Resonant Nonlinear Wave Research Articles

Observation of multiple attractors and diffusive transport in a periodically driven Klein-Gordon chain.

We consider a Klein-Gordon chain that is periodically driven at one end and has dissipation at one or both boundaries. An interesting numerical observation in a recent study [Prem et al., Phys. Rev. B 107, 104304 (2023)2469-995010.1103/PhysRevB.107.104304] was that for driving frequency in the phonon band, there is a range of values of the driving amplitude F_{d}∈(F_{1},F_{2}) over which the energy current remains constant. In this range the system exhibits a traveling wave solution termed a "resonant nonlinear wave" (RNW). It was noted that the RNW mode occurs over a range (F_{1},F_{2}) and shrinks with increasing system size, N. Remarkably, we find that the RNW mode is in fact a stable solution even for F_{d}>F_{2}, and that in this regime there exist two attractors, both with finite basins of attraction. We improve the perturbative treatment for the RNW mode, presented in the earlier work, by including the contributions of third harmonics. We also consider the effect of thermal noise at the boundaries and find that the RNW mode is stable for small temperatures. Corresponding to the two attractors for large F_{d} at zero temperature, the system can now be in two nonequilibrium steady states. Finally, we present results for a different driving protocol [Komorowski et al., Commun. Math. Phys. 400, 2181 (2023)10.1007/s00220-023-04654-4] where F_{d} is taken to scale with system size as N^{-1/2} and dissipation is only at the nondriven end. We find that the steady state for this case can be characterized by Fourier's law. We point out interesting differences that occur because of our dynamics being nonlinear and Hamiltonian. Our results suggest the intriguing possibility of observing the high-current-carrying RNW phase in experiments by careful preparation of initial conditions.

A Hamiltonian theory for nonlinear resonant wave–particle interaction in weakly inhomogeneous magnetic field

We develop a Hamiltonian theory for the nonlinear resonant interactions between energetic particles and nonlinear frequency chirping waves in the weakly inhomogeneous magnetic field. A canonical transformation is constructed to separate the fast and slowly varying scales, and the Hamiltonian of the resonant particle is transformed to the local resonance reference frames. The Vlasov equation of the local distribution function moving at the local resonance velocity is obtained using Liouville's theorem. The evolution for the slowly varying wave envelope is derived from the Ampère's law with both cold plasma and energetic particle currents. The Vlasov equation coupled with the wave envelope equation self-consistently describes the dynamics of the deeply trapped resonant particles and the slowly varying coherent wave envelope. The application of the theory to the frequency chirping chorus wave in magnetospheric plasmas is also discussed.

Sum-frequency triad interactions among surface waves propagating through an ice sheet

We study nonlinear resonant wave–wave interactions which occur when ocean waves propagate into a thin floating ice sheet. Using multiple-scale perturbation analysis, we obtain theoretical predictions of the wave amplitude evolution as a function of distance travelled past the ice edge for a semi-infinite ice sheet. The theoretical predictions are supported by a high-order spectral (HOS) method capable of simulating nonlinear interactions in both open water and the ice sheet. Using the HOS method, the amplitude evolution predictions are extended to multiple (coupled) triad interactions and a single ice sheet of finite length. We relate the amplitude evolution to mechanisms with strong frequency dependence – ice bending strain, related to ice breakup, as well as wave reflection and transmission. We show that, due to sum-frequency interactions, the maximum strain in the ice sheet can be more than twice that predicted by linearised theory. For an ice sheet of finite length, we show that nonlinear wave reflection and transmission coefficients depend on a parameter in terms of wave steepness and ice length, and can have values significantly different than those from linear theory. In particular, we show that nonlinear sum-frequency interactions can appreciably decrease the total wave energy transmitted past the ice sheet. This work has implications for understanding the occurrence of ice breakup, wave attenuation due to scattering in the marginal ice zone and the resulting ice floe size distribution.

