Oscillatory Wave Trains Research Articles

Spin-piston problem for a ferromagnetic thin film: Shock waves and solitons

The unsteady, nonlinear magnetization dynamics induced by spin injection in an easy-plane ferromagnetic channel subject to an external magnetic field are studied analytically. Leveraging a dispersive hydrodynamic description, the Landau-Lifshitz equation is recast in terms of hydrodynamic-like variables for the magnetization's perpendicular component (spin density) and azimuthal phase gradient (fluid velocity). Spin injection acts as a moving piston that generates nonlinear, dynamical spin textures in the ferromagnetic channel with downstream quiescent spin density set by the external field. In contrast to the classical problem of a piston accelerating a compressible gas, here, variable spin injection and field lead to a rich variety of nonlinear wave phenomena from oscillatory spin shocks to solitons and rarefaction waves. A full classification of solutions is provided using nonlinear wave modulation theory by identifying two key aspects of the fluid-like dynamics: subsonic/supersonic conditions and convex/nonconvex hydrodynamic flux. Familiar waveforms from the classical piston problem such as rarefaction (expansion) waves and shocks manifest in their spin-based counterparts as smooth and highly oscillatory transitions, respectively. The spin shock is an example of a dispersive shock wave, which arises in many physical systems. New features without a gas dynamics counterpart include composite wave complexes with "contact" spin shocks and rarefactions. Magnetic supersonic conditions lead to two pronounced piston edge behaviors including a stationary soliton and an oscillatory wavetrain. These coherent wave structures have physical implications for the generation of high frequency spin waves from pulsed injection and persistent, stable stationary and/or propagating solitons in the presence of magnetic damping. The analytical results are favorably compared with numerical simulations.

Retrograde ERK activation waves drive base-to-apex multicellular flow in murine cochlear duct morphogenesis.

A notable example of spiral architecture in organs is the mammalian cochlear duct, where the morphology is critical for hearing function. Genetic studies have revealed necessary signaling molecules, but it remains unclear how cellular dynamics generate elongating, bending, and coiling of the cochlear duct. Here, we show that extracellular signal-regulated kinase (ERK) activation waves control collective cell migration during the murine cochlear duct development using deep tissue live-cell imaging, Förster resonance energy transfer (FRET)-based quantitation, and mathematical modeling. Long-term FRET imaging reveals that helical ERK activation propagates from the apex duct tip concomitant with the reverse multicellular flow on the lateral side of the developing cochlear duct, resulting in advection-based duct elongation. Moreover, model simulations, together with experiments, explain that the oscillatory wave trains of ERK activity and the cell flow are generated by mechanochemical feedback. Our findings propose a regulatory mechanism to coordinate the multicellular behaviors underlying the duct elongation during development.

Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains.

We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [Zemskov et al., Phys. Rev. E 77, 036219 (2008) and Phys. Rev. E 95, 012203 (2017) for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.

Operation of a Gyromagnetic Line at Low and High Voltages With Simultaneous Axial and Azimuthal Biases

A great interest has been devoted to the study of nonlinear transmission lines (NLTLs) for radio frequency (RF) generation since they have been used with great success in RF generation by producing a train of oscillatory waves along the line and at its output. There are two configurations of NLTLs. The first one is a dispersive line consisting of LC sections with nonlinear components, and the second one is a continuous ferrite loaded nondispersive line generally biased by an axial magnetic field, known as gyromagnetic line. In this paper, the focus of the study is on the second one, since gyromagnetic lines can operate in a broader frequency range (0.3–2.0 GHz) with higher conversion efficiency (20.0%) when compared to lumped NLTLs, generally limited up to 300 MHz with less than 10.0% of efficiency, because of their dielectric losses and stray impedances on line structure. Different models have been used along the years by several authors with different approaches to study the gyromagnetic phenomenon by means of numerical simulations based on analytical models to predict the precession movement of the electron magnetic dipole of the ferromagnetic material. Thus, the goal of this paper is to analyze the gyromagnetic NLTL behavior through the effects on the line operation. The novelty herein is to use two biases simultaneously to study continuous gyromagnetic NLTLs, focusing on the pulse rise time compression and RF generation caused by the precession of the magnetic dipole. An experimental setup is described and tested for low- and high-voltage operation for a 20-cm gyromagnetic line loaded with NiZn ferrite beads. The novelty of this paper is to use two biases simultaneously to study continuous gyromagnetic NLTLs, such techniques can be useful in the design of continuous lines for RF applications in space and mobile defense platforms of compact size.

