Three-dimensional Nonlinear Wave Research Articles

Investigation of the wave solutions of two space–time fractional equations in physics

In this study, two fractional differential equations describing the characteristics of three-dimensional nonlinear ion-acoustic waves (IAWs) in plasmas are considered with the conformable fractional derivative (CFD). 3+1-dimensional space–time fractional Jimbo–Miwa (JM) equation and 3+1-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov (mKdV-ZK) equation are reduced into ordinary differential equations and an investigation into the exact solutions is carried out through the modified exponential function method (MEFM). These exact wave solutions are beneficial in interpreting the interior structure and characteristics of physical phenomena modeled by the aforementioned nonlinear evolution equations (NLEEs) arising in magnetized plasma. Various analytical methods were applied to such equations. Therefore, it is vital to determine the proper technique for the present problem. The proposed scheme has demonstrated a different spectrum of wave structures. The portrait of the exact solutions permits the exploration of plasmas, plasma models, and optics. The software system Mathematica is employed and the rational, exponential, trigonometric, and hyperbolic function solutions are acquired. The graphical results are presented for the physical nature of the solutions. The results are compared with the integer-order models.

A GPU-Accelerated Method for 3D Nonlinear Kelvin Ship Wake Patterns Simulation

The study of ship waves is important for ship detection, coastal erosion and wave drag. This paper proposed a highly paralleled numerical computation method for efficiently simulating three-dimensional nonlinear kelvin waves. First, a numerical model for nonlinear ship waves is established based on potential flow theory, the Jacobian-free Newton–Krylov (JFNK) method and the boundary integral method. To reduce the amount of data stored in the JFNK method and improve the computational efficiency, a banded preconditioner method is then developed by formulating the optimal bandwidth selection rule. After that, a Graphics Process Unit (GPU)-based parallel computing framework is designed, and we used the Compute Unified Device Architecture (CUDA) language to develop a GPU solution. Finally, numerical simulations of 3D nonlinear ship waves under multiple scales are performed by using the GPU and CPU solvers. Simulation results show that the proposed GPU solver is more efficient than the CPU solver with the same accuracy. More than 66% GPU memory can be saved, and the computational speed can be accelerated up to 20 times. Hence, the computation time for Kelvin ship waves simulation can be significantly reduced by applying the GPU parallel numerical scheme, which lays a solid foundation for practical ocean engineering.

Sixth-order finite difference schemes for nonlinear wave equations with variable coefficients in three dimensions

First, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.

On some recently derived exact solutions to the Euler equations

We present some explicit solutions with radial character in the setting of Eulerian framework to the three-dimensional nonlinear water waves in arctic regions. In particular, we focus on the motion of water waves with a moving bed and the pressure varying on free surface.

Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators.

Two-component three-dimensional atomic Bose-Einstein condensates supporting complex stable patterns

We report the computational discovery of complex, topologically charged, and spectrally stable states in three-dimensional multicomponent nonlinear wave systems of nonlinear Schr\"odinger type. While our computations relate to two-component atomic Bose-Einstein condensates in parabolic traps, our methods can be broadly applied to high-dimensional nonlinear systems of partial differential equations. The combination of the so-called deflation technique with a careful selection of initial guesses enables the computation of an breadth of patterns, including ones combining vortex lines, rings, stars, and vortex labyrinths. Despite their complexity, they may be dynamically robust and amenable to experimental observation, as confirmed by Bogoliubov--de Gennes spectral analysis and numerical evolution simulations.

Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus {mathbb {T}}^3. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on {mathbb {T}}^3 by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing Phi ^4_3-model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.

A flexible fully nonlinear potential flow model for wave propagation over the complex topography of the Norwegian coast

Coastal wave propagation and transformation are complicated due to the significant variations of water depth and irregular coastlines, which are typically present at the Norwegian fjords. A potential flow model provides phase-resolved solutions with low demands on computational resources. Many potential flow models are developed for offshore waves and lack of numerical treatments of coastal conditions. In the presented work, several modifications are introduced to a fully nonlinear potential flow model with a σ-grid for the purpose of coastal wave modelling: Shallow water breaking criteria are included in addition to deepwater breaking algorithms to approximate breaking waves over a complete range of water depth. A new coastline algorithm is introduced to detect complex coastlines and ensure robust simulations near the coast. The algorithm is compatible with structured grid arrangement in the horizontal plane and allows for high-order discretisation schemes for the free surface boundary conditions for an accurate representation of complex free surfaces. A parallelised solver for the Laplace equation is utilised to ensure fast simulations for large domains with multi-core infrastructures.The proposed model is validated against theories and experiments for various two- and three-dimensional nonlinear wave propagation and transformation cases that represent typical coastal conditions. The simulations show a good representation of nonlinear waves, and the results compare well with experiments. Furthermore, two large-scale engineering scenarios are simulated, where the applicability of the coastline algorithm and the parallel commutation capability of the model are demonstrated.