Dynamics and transport in the boundary-driven dissipative Klein-Gordon chain

Motivated by experiments on chains of superconducting qubits, we consider the dynamics of a classical Klein-Gordon chain coupled to coherent driving and subject to dissipation solely at its boundaries. As the strength of the boundary driving is increased, this minimal classical model recovers the main features of the "dissipative phase transition" seen experimentally. Between the transmitting and non-transmitting regimes on either side of this transition (which support ballistic and diffusive energy transport respectively), we observe additional dynamical regimes of interest. These include a regime of superdiffusive energy transport at weaker driving strengths, together with a "resonant nonlinear wave" regime at stronger driving strengths, which is characterized by emergent translation symmetry, ballistic energy transport, and coherent oscillations of a nonlinear normal mode. We propose a non-local Lyapunov exponent as an experimentally measurable diagnostic of many-body chaos in this system, and more generally in open systems that are only coupled to an environment at their boundaries.

Collisional broadening of nonlinear resonant wave–particle interactions

A general procedure for understanding plasma behaviour when resonant wave–particle interactions are the sole destabilizing and transport mechanism or only heating and/or current drive source is highlighted without recourse to involved numerical or analytical treatments. These phenomena are characterized by transport that appears to be collisionless even though collisions play a central role in narrow collisional boundary layers. The order of magnitude estimates, which include nonlinear effects, are shown to provide expressions in agreement with the principal results of recent toroidal Alfvén eigenmode (TAE), toroidal magnetic field ripple, and heating and current drive treatments. More importantly, the retention of nonlinearities leads to new estimates of the alpha particle energy diffusivity at saturation for TAE modes, and the ripple threshold at which superbanana plateau evaluations of alpha particle transport are modified by nonlinear radial drift effects. In addition, the estimates indicate when quasilinear descriptions for heating and current drive will begin to fail. The phenomenological procedure demonstrates that in magnetic fusion relevant plasmas, narrow collisional boundary layers must be retained for resonant wave–particle interactions as they enhance the role of collisions, and make stochastic particle motion unlikely to be more important than other nonlinear processes.

Level III Reliability Design of an Armor Block of Rubble Mound Breakwater Using Probabilistic Model of Wave Height Optimized for the Korean Sea Wave Conditions and Non-Gaussian Wave Slope Distribution

In this study, a Level III reliability design of an armor block of rubble mound breakwater was developed using the optimized probabilistic wave height model for the Korean marine environment and Van der Meer equation. To demonstrate what distinguishes this study from the others, numerical simulation was first carried out, assuming that wave slope follows Gaussian distribution recommended by PIANC. Numerical results showed that Gaussian wave slope distribution overpredicted the failure probability of armor block, longer and shorter waves, and on the contrary, underpredicted waves of the medium period. After noting the limitations of Gaussian distribution, some efforts were made to develop an alternative for Gaussian distribution. As a result, non-Gaussian wave slope distribution was analytically derived from the joint distribution of wave amplitude and period by Longuet–Higgins using the random variables transformation technique. Numerical results showed that non-Gaussian distribution could effectively address the limitations of Gaussian distribution due to its capability to account for the nonlinear resonant wave–wave interaction and its effects on the wave slope distribution that significantly influences the armor block’s stability. Therefore, the non-Gaussian wave slope distribution presented in this study could play an indispensable role in addressing controversial issues such as whether or not enormous armor blocks like a Tetrapod of 100 t frequently mentioned in developing countermeasures against rough seas due to climate change is too conservatively designed.

Dynamical behaviors and oblique resonant nonlinear waves with dual-power law nonlinearity and conformable temporal evolution

<p style='text-indent:20px;'>In this article, the oblique resonant traveling waves and dynamical behaviors of (2+1)-dimensional Nonlinear Schrödinger equation along with dual-power law nonlinearity, and fractal conformable temporal evolution are reported. The considered equation is converted to an ordinary differential equation by taking the traveling variable wave transform and properties of Khalil's conformable derivative into account. The modified Kudryashov method is implemented to divulge the oblique resonant traveling wave of such an equation. It is found that the obliqueness is only affected on width, but not on amplitude and phase patriots of resonant nonlinear propagating wave dynamics. The research outcomes are very helpful for analyzing the obliquely propagating nonlinear resonant wave phenomena and their dynamical behaviors in several nonlinear systems having Madelung fluids and optical bullets.