Time-linearized, compact methods for the inviscid GRLW equation subject to initial Gaussian conditions

• Effects of the width of initial Gaussian conditions on wave propagation are studied. • Roles of nonlinearity and local advection on wave dynamics are investigated. • Time-linearization methods for the GRLW equation are presented. • Rarefaction waves may appear behind solitary waves. • Localized dispersive oscillations may occur behind the leading wave. The objective of this paper is three-fold. First, four time-linearization methods that are second- and fourth-order accurate in time and space, respectively, are presented and used to study the dynamics of the modified and generalized regularized-long wave equations (mRLW and GRLW equations, respectively). Two of the methods use the conservation-law form of the equations and treat the wave amplitude and its second-order spatial derivative and the linear and nonlinear advection fluxes as unknowns, whereas the other two employ the non-conservation-law form of the equations and consider the wave amplitude and its first- and second-order spatial derivatives as unknowns. The methods employ three-point fourth-order accurate Padé discretizations for the first- and second-order derivatives, are second-order accurate in time, and yield linear systems of blocktridiagonal matrices. Second, the accuracy of these methods is assessed by comparing their results with those of the exact solution of the mRLW equation. It is reported that the four methods predict nearly the same values of the three invariants and have the same accuracy, and that an accurate prediction of the invariants may not correspond to small errors in space and time. Third, the dynamics of the inviscid GRLW equation is studied first qualitatively in terms of length and time scales and then numerically as a function of the linear advection speed, the exponent of the nonlinear advection flux, the dispersion coefficient and the amplitude and width of the initial bell-shaped or Gaussian conditions. It is shown that wide initial conditions result in wave steepening and breakup and the formation of solitary waves whose amplitude and speed decrease as the time for their formation increases. For narrow initial conditions, it is shown that only a single solitary wave may form. Behind this wave and depending on the parameters that characterized the inviscid GRLW equation, rarefaction or negative amplitude waves that propagate towards the upstream boundary or a train of localized oscillatory waves that do not emerge from the trailing edge of the leading solitary wave may be formed. These oscillatory waves exhibit the characteristics of, but are not dispersive shock waves and their amplitude and frequency increases as the width of the initial conditions is decreased. The results presented here do not only complement previous work by the authors, they also show that the dynamics of the inviscid GRLW equation undergoes new and interesting phenomena as the width of the initial conditions is decreased.

Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion.

We study waves with exponentially decaying oscillatory tails in a reaction-diffusion system with linear cross diffusion. To be specific, we consider a piecewise linear approximation of the FitzHugh-Nagumo model, also known as the Bonhoeffer-van der Pol model. We focus on two types of traveling waves, namely solitary pulses that correspond to a homoclinic solution, and sequences of pulses or wave trains, i.e., a periodic solution. The effect of cross diffusion on wave profiles and speed of propagation is analyzed. We find the intriguing result that both pulses and wave trains occur in the bistable cross-diffusive FitzHugh-Nagumo system, whereas only fronts exist in the standard bistable system without cross diffusion.

Shock wave generation triggered by a weak background in optical fibers

We experimentally report the observation of dispersive shock waves from a short pulse superimposed onto a small continuous wave background in optical fibers. We show that the background allows us to strongly enhance the extension and contrast of the oscillatory wave train inherent to the dispersive shock. More than seven periods of oscillations with high contrast are observed experimentally and confirmed with numerical simulations. The dynamics of the process are simply explained from spectro-temporal representations.

Nonlinear ring waves in a two-layer fluid

Surface and interfacial weakly-nonlinear ring waves in a two-layer fluid are modelled numerically, within the framework of the recently derived 2+1-dimensional cKdV-type equation. In a case study, we consider concentric waves from a localised initial condition and waves in a 2D version of the dam-break problem, as well as discussing the effect of a piecewise-constant shear flow. The modelling shows, in particular, the formation of 2D dispersive shock waves and oscillatory wave trains.

