Nonlinear Wave Interactions Research Articles

Infragravity waves at the Hambantota port located in the North Indian Ocean

Infragravity (IG) waves are the primary cause of harbor oscillations. The Hambantota port located in the North Indian Ocean is taken as an example to study the IG waves in the vicinity and interior of a complex-shaped real harbor and to reveal the energy transfer of IG waves toward the harbor. The research employs a combination of in-situ wave measurement and numerical simulations conducted using the spectral wave model WAVEWATCH III. The latter could provide a long-term estimation of the level of IG wave energy over a broad geographical region, though it cannot reflect the complex mechanism of IG waves near/at the shore. Results indicate that, at the sensor near the harbor entrance, the free IG wave heights HFIG generally exceeds the bound IG wave height HBIG. However, during energetic sea states (significant wave height HS > 2.5 m in this study), HBIG increases significantly and occasionally exceeds HFIG. The spatial distributions of HS, HBIG, and HFIG indicate that the narrow continental shelf surrounding Sri Lanka has a marginal effect on attenuating swell energy. Consequently, the Hambantota port experiences a substantial impact from both swell and accompanying IG waves. If solely considering the maximum acceptable HS, the downtime for large ships would be greatly underestimated. The process of wave propagation into the harbor enables the energy transfer from bound IG waves to free IG waves that finally excite harbor oscillations. Albeit the tidal range is low, the tidal level has a noteworthy impact on harbor oscillations. The IG spectral density during low tides is generally greater than that during high tides due to the enhanced nonlinear interactions of short waves under low-tide conditions.

A spectral coupled boundary element method for the simulation of nonlinear surface gravity waves

The challenge of simulating the broad open sea with limited computational resources has long been of interest in ocean engineering research. In view of this issue, this paper proposes a fully nonlinear potential flow method named the spectral coupled boundary element method (SCBEM). By leveraging the approach of domain decomposition, SCBEM achieves significantly reduced computational cost and an order of magnitude increase in computational domain compared to the conventional boundary element method (BEM). The SCBEM encompasses the marine structure with only a tiny BEM domain and employs a high-order spectral layer to simulate the broad water outside the BEM domain. The performance of the SCBEM is evaluated through comparison with the wave damping approach and literature results for regular waves, modulated wave trains, focused waves, and diffraction of a vertical cylinder. The numerical results demonstrate the effectiveness and accuracy of the SCBEM in simulating a wide range of wavelengths and nonlinear wave interactions.

An Efficient Meshfree Numerical Technique to Solve Fractional Schamel–KdV Equation for Ion-Acoustic Solitary Waves in Dusty Plasma

The Schamel–KdV equation plays a vital role in studying the effect of electron trapping on the nonlinear interaction of ion-acoustic waves in plasma and dusty plasma. This work presents a numerical scheme-based radial basis function-finite difference (RBF-FD) method to solve the time-fractional Schamel–KdV equation. The advantages of using this method are their meshfree nature and flexibility in dealing with complex geometries. In the discretization process, a finite difference (FD) technique is used to discretize the temporal derivative, while the multiquadric (MQ) RBF is used to approximate the spatial derivatives. The theoretical convergence and stability analysis of the time-discrete scheme are also established. In addition, numerical experiments are executed through some illustrated problems, and the obtained results are compared with that acquired by the tanh method and Kudryashov method solutions to show the high accuracy and plausibility of the proposed technique. Also, the graphical representations are given to demonstrate the physical interpretation of the resulting wave structures.

The nonlinear wave interaction with a two-dimensional large-scale floating elastic plate

A two-dimensional fully nonlinear numerical wave tank is developed based on the higher order boundary element method and modal expansion to simulate the nonlinear interaction between waves and a finite horizontal floating elastic plate with draft. The large second harmonic displacement of a large-scale elastic plate near a second-order particular frequency is successfully calculated by this model. By the comparison with the results in a frequency model that involve perturbation expansion method, it is found that the large values of the second-order waves in a finite elastic plate near the particular frequency are not introduced by perturbation expansion. It is different from the case in infinite plate. The effect of the third-order particular triple frequency on the hydroelastic problem is small compared to the effect of the second-order particular frequency.