Steady-state multiple near resonances of periodic interfacial waves with rigid boundary

Steady-state resonant interfacial waves in a two-layer fluid within a frictionless duct are investigated theoretically. A combination of the homotopy analysis method (HAM) and Galerkin’s method is used to search for accurate steady-state resonant solutions with multiple near resonances. In the HAM, a piecewise parameter in the auxiliary linear operators is introduced to remove the small divisors caused by nearly resonant components. Convergent series solutions are then provided to the Galerkin iterations to accelerate the convergence rate. It is found that weakly nonlinear steady-state resonant waves form a continuum in the parameter space. As nonlinearity (wave steepness) increases, energy appears to be progressively shifted to sideband frequency components, effectively broadening the spectrum. The corresponding interfacial wave profile exhibits an almost fixed spatial pattern of repeated relatively high frequency, high-amplitude bursts followed by low-amplitude, longer waves. On examining the influence of density ratio, though changing slightly, the upper layer enlarges the amplitude of components near primary ones, which reduces the amplitude of higher frequency components, enlarges the wave steepness, and reduces the horizontal velocity in the wave field. Our results indicate that steady-state systems with resonant interactions among periodic interfacial wave components could occur naturally in the ocean. All these should enhance our understanding of periodic resonant interfacial waves.

David J. Benney: Nonlinear Wave and Instability Processes in Fluid Flows

David J. Benney (1930–2015) was an applied mathematician and fluid dynamicist whose highly original work has shaped our understanding of nonlinear wave and instability processes in fluid flows. This article discusses the new paradigm he pioneered in the study of nonlinear phenomena, which transcends fluid mechanics, and it highlights the common threads of his research contributions, namely, resonant nonlinear wave interactions; the derivation of nonlinear evolution equations, including the celebrated nonlinear Schrödinger equation for modulated wave trains; and the significance of three-dimensional disturbances in shear flow instability and transition.

A nonlinear flow-transition criterion for the onset of slugging in horizontal channels and pipes

In this work, the interfacial instability and transition of a two-fluid flow from a stratified state to large amplitude waves or slugs is considered. By combining an asymptotic approximation of the linear Orr-Sommerfeld analysis with nonlinear resonant wave interaction theory, a novel nonlinear slug-transition criterion is derived. This criterion corresponds to a bounding condition on the upper fluid’s velocity in order to limit the amount of energy (provided by the linear instability) which is transferred to long waves through resonant wave interactions. It is proposed that such a condition can predict the formation of large-amplitude long waves and/or slugs. Quantitative comparisons of the onset of slugging are made between the prediction by the nonlinear transition criterion and the experimental measurements carried out in a horizontal square channel. Good agreement is observed. An additional heuristic model is developed which generalizes the transition criterion to flow through horizontal pipes. Comparisons are made for flows through different pipe diameters and over a wide range of fluid properties. Good agreement between the present theoretical predictions and the experimental measurements is also observed.

Universality of sea wave growth and its physical roots

Assuming resonant nonlinear wave interactions to be the dominant physical mechanism of growing wind-driven seas we propose a concise relationship between instantaneous wave steepness and time or fetch of wave development expressed in dimensionless wave periods or lengths. This asymptotic physical law derived from the first principles of the theory of weak turbulence does not contain wind speed explicitly. The validity of this law is illustrated by results of numerical simulations, in situ measurements of growing wind seas and wind-wave tank observations. The impact of this new view of sea-wave physics is discussed in the context of conventional approaches to wave modelling and forecasting.

Nonlinear ultrasonic waves for structural monitoring: Thermal stress measurement and guided wave management

This presentation will cover two aspects of ultrasonic nonlinear wave propagation in solids. First, a new model is proposed to justify the existence of wave nonlinearities in constrained solids subjected to thermal excursions. This problem is solved on the basis of the interatomic potential of the solid that indicates a “residual” strain energy, due to the prevented thermal expansion, that is at least cubic as a function of strain. This study finds applications in the monitoring of thermal stresses in buckling-prone structures, such as continuously welded railroad tracks and pipelines. Experimental tests conducted on railroad tracks with realistic support will be also presented. Second, in the case of waveguides the efficiency of nonlinear ultrasonic testing based on higher-harmonic generation strongly relies on the correct identification of favorable combinations of primary and resonant double-harmonic nonlinear wave modes. This presentation will identify these combinations of wave modes in complex waveguides by extending the classical Semi-Analytical Finite Element formulation to the nonlinear regime, and implementing it into a highly flexible commercial Finite Element code. The proposed algorithm is benchmarked for four case-studies, including a railroad track, a viscoelastic plate, a composite quasi-isotropic laminate, and a reinforced concrete slab.

