While the peak height and width of the rogue wave turn out to be insensitive to the linear potential, the trajectory of its center-of-mass can be manipulated with an arbitrary time-dependent slope of the linear potential. With a quadratic potential, the peak height and width of the rogue wave can be arbitrarily manipulated to result, for a special case, in a very intense pulse.     

Rogue waves in Lugiato-Lefever equation with variable coefficients

In this paper, we theoretically investigate the generation of optical rogue waves from a Lugiato-Lefever equation with variable coefficients by using the nonlinear Schrödinger equation-based constructive method. Exact explicit rogue-wave solutions of the Lugiato-Lefever equation with constant dispersion, detuning and dissipation are derived and presented. The bright rogue wave, intermediate rogue wave and the dark rogue wave are obtained by changing the value of one parameter in the exact explicit solutions corresponding to the external pump power of a continuous-wave laser.     

Dynamic multistable states in a system of magnetic dipoles

Bistable and multistable states induced by a fluctuating magnetic field in a system of four magnetic dipoles were studied numerically. The possibility of performing switchings and cyclic transitions between the different oscillatory regimes by changing field parameters is examined, along with the possibility of creating conditions under which the system is sensitive to noise signals.     

Rogue waves, dissipation, and downshifting

We investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one-dimensional Dysthe equation. Significantly, rogue waves do not develop after the downshifting becomes permanent. Thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves. Using the inverse spectral theory of the NLS equation, simulations of the damped Dysthe equation for sea states characterized by JONSWAP spectrum consistently show that rogue wave events are well-predicted by proximity to homoclinic data, as measured by the spectral splitting distance δ. The cut off distance δ_cutoff decreases as the strength of the damping increases, indicating that for stronger damping the JONSWAP initial data must be closer to homoclinic data for rogue waves to occur.     

Vibrational resonance in a multistable system with position-dependent mass

The occurrence of vibrational resonance(VR) in a dual-frequency-driven multistable system with a spatially varying mass modelling particle with position-dependent mass(PDM) and evolving in a one-dimensional symmetric periodic potential has been investigated and reported in this paper. We numerically compute and analyze the response amplitude, the effects of the PDM parameters(m₀, a) on the potential structure, the occurrence of VR and the bifurcation of the equilibrium points. It is s...     

Rogue waves in multi‐pair plasma medium

The non‐linear propagation of ion acoustic (IA) waves, which is governed by the non‐linear Schrödinger equation, in multi‐pair plasmas (MPPs) containing adiabatic positive and negative ion fluids as well as non‐extensive (q‐distributed) electrons and positrons is theoretically investigated. It is observed that the MPP under consideration supports two types of modes, namely fast and slow IA modes, and the modulationally stable and unstable parametric regimes for the fast and slow IA modes are determined by the sign of the ratio of the dispersive coefficient to the non‐linear one. It is also found that the modulationally unstable regime generates highly energetic IA rogue waves (IARWs), and the amplitude as well as the width of the IARWs decreases with increase in the value of q (for both q > 0 and q < 0 limits). These new striking features of the IARWs are found to be applicable in the space (i.e., D‐region [], and F‐region [H⁺, H^?] of the Earth's ionosphere) and laboratory MPPs (i.e., fullerene [C⁺, C^?]).     

Controlling vibrational resonance in a multistable system by time delay

The phenomenon of vibrational resonance in a delayed multistable system that is excited by biharmonic signals is investigated in the present paper. Different from the former theory, the appearance and the disappearance of the vibrational resonance are controlled by adjusting the time delay parameter instead of modulating the amplitude of the high-frequency signal. The motion of the orbit within or between the different potential wells can also be controlled. Furthermore, based on both the methods of numerical simulation and analytical analysis, the behavior of delay-induced multiple vibrational resonance and its mechanism are investigated and discussed. The multiple vibrational resonance, which is quantified by the response amplitude at the low-frequency, is found to be periodic in the delay parameter with two periods, i.e., the periods of the two driven signals. The method used in this paper gives a new way for controlling vibrational resonance in a multistable system.     