Multiplexing Linear and Nonlinear Bragg Diffractions through Volume Gratings Fabricated by Femtosecond Laser Writing in Lithium Niobate Crystal

The femtosecond-laser-writing technique provides a flexible method for fabrication of nonlinear photonic crystals in three dimensions, providing structures that enable efficient complex nonlinear wave interactions and modulation for applications including nonlinear holography, nonlinear beam shaping, and waveguide-integrated wavelength conversion. However, the tightly focused laser pulse inevitably causes structural modification and then changes the local refractive index, resulting in additional linear modulation of the interacting waves. Here, we use the same periodic distributions of the refractive index and the second-order nonlinear coefficient for grating arrays engineered in lithium niobate crystals by femtosecond laser writing to achieve polarization-dependent linear and nonlinear Bragg diffractions simultaneously. The experimental results show that the linear and nonlinear diffraction efficiencies range up to 31% and 2.9 × 10−5, respectively, for grating arrays with dimensions of 100 μm (x) × 100 μm (y) × 100 μm (z). This work paves the way toward the realization of the multiplexing of linear and nonlinear optical modulations in a single structure for potential applications that include multidimensional optical data storage and optical coding.

Planetary-scale MLT waves diagnosed through multi-station methods: a review

Most experimental investigations on planetary-scale waves in the mesosphere and lower thermosphere (MLT) region are based on single-station or -satellite spectral analysis methods, which suffer from intrinsic spectral aliasing/ambiguity. To overcome the aliasing, the author has developed and utilized dual- and multi-station spectral methods in a series of recent works. These methods were implemented on meteor radar observations and surface magnetometer observations. In the implements, a variety of waves were discovered or investigated in terms of seasonal variations and responses to sudden stratospheric warming events, such as lunar and solar tides (migrating and non-migrating), Rossby wave normal modes, ultra-fast Kelvin waves, and secondary waves of wave–wave nonlinear interactions between the previous waves. The current paper illustrates these methods using synthetic data, comparatively reviews the methods and results in plain language, and proposes a new representation, termed the adjusted Feynman diagram, to summarize the nonlinear interactions and explain their implications.Graphical

Two Novel Computational Techniques for Solving Nonlinear Time-Fractional Lax’s Korteweg-de Vries Equation

This article investigates the seventh-order Lax’s Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV equation was constructed to represent a wide range of physical processes involving the evolution and interaction of nonlinear waves. In the Caputo sense, the fractional derivative is considered. We employed the Yang transform, the Adomian decomposition method and the homotopy perturbation method to obtain the solution to the time-fractional Lax’s Korteweg–de Vries problem. We examined and compared a particular example with the actual result to verify the approaches. By utilizing these methods, we can construct recurrence relations that represent the solution to the problem that is being proposed, and we are then able to present graphical representations that enable us to visually examine all of the results in the proposed case for different fractional order values. Furthermore, the results of the current approach exhibit a good correlation with the precise solution to the problem being studied. Furthermore, the present study offers an example of error analysis. The numerical outcomes obtained by applying the provided approaches demonstrate that the techniques are easy to use and have superior computational performance.

Examination of High-Density Plasma Heating on the GAMMA 10/PDX Central Cell with 3D Wave Analysis Code

In GAMMA 10/PDX, ion heating is achieved using slow wave in the ion cyclotron frequency absorbed near the resonance layer. However, in high-density plasma, the slow wave is less likely to be excited due to the shielding effect. To overcome this challenge, researchers have investigated difference frequency (DF) waves excited by the nonlinear interaction of fast waves have been investigated. The fast waves can propagate to the core plasma even in high-density plasma making DF-waves an efficient option for ion heating. This study has suggested that a DF-wave with left-handed polarization, which is excited as a slow wave, is effective for ion heating in high-density plasma. The characteristics of the DF-wave were analyzed using a three-dimensional (3D) wave analysis code, TASK/WF3D, to investigate the ion resonance absorption of DF-wave heating. The results showed that when the antenna is positioned inside the core plasma to simulate DF-wave heating, ion resonance absorption is higher in high-density plasma than in low-density plasma. These results indicate that DF-wave heating is effective in high-density plasmas.