Propagation of rarefaction pulses in discrete materials with strain-softening behavior.

We report on the dynamic behavior of strongly nonlinear discrete materials with anomalous strain-softening behavior. Rarefaction solitary waves found in numerical calculations agree well with the exact solution to the continuum wave equation. Compression pulses generated by impact quickly disintegrate into a leading rarefaction solitary wave followed by an oscillatory wave train containing localized excitations. Such behavior is favorable for metamaterials design of shock absorption layers as well as tunable information transmission lines for scrambling of acoustic information.

Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability

The Benjamin-Feir instability describes the instability of a uniform oscillatory wave train in an irrotational flow subject to small perturbation of wave number, amplitude and frequency. Their instability analysis is based on the perturbation around the second order Stokes wave which satisfies the dynamic and kinematic free-surface boundary conditions up to the second order. In the same irrotational flow and perturbation framework of the Benjamin-Feir analysis, the perturbation in the present paper is around a nonlinear oscillatory wave train which solves exactly the dynamic free-surface boundary condition and satisfies the kinematic free-surface boundary condition up to the third order. It is shown that the nonlinear oscillatory wave train is stable with respect to the perturbation when the irrotational flow involves small Rayleigh energy dissipation.

ANALYSIS OF SOLITON FISSION OVER A SUBMERGED STRUCTURE USING "NONLINEAR FOURIER TRANSFORM (NLFT)†

When a single incident solitary wave passes over a submerged reef, it disintegrates into a train of solitons (soliton fission), followed by a train of oscillatory waves. One of the major problems in the analysis of the recorded time series is the uncertain identification of the number of solitons N in the transmitted wave train behind the reef due to the difficulties to distinguish between solitons and oscillatory waves, especially in the case of breaking waves. With the "nonlinear Fourier transform (NLFT)†, an application of the inverse scattering transform (IST) of the Korteweg-de Vries equation, a powerful analysis method is proposed to analyse nonlinear wave processes. Application of the NLFT to the transmitted waves of systematic numerical tests with breaking and non-breaking solitary waves behind a submerged structure (reef) with finite width br allows to separate distinctly solitons and oscillatory waves. The paper gives an overview over the first NLFT analysis results for the determination of the number of solitons N behind the reef. The influence of relative submergence depth dr/h, relative reef width br/Li and relative incident wave height Hi/dr is examined. First recommendations for the distinct identification of the number of solitons arising from the fission using NLFT will also be provided, including the limitations of the method.

A weakly nonlinear evolution model for long internal waves in a large lake

A wind-forced weakly nonlinear weakly dispersive evolution model is derived for a continuously stratified circular lake of slowly varying depth under the effect of the Earth's rotation. The model was numerically integrated to investigate the evolution of long internal waves of vertical mode one for various sets of environmental parameters. It is demonstrated that the Kelvin wave steepens as it propagates, and the steepened front subsequently generates a train of oscillatory waves. It is demonstrated that Poincaré waves do not steepen, but their amplitude is modulated in an oscillatory manner with time, exhibiting a pseudo recurrence character. The model was applied to the wind forced problem, confirming that Kelvin and Poincaré waves are the dominant response. Energy partition among Kelvin and Poincaré wave modes is estimated as a function of wind-forcing parameters. For large lakes, the most significant wave amplitude is found in the Kelvin wave mode, but the gross field energy is most significantly contained in Poincaré wave modes.

Numerical solution of Boussinesq systems of KdV–KdV type: II. Evolution of radiating solitary waves

In this paper we consider a coupled KdV system of Boussinesq type and its symmetric version. These systems were previously shown to possess generalized solitary waves consisting of a solitary pulse that decays symmetrically to oscillations of small, constant amplitude. We solve numerically the periodic initial-value problem for these systems using a high order accurate, fully discrete, Galerkin-finite element method. (In the case of the symmetric system, it is possible to prove rigorous, optimal-order, error estimates for this scheme.) The numerical scheme is used in an exploratory fashion to study radiating solitary-wave solutions of these systems that consist, in their simplest form, of a main, solitary-wave-like pulse that decays asymmetrically to small-amplitude, outward-propagating, oscillatory wave trains (ripples). In particular, we study the generation of radiating solitary waves, the onset of ripple formation and various aspects of the interaction and long time behaviour of these solutions.