Rogue waves for Kadomstev-Petviashvili solutions in a warm dusty plasma with opposite polarity

Kadomstev-Petviashvili (KP) equation is derived using reductive perturbation method. This equation transformed into a nonlinear Schrödinger equation (NLS) by using appropriate variable transformations. When the carrier wave frequency is much smaller than the dust plasma frequency, the DA waves generating modulated wave packets in the form of rogue waves. The dependence of rogue wave profile on system plasma parameters investigated numerically. The parameters in this model are within the ranges corresponding to upper mesosphere, cometary tails and Jupiter’s magnetosphere.     

Rogue waves of a (3+1)-dimensional BKP equation

We investigate certain rogue waves of a (3+1)-dimensional BKP equation via the Kadomtsev-Petviashili hierarchy reduction method. We obtain semi-rational solutions in the determinant form, which contain two special interactions: (i) one lump develops from a kink soliton and then fuses into the other kink one; (ii) a line rogue wave arises from the segment between two kink solitons and then disappears quickly. We find that such a lump or line rogue wave only survives in a short time and localizes in both space and time, which performs like a rogue wave. Furthermore, the higher-order semi-rational solutions describing the interaction between two lumps (one line rogue wave) and three kink solitons are presented.     

Rogue internal waves in the ocean: Long wave model

Rogue waves can be categorized as unexpectedly large waves, which are temporally and spatially localized. They have recently received much attention in the water wave context, and also been found in nonlinear optical fibers. In this paper, we examine the issue of whether rogue internal waves can be found in the ocean. Because large-amplitude internal waves are commonly observed in the coastal ocean, and are often modeled by weakly nonlinear long wave equations of the Korteweg-de Vries type, we focus our attention on this shallow-water context. Specifically, we examine the occurrence of rogue waves in the Gardner equation, which is an extended version of the Korteweg-de Vries equation with quadratic and cubic nonlinearity, and is commonly used for the modelling of internal solitary waves in the ocean. Importantly, we choose that version of the Gardner equation for which the coefficient of the cubic nonlinear term and the coefficient of the linear dispersive term have the same sign, as this allows for modulational instability. From numerical simulations of the evolution of a modulated narrow-band initial wave field, we identify several scenarios where rogue waves occur.     

Rogue waves: New forms enabled by GPU computing

A method to determine parameters governing periodic Riemann theta function rogue-wave solutions to the nonlinear Schrödinger equation is presented. A map of parameter values leading to candidate solutions is developed. In addition to candidate solutions, an overview of qualitative aspects of the solution space can be gained from this map. Based on these findings, several new extreme wave solutions are presented. Although the computations required to determine the map are quite demanding, it is shown that these computations can be efficiently accelerated with a parallel computing architecture. A general purpose computing on a graphics processor unit (GPGPU) implementation yielded a 400× acceleration over a single threaded high level implementation. This acceleration enabled exploration and examination of the solution space, which otherwise would not have been possible. In addition, the solution methodology presented here can be extended to explore other classes of solutions.     

Output response identification in a multistable system for piezoelectric energy harvesting

In this paper we examine in detail the multiple responses of a novel vibrational energy harvester composed of a vertical bistable beam whose complex non-linear behavior is tuned via magnetic interaction. The beam was excited horizontally by a harmonic inertial force while mechanical vibrational energy is converted to electrical power through a piezoelectric element. The bistable laminate beam coupled to the piezoelectric transducer showed a variety of complex responses in terms of the beam displacement and harvested power output. The range of vibration patterns in this non-linear system included single-well oscillations and snap-through vibrations of periodic and chaotic character. Harvested power was found to be strongly dependent on the vibration pattern with nonlinearities providing a broadband response for energy harvesting. Wavelet analysis of measured voltage, displacement and velocity time histories indicated the presence of a variety of nonlinear periodic and also chaotic phenomena. To measure the complexity of response time series we applied phase portraits and determine stroboscopic points and multiscale entropy. It is demonstrated that by changing parameters such as the magnetic interaction, the characteristics of the bistable laminate harvester, such as the natural frequency, bandwidth, vibration response and peak power can be readily tailored for harvesting applications.