Simulating Landslide Generated Tsunamis in Palu Bay, Sulawesi, Indonesia

The results of an extensive series of numerical experiments of the GNOM-LS model for modelling the physical and energy characteristics of tsunami waves generated by landslides are presented. Based on the published data on the tsunami on 28 September 2018 in Palu Bay, we analysed the sensitivity of the distribution of wave heights along the coastline formed by the landslide system, depending on the characteristics of these landslides and model parameters. The complexity of the work lies in the lack of a holistic picture of the initial information about landslides, their number and accurate measurement data on the height of the waves of the event. We attempted to restore these conditions by comparing numerical simulations for various initialisations of the landslide system with available observational data. It is revealed that the simulated system has a very high sensitivity to the initial conditions and characteristics of landslides. An essential task of the work is interpreting a complex picture of the nonlinear interaction of tsunami waves with minor changes in the initial characteristics of landslides. Based on the numerical simulation of single landslides and a complete system of landslides, an analysis of the complex structure of the nonlinear interaction of tsunami waves is carried out.

Surface waves and currents in aquatic vegetation

A multiscale asymptotic theory is formulated for surface gravity waves and currents in finite-depth water with a vegetation canopy that provides a drag force on both flows with known drag coefficients. It assumes that the density is uniform and the depth is uniform pro tem and that the wave frequency is fast compared to the current advective rate. It is a quasi-linear theory in which the wave dynamics is independent of current and drag to leading order but provides perturbative corrections, and in which wave nonlinear interactions are neglected while quadratic wave-averaged wave fluxes and quadratic wave-drag effects are retained. The primary surface wave is modified by drag and current interactions, and the wave-averaged current momentum balance includes a wave-augmented drag force and several vortex forces due to Earth's rotation, current vorticity, Stokes drift and drag-induced wave vorticity. The wave-averaged current equations derived here are a suitable basis for future large-eddy simulation and submesoscale circulation computational models.

Hirota bilinear method and multi-soliton interaction of electrostatic waves driven by cubic nonlinearity in pair-ion–electron plasmas

Multi-soliton interaction of nonlinear ion sound waves in a pair-ion–electron (PIE) plasma having non-Maxwellian electrons including Kappa, Cairns, and generalized two spectral index distribution functions is studied. To this end, a modified Korteweg–de Vries (mKdV) equation is obtained to investigate the ion-acoustic waves in a PIE plasma at a critical plasma composition. The effects of temperature and density ratios and the non-Maxwellian electron velocity distributions on the overtaking interaction of solitons are explored in detail. The results reveal that both hump (positive peak) and dip (negative peak) solitons can propagate for the physical model under consideration. Two and three-soliton interactions are presented, and the novel features of interacting compressive and rarefactive solitons are highlighted. The present investigation may be useful in laboratory plasmas where PIE plasmas have been reported.

Nonlinear wave interactions in geochemical modeling

This paper is concerned with the study of the main wave interactions in a system of conservation laws in geochemical modeling. We study the modeling of the chemical complexes on the rock surface. The presence of stable surface complexes affects the relative permeability. We add terms representing surface complexes to the accumulation function in the model presented in [14]. This addition allows to take into account the interaction of ions with the rock surface in the modeling of the oil recovery by the injection of carbonated water. Compatibility hypotheses with the modeling are made on the coefficients of the system to obtain meaningful solutions. We developed a Riemann solver taking into account the complexity of the interactions and bifurcations of nonlinear waves. Such bifurcations occur at the inflection and resonance surfaces. We present the solution of a generalized eigenvalue problem in a (n+1)-dimensional case, which allows the construction of rarefaction curves. A method to find the discontinuous solutions is also presented. We find the solution path for some examples.

Nonlinear interaction of the fast magnetosonic wave with the extraordinary mode laser and ion-scaled turbulence generation in the magnetized plasma

This study introduces a nonlinear wave-wave coupling model to understand the nature of the turbulence generation at the ion scale observed in the interaction of high-intensity laser with plasma. For this nonlinear coupling of waves, the fast magnetosonic wave and the extraordinary mode laser are taken into account, along with the nonlinear ponderomotive force. The model dynamical equations have been developed for the extraordinary mode laser and the fast magnetosonic wave. Laboratory simulations have been employed to solve the obtained model equations utilizing the finite difference and pseudo-spectral methods. The simulation results show the initiation of the nonlinear progression of the laser beam at an early time and its turbulent nature at a later time. The background density perturbation, along with the cavity and hump caused by the ponderomotive force, has also been studied. The simulation results demonstrate that the propagation angle of the fast magnetosonic wave affects the magnitudes of the localization of the laser beam (pump wave) and the perturbed background density. The turbulent power spectra (ensemble-averaged) have also been studied for the different propagation angles of the fast magnetosonic wave. The turbulent spectra demonstrate that the plasma turbulence is MHD-type at small-scale lengths and ion-dominated at larger-scale lengths. This study also observed that the simulation results have some resemblance with the experimental results reported by Chatterjee et al.12 in the intense laser-plasma interaction experiment at a laboratory astrophysical scale.