Nucleation, Drift, and Decay of Phase Bubbles in Period-2 Oscillatory Wave Trains in a Reaction-Diffusion System

This is a report on experimental observations of phase bubbles, simply closed boundaries between domains oscillating 2pi out of phase, associated with period-2 oscillatory traveling waves in a Belousov-Zhabotinsky reaction-diffusion system. The bubbles nucleate spontaneously through a fast localized phase slip, drift radially away from a neighboring spiral wave core in an oscillatory fashion, and gradually shrink to disappear. Their oscillatory drift along the radial direction is a consequence of "period adaptation," while their lateral shrinkage is an attribute of local curvature. Similar dynamic structures can be reproduced in a simple, three-species reaction-diffusion model that supports period-2 oscillatory wave trains.

Nonlinear geometric optics for contact discontinuities and shock waves in one-dimensional Euler system for gas dynamics

The aim of this paper is to study the rigorous theory of nonlinear geometric optics for a contact discontinuity and a shock wave to the Euler system for one-dimensional gas dynamics. For the problem of a contact discontinuity and a shock wave perturbed by a small amplitude, high frequency oscillatory wave train, under suitable stability assumptions, we obtain that the perturbed problem has still a shock wave and a contact discontinuity, and we give their asymptotic expansions.

Asymptotic solitons on a non-zero mean level

Abstract The collision of solitary waves for a higher-order modified Korteweg-de Vries (mKdV) equation is examined. In particular, the collision between solitary waves with sech-type and algebraic (which only exist on a non-zero mean level) profiles is considered. An asymptotic transformation, valid if the higher-order coefficients satisfy a certain algebraic relationship, is used to transform the higher-order mKdV equation to an integrable member of the mKdV hierarchy. The transformation is used to show that the higher-order collision is asymptotically elastic and to derive the higher-order phase shifts. Numerical simulations of both elastic and inelastic collisions are performed. For the example covered by the asymptotic theory the numerical results confirm that the collision is elastic and the theoretical predictions for the higher-order phase shifts. For the example not covered by the asymptotic theory the numerical results show that the collision is inelastic; an oscillatory wavetrain is produced by the colliding solitary waves. The higher-order phase shift for the faster (sech-type) solitary wave is found. For the slower (algebraic) wave, however, the shed radiation never completely separates from the solitary wave. Interaction with the shed radiation causes the phase shift to evolve long after collision is over, with no final higher-order phase shift able to be determined. An asymptotic mass-conservation law is to test the accuracy of the finite-difference scheme for the numerical solutions. It is shown that, in general, mass is not conserved by the higher-order mKdV equation, but varies during the interaction of the solitary waves.

Stabilizing the Benjamin–Feir instability

The Benjamin–Feir instability is a modulational instability in which a uniform train of oscillatory waves of moderate amplitude loses energy to a small perturbation of other waves with nearly the same frequency and direction. The concept is well established in water waves, in plasmas and in optics. In each of these applications, the nonlinear Schrodinger equation is also well established as an approximate model based on the same assumptions as required for the derivation of the Benjamin–Feir theory: a narrow-banded spectrum of waves of moderate amplitude, propagating primarily in one direction in a dispersive medium with little or no dissipation. In this paper, we show that for waves with narrow bandwidth and moderate amplitude, any amount of dissipation (of a certain type) stabilizes the instability. We arrive at this stability result first by proving it rigorously for a damped version of the nonlinear Schrodinger equation, and then by confirming our theoretical predictions with laboratory experiments on waves of moderate amplitude in deep water. The Benjamin–Feir instability is often cited as the first step in a nonlinear process that spreads energy from an initially narrow bandwidth to a broader bandwidth. In this process, sidebands grow exponentially until nonlinear interactions eventually bound their growth. In the presence of damping, this process might still occur, but our work identifies another possibility: damping can stop the growth of perturbations before nonlinear interactions become important. In this case, if the perturbations are small enough initially, then they never grow large enough for nonlinear interactions to become important.