Influence of fiber orientation on the inherent acoustic nonlinearity in carbon fiber reinforced composites.

This paper presents the study of non-classical nonlinear response of fiber-reinforced composites. Nonlinear elastic wave methods such as nonlinear resonant ultrasound spectroscopy (NRUS) and nonlinear wave modulation spectroscopy have been used earlier to detect damages in several materials. It was observed that applying these techniques to composites materials becomes difficult due to the significant inherent baseline nonlinearity. Understanding the non-classical nonlinear nature of the composites plays a vital role in implementing nonlinear acoustic techniques for material characterization as well as qualitative nondestructive testing of composites. Since fiber reinforced composites are orthotropic in nature, the baseline response variation with fiber orientation is very important. This work explores the nature of the inherent nonlinearity by performing nonlinear resonant spectroscopy (NRS) in intact unidirectional carbon/epoxy samples with different fiber orientations with respect to major axis of the sample. Factors such as frequency shifts, modal damping ratio, and higher harmonics were analyzed to explore the non-classical nonlinear nature of these materials. Conclusions were drawn based on the experimental observations.

Nonlinear spatio-temporal instability regime for electrically forced viscous jets

This paper considers the problem of nonlinear instability in electrically driven viscous axisymmetric jets with respect to spatial and temporal growing disturbances in the presence of a uniform or non-uniform applied electric field. The mathematical modeling for the jets, which uses the original electrohydrodynamics equations (Melcher and Taylor, 1969) [8], is based on the nonlinear mechanics that govern the liquid jet due to tangential electric field effects. At the linear stage, we found that a particular jet of fluid could exhibit the Rayleigh and Conducting flow Instabilities for the spatial and temporal evolution of the disturbance. For the nonlinear regime of the problem, we studied the resonant instability and nonlinear wave interactions of certain modes that satisfy the dyad resonant condition. The nonlinear wave interactions in the jet provided a significant change in the fluid flow properties that extend notably the available understanding of the problem at the linear stage. It was found that the nonlinear resonant instability provides an amplifying effect on the magnitude of the disturbances which evolves the jet to reduce significantly its radius at a shorter axial location. For the case of higher viscosity fluid, the electric field in the jet was found to be increasing spatially and temporally when nonlinear wave interactions were taken into account during the resonant instability. The resulting nonlinear solutions for the jet thickness, jet׳s electric field, jet׳s surface charge and jet velocity are presented and discussed.

Sub-harmonic resonant wave interactions in the presence of a linear interfacial instability

This work considers the role of nonlinear sub-harmonic resonant wave interactions in the development of interfacial waves, which may be under the influence of a linear interfacial instability, in an inviscid two-fluid stratified flow through a horizontal channel. We begin by examining the case of resonant interactions between one linearly unstable mode and its sub-harmonic that may be linearly stable or unstable. Using the method of multiple scales, we derive the nonlinear interaction equations governing the time evolution of interacting wave amplitudes. These nonlinear equations account for the combined effects of both the nonlinear resonant interaction and the linear instability. We show that through this nonlinear coupling, the linearly stable sub-harmonic mode can achieve faster than exponential growth. It is found that such a mechanism is capable of generating large-amplitude long waves that are stable by linear stability analysis. The analytical predictions are cross validated by comparisons with direct nonlinear numerical simulations based on a more general perturbation based spectral method. Good agreement between the analytical and numerical solutions is observed. The more complicated case where a single mode is simultaneously involved in multiple (sub-harmonic and triad) resonances is also investigated numerically. The results demonstrate that chains of resonances can permit the energy generated by the linear instability, among high wavenumber components, to be passed across the spectrum to the longest wave components creating an efficient mechanism for the generation of large-amplitude long waves from unstable short waves.