Analysis of nonlinear water wave interaction solutions and energy exchange between different wave modes

In this study, we consider the ideal fluid model of an inviscid fluid, assuming that the fluid motion is adiabatic; the flow is irrotational, that is, the individual fluid particles do not rotate; vorticity ω̃=0; and the flow is incompressible, in which the density of fluid particles does not vary significantly with fluid motion and can be considered constant throughout the fluid volume and throughout the motion. We start with equations representing continuity, conservation of momentum, conservation of entropy, and streamline equations, respectively. It is then reduced to a standard system of equations describing motion in two dimensions, defined by the Laplace equation with appropriate kinematic and dynamic boundary conditions, in terms of velocity potential and surface elevation. Finally, the one-dimensional nonlinear Korteweg–De Vries (KdV) equation is derived. Then, we further investigate the interaction of multiple periodic waves using the KdV equation and explain the interaction wave energy transfer procedure between the primary and higher order harmonics, and the Phillips [“On the dynamics of unsteady gravity waves of finite amplitude. I. The elementary interactions,” J. Fluid Mech. 9, 193–217 (1960)] wave resonance criterion is employed for capturing the periodic wave interaction whose energy conversion is analyzed via Fourier spectra. It is also found that for solitons, multiple collisions of different solitons eventually regain their original shape and that higher-energy solitons have faster velocities than lower-energy solitons, which, to the best of our knowledge, is overlooked.

Nonlinear wave interactions on the surface of a conducting fluid under vertical electric fields

In this paper, we are concerned with capillary–gravity waves propagating on a two-dimensional conducting fluid under the effect of an electric field imposed in the direction perpendicular to the undisturbed free surface. This work aims to investigate the impact of the electric fields on the nonlinear wave interactions in this context. The full system is mathematically difficult to solve since it is a nonlinear, two-layered, free boundary problem, and the interface dynamics results from the strong coupling between the Euler equations for the lower fluid layer and an electric contribution from the upper gas layer. To investigate electrohydrodynamic wave interactions, we propose a novel numerical scheme based on a time-dependent conformal map and an interpolation technique to conduct unsteady simulations of the fully nonlinear electrified Euler equations of the two-layered problem. To gain analytical insights, we first derive weakly nonlinear envelope equations based on the method of multiple scales for the resonant triad, long-short wave interaction, and modulation of a single-mode wavetrain (a special case of resonant quartet interaction). Summative remarks are made to illustrate the transition from 3-wave interactions to 4-wave interactions and vice versa, occurring when the coefficients of the associated nonlinear Schrödinger equation become singular. The fully nonlinear results obtained by the prescribed numerical method are compared with the predictions by the weakly nonlinear theories, and good agreement is achieved.

Dynamics control in four-wave mixing processes in optical fiber

A nonlinear interaction of waves in a dispersive medium manifests itself in a four-wave mixing process that can be described as an evolution of waves’ parameters on a phase plane in a form of closed orbits. Here we propose a method to control these trajectories and to switch from one state to another in an optimal manner by implementing an abrupt change of the average power. The method is confirmed experimentally by the reconstruction of a fundamental four-wave mixing dynamics in an idealized model using iterative propagation in a short segment of fiber.