Asymptotic solitons for a third-order Korteweg–de Vries equation

Abstract Solitary wave interaction for a higher-order version of the Korteweg–de Vries (KdV) equation is considered. The equation is obtained by retaining third-order terms in the perturbation expansion, where for the KdV equation only first-order terms are retained. The third-order KdV equation can be asymptotically transformed to the KdV equation, if the third-order coefficients satisfy a certain algebraic relationship. The third-order two-soliton solution is derived using the transformation. The third-order phase shift corrections are found and it is shown that the collision is asymptotically elastic. The interaction of two third-order solitary waves is also considered numerically. Examples of an elastic and an inelastic collision are both considered. For the elastic collision (which satisfies the algebraic relationship) the numerical results confirm the theoretical predictions, in particular there is good agreement found when comparing the third-order phase shift corrections. For the inelastic collision (which does not satisfy the algebraic relationship) an oscillatory wavetrain is produced by the interacting solitary waves. Also, the third-order phase shift corrections are found numerically for a range of solitary wave amplitudes. An asymptotic mass-conservation law is used to test the finite-difference scheme for the numerical solutions. In general, mass is not conserved by the third-order KdV equation, but varies during the interaction of the solitary waves.

A shallow fault-zone structure illuminated by trapped waves in the Karadere-Duzce branch of the North Anatolian Fault, western Turkey

SUMMARY We discuss the subsurface structure of the Karadere‐Duzce branch of the North Anatolian Fault based on analysis of a large seismic data set recorded by a local PASSCAL network in the 6 months following the Mw = 7.4 1999 Izmit earthquake. Seismograms observed at stations located in the immediate vicinity of the rupture zone show motion amplification and long-period oscillations in both P- and S-wave trains that do not exist in nearby off-fault stations. Examination of thousands of waveforms reveals that these characteristics are commonly generated by events that are well outside the fault zone. The anomalous features in fault-zone seismograms produced by events not necessarily in the fault may be referred to generally as fault-zone-related site effects. The oscillatory shear wave trains after the direct S arrival in these seismograms are analysed as trapped waves propagating in a low-velocity fault-zone layer. The time difference between the S arrival and trapped waves group does not grow systematically with increasing source‐receiver separation along the fault. These observations imply that the trapping of seismic energy in the Karadere‐Duzce rupture zone is generated by a shallow fault-zone layer. Traveltime analysis and synthetic waveform modelling indicate that the depth of the trapping structure is approximately 3‐4 km. The synthetic waveform modelling indicates further that the shallow trapping structure has effective waveguide properties consisting of thickness of the order of 100 m, a velocity decrease relative to the surrounding rock of approximately 50 per cent and an S-wave quality factor of 10‐15. The results are supported by large 2-D and 3-D parameter space studies and are compatible with recent analyses of trapped waves in a number of other faults and rupture zones. The inferred shallow trapping structure is likely to be a common structural element of fault zones and may correspond to the top part of a flower-type structure. The motion amplification associated with fault-zone-related site effects increases the seismic shaking hazard near fault-zone structures. The effect may be significant since the volume of sources capable of generating motion amplification in shallow trapping structures is large.

Asymptotic solitons for a higher-order modified Korteweg-de Vries equation.

Solitary wave interaction for a higher-order modified Korteweg-de Vries (mKdV) equation is examined. The higher-order mKdV equation can be asymptotically transformed to the mKdV equation, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive the higher-order two-soliton solution and it is shown that the interaction is asymptotically elastic. Moreover, the higher-order phase shifts are derived using the asymptotic theory. Numerical simulations of the interaction of two higher-order solitary waves are also performed. Two examples are considered, one satisfies the algebraic relationship derived from the asymptotic theory, and the other does not. For the example which satisfies the algebraic relationship the numerical results confirm that the collision is elastic. The numerical and theoretical predictions for the higher-order phase shifts are also in strong agreement. For the example which does not satisfy the algebraic relationship, the numerical results show that the collision is inelastic; an oscillatory wavetrain is produced by the interacting solitary waves. Also, the higher-order phase shifts for this inelastic example are tabulated, for a range of solitary wave amplitudes. An asymptotic mass-conservation law is derived and used to test the finite-difference scheme for the numerical solutions. It is shown that, in general, mass is not conserved by the higher-order mKdV equation, but varies during the interaction of the solitary waves.