Modeling of nonlinear guided waves and applications to structural health monitoring

Research efforts on nonlinear guided wave propagation have increased dramatically in the last few decades because of the large sensitivity of nonlinear waves to structural condition (defects, quasi-static loads, instability conditions, etc...). However, the mathematical framework governing the nonlinear guided wave phenomena becomes extremely challenging in the case of waveguides that are complex in either materials (damping, anisotropy, heterogeneous, etc...) or geometry (multilayers, geometric periodicity, etc…). The present work develops predictions of nonlinear second-harmonic generation in complex waveguides by extending the classical Semi-Analytical Finite Element formulation to the nonlinear regime, and implementing it into a highly flexible, yet very powerful, commercial Finite Element code. Results are presented for the following cases: a railroad track, a viscoelastic plate, a composite quasi-isotropic laminate, and a reinforced concrete slab. In these cases, favorable combinations of primary wave modes and resonant double-harmonic nonlinear wave modes are identified. Knowledge of such combinations is important to the implementation of structural monitoring systems for these structures based on higher-harmonic wave generation. The presentation will also present a specific application of nonlinear guided waves for the monitoring of thermal stresses in rail tracks to prevent buckling.

Deterministic learning of completely resonant nonlinear wave systems with Dirichlet boundary conditions

In this paper, we investigate the approximation of completely resonant nonlinear wave systems via deterministic learning. The plants are distributed parameter systems (DPS) describing homogeneous and isotropic elastic vibrating strings with fixed endpoints. The purpose of the paper is to approximate the infinite-dimensional dynamics, rather than the parameters of the wave systems. To solve the problem, the wave systems are first transformed into finite-dimensional dynamical systems described by ordinary differential equation (ODE). The properties of the finite-dimensional systems, including the convergence of the solution, as well as the dominance of partial system dynamics according to point-wise measurements, are analyzed. Based on the properties, second, by using the deterministic learning algorithm, an approximately accurate neural network (NN) approximation of the the finite-dimensional system dynamics is achieved in a local region along the recurrent trajectories. Simulation studies are included to demonstrate the effectiveness of the proposed approach.

Asymptotic approach for the nonlinear equatorial long wave interactions

In the present work we use an asymptotic approach to obtain the long wave equations. The shallow water equation is put as a function of an external parameter that is a measure of both the spatial scales anisotropy and the fast to slow time ratio. The values given to the external parameters are consistent with those computed using typical values of the perturbations in tropical dynamics. Asymptotically, the model converge toward the long wave model. Thus, it is possible to go toward the long wave approximation through intermediate realizable states. With this approach, the resonant nonlinear wave interactions are studied. To simplify, the reduced dynamics of a single resonant triad is used for some selected equatorial trios. It was verified by both theoretical and numerical results that the nonlinear energy exchange period increases smoothly as we move toward the long wave approach. The magnitude of the energy exchanges is also modified, but in this case depends on the particular triad used and also on the initial energy partition among the triad components. Some implications of the results for the tropical dynamics are disccussed. In particular, we discuss the implications of the results for El Niño and the Madden-Julian in connection with other scales of time and spatial variability.

Resonant instability and nonlinear wave interactions in electrically forced jets

We investigate the problem of linear temporal instability of the modes that satisfy the dyad resonance conditions and the associated nonlinear wave interactions in jets driven by either a constant or a variable external electric field. A mathematical model, which is developed and used for the temporally growing modes with resonance and their nonlinear wave interactions in electrically driven jet flows, leads to equations for the unknown amplitudes of such waves. These equations are solved for both water and glycerol jet cases, and the expressions for the dependent variables of the corresponding modes are determined. The results of the generated data for these dependent variables versus time indicate, in particular, that the instability resulted from the nonlinear interactions of such modes is mostly quite strong but can also lead to significant reduction in the jet radius.

Convergence of Fourier spectral method for resonant long-short nonlinear wave interaction

In this paper, the evolution equations with nonlinear term describing the resonance interaction between the long wave and the short wave are studied. The semi-discrete and fully discrete Crank-Nicholson Fourier spectral schemes are given. An energy estimation method is used to obtain error estimates for the approximate solutions. The numerical results obtained are compared with exact solution and found to be in good agreement.