МОДЕЛЮВАННЯ РОЗПОДІЛЕНОГО НЕЛІНІЙНО-ОПТИЧНОГО ПІДСИЛЕННЯ У ТЕЛЕКОМУНІКАЦІЙНИХ ВОЛОКНАХ TRUEWAVERS ТА G.652

Common techniques for modeling of the nonlinear optical amplifiers, specifically based on the effect Raman effect, despite the simplicity of its technical implementation, are quite complex and cumbersome due to the necessity of theoretically describing the nonlinear interaction of many pump and signal optical waves using the corresponding systems of high-order differential equation. The fundamental limitations of such methodologies lie not only in the number of equations but also in the existence of special points in each equation describing the dynamics of signal waves (these points correspond to threshold values of pump power). Under these conditions, the solutions instabilities of the differential equations and their ambiguity sometimes have a detrimental impact on modeling results. An alternative modeling approach using the specified pump approximation requires particular investigations of its practical applicability to the analysis of distributed optical amplification in telecommunications fibers. Therefore, there is a need to determine the criterion for the applicability of the specified pump approximation for the development of a simplified model of nonlinear optical signal amplification under the conditions of their propagation along standard silica fibers. Among the undeniable advantages of analytical solutions obtained based on the simplified model there is the elimination of the mentioned constraints of solving systems of many differential equations. This article is devoted to modeling the dynamics of distributed Raman amplification, as function the length of standard single-mode optical fiber (SMF) and pump power. In justifying the proposed modeling methodology, it is shown that the parameters of active fibers and fiber Raman amplifier (hereinafter referred to as FRA) schemes correspond to the criterion for the applicability of the specified pump approximation. The paper presents an analytical expression for the criterion of the applicability of the specified pump approximation in the form of an upper limit of the signal power, which cannot exceed the amplified signal. Comparative analysis results of the distributed FRA parameters depending on the length of active fibers, such as G.652 and TrueWaveRS, are presented.

Nonlinear edge wave generation in aluminum plates with microstructural damage

Free edges are omnipresent in engineering components, such as I-beam flanges and aircraft wings. These components are subject to different types of damage with various scales, which can reduce their designed service life span. Nonlinear ultrasonic guided waves have attracted substantial research attention in the field of structural health monitoring (SHM), for identifying and tracking micro-defects in the early stage. Guided waves propagating along free edges are called edge waves. A number of studies have demonstrated that edge waves have the potential to characterise the defects near the structural edges. However, these studies have extensively studied the linear features of edge waves, typified as group and phase velocity, attenuation, and mode conversion. The nonlinear interactions of the edge waves and structural defects have not been fully understood. This study presents numerical and experimental analyses of nonlinear edge waves on the edge of an aluminum plate with microstructural damage. The focus has been directed to the higher harmonic generation. A three-dimensional (3D) finite element model (FEM) was developed and the material nonlinearity was simulated by Murnaghan's strain energy equation. Subsequently, the accuracy of the numerical model was validated by comparing numerical results with the experimental data. After that, the experimentally validated FEM was employed to carry out a parametric study to investigate the propagation characteristics of the nonlinear edge waves on aluminum plates with different fatigue levels. The results demonstrate that it is promising to use edge waves for detecting micro-defects in the free edges of metallic structures.

Фазовые эффекты при вынужденном рассеянии Мандельштама-Бриллюэна

In the study of stimulated Brillouin scattering (SBS) in the constant intensity approximation, the spatial behavior of the intensity of the Stokes scattering wave in a medium was investigated. It was found that as a result of nonlinear interaction of waves, the period of spatial beats changes. The intensity of the Stokes scattering component is considered as a function of the phase mismatch, pump and acoustic wave intensities. It is found that the efficiency of the scattering of the backward Stokes wave is affected by the total length of the nonlinear medium. According to the analytical expressions obtained in this work, the choice of the optimal parameters of the problem makes it possible to implement the regime of efficient generation of the Stokes scattering component in SBS. The process of generation and amplification of the Stokes scattering component is compared with experiment. By varying the pump intensity, one can control and manipulate the intensity of the output radiation of the Stokes component.

Phase effects at stimulated Brillouin scattering

In the study of stimulated Brillouin scattering (SBS) in the constant intensity approximation, the spatial behavior of the intensity of the Stokes scattering wave in a medium was investigated. It was found that as a result of nonlinear interaction of waves, the period of spatial beats changes. The intensity of the Stokes scattering component is considered as a function of the phase mismatch, pump and acoustic wave intensities. It is found that the efficiency of the scattering of the backward Stokes wave is affected by the total length of the nonlinear medium. According to the analytical expressions obtained in this work, the choice of the optimal parameters of the problem makes it possible to implement the regime of efficient generation of the Stokes scattering component in SBS. The process of generation and amplification of the Stokes scattering component is compared with experiment. By varying the pump intensity, one can control and manipulate the intensity of the output radiation of the Stokes component. Keywords: SBS, backward Stokes scattering wave, constant intensity approximation.