Wave-GAN: A deep learning approach for the prediction of nonlinear regular wave loads and run-up on a fixed cylinder

Machine learning techniques have inspired reduced-order solutions in the fluid mechanics field that show benefits of unprecedented capability and efficiency. Targeting ocean-wave problems, this work has developed a novel data-driven computational approach, named Wave-GAN. This new tool is based upon the conditional Generative Adversarial Network (GAN) principle, and it provides the ability to predict three-dimensional nonlinear wave loads and run-up on a fixed structure. The paper presents the principle of Wave-GAN and an application example of regular waves interacting with a vertical fixed cylinder. Computational Fluid Dynamics (CFD) is used to provide training and testing datasets for the Wave-GAN deep learning network. Upon verification, Wave-GAN proved the ability to provide accurate results for predicting wave load and run-up for wave conditions that were not informed during training. Yet the CFD-comparative results were only obtained within seconds by the deep learning tool. The promising results demonstrate Wave-GAN's outstanding potential to act as a pioneering sample of applying machine learning techniques to wave-structural interaction problems. It is envisioned that the new approach could be extended to more complex shapes and wave conditions to facilitate the various design stages of marine and offshore engineering applications such as monopiles. As a result, enhanced reliability is expected to optimise structural performance and prevent environmental disasters.

On the breaking inception of unsteady water wave packets evolving in the presence of constant vorticity

Abstract

Some Explicit Solutions to the Three-Dimensional Nonlinear Water Wave Problem

We present some explicit solutions (given in Eulerian coordinates) to the three-dimensional nonlinear water wave problem. The velocity field of some of the solutions exhibits a non-constant vorticity vector. An added bonus of the solutions we find is the possibility of incorporating a variable (in time and space) surface pressure which has a radial structure. A special type of radial structure of the surface pressure (of exponential type) is one of the features displayed by hurricanes, cf. Overland (Earle, Malahoff (eds) Overland in ocean wave climate, Plenum Pub. Corp., New York, 1979).

A Note on a Three-Dimensional Bäcklund Transformation

The Backlund transformation (BT) for a three-dimensional nonlinear wave equation and its nonlinear superposition formula are studied in this note. We prove that the three dimensional Backlund transformation obtained by Leibbrandt, et al. can be decomposed into three two-dimensional BTs. Some results on the N-dimensional Liouville equation are also discussed in the article.

Investigation of ion - acoustic wave dynamics in unmagnetized grain plasmas

In this work, we study an extended (3+1)-dimensional Jimbo-Miwa equation. We analyze the dynamics of three-dimensional nonlinear ion-acoustic waves in an unmagnetized plasma. We furnish the Lie symmetry technique to investigate the symmetry reductions of the examined model. The arbitrary functions involved in the Lie vectors are secured by using an optimization process, that leads to a reduction of the number of vectors in the commutator table. We next apply the singular manifold method (SMM) to the obtained reduced equations. This will result in a Schwarzian derivative whose solution will lead to new traveling wave solutions.

Harnessing a multi-dimensional fibre laser using genetic wavefront shaping

The multi-dimensional laser is a fascinating platform not only for the discovery and understanding of new higher-dimensional coherent lightwaves but also for the frontier study of the complex three-dimensional (3D) nonlinear dynamics and solitary waves widely involved in physics, chemistry, biology and materials science. Systemically controlling coherent lightwave oscillation in multi-dimensional lasers, however, is challenging and has largely been unexplored; yet, it is crucial for both designing 3D coherent light fields and unveiling any underlying nonlinear complexities. Here, for the first time, we genetically harness a multi-dimensional fibre laser using intracavity wavefront shaping technology such that versatile lasing characteristics can be manipulated. We demonstrate that the output power, mode profile, optical spectrum and mode-locking operation can be genetically optimized by appropriately designing the objective function of the genetic algorithm. It is anticipated that this genetic and systematic intracavity control technology for multi-dimensional lasers will be an important step for obtaining high-performance 3D lasing and presents many possibilities for exploring multi-dimensional nonlinear dynamics and solitary waves that may enable new applications.

Experimental investigation of weakly three-dimensional nonlinear wave interactions

Abstract The properties of surface waves coalescing and spreading directionally is presented where two identical wave trains propagate at an angle to each other and hence interact in an “X” configuration ranging from near-linear to violent breaking. An event of this type is believed to be one of the foundations of the formation of extreme waves in the open sea. The study involved measuring the wave surface elevation along the channels and hence both frequency spectra and energy were calculated. Comparisons between these data confirm that the directional spread has a significant effect on the nonlinear wave–wave interactions, especially for the wave profile and energy loss due to breaking. In general, the greater the approach angle of the two wave trains, the more cross-channel variation there is following breaking and the larger the energy loss during breaking. It is believed that the observed results will provide a helpful preliminary study of full three-dimensional interactions.

Mean flow generation by three-dimensional nonlinear internal wave beams

We study the generation of resonantly growing mean flow by weakly nonlinear internal wave beams. With a perturbational expansion, we construct analytic solutions for three-dimensional internal wave beams, exact up to first-order accuracy in the viscosity parameter. We specifically focus on the subtleties of wave beam generation by oscillating boundaries, such as wave makers in laboratory set-ups. The exact solutions to the linearized equations allow us to derive an analytic expression for the mean vertical vorticity production term, which induces a horizontal mean flow. Whereas mean flow generation associated with viscous beam attenuation – known as streaming – has been described before, we are the first to also include a peculiar inviscid mean flow generation in the vicinity of the oscillating wall, resulting from line vortices at the lateral edges of the oscillating boundary. Our theoretical expression for the mean vertical vorticity production is in good agreement with earlier laboratory experiments, for which the previously unrecognized inviscid mean flow generation mechanism turns out to be significant.

Exact Pollard-like internal water waves

In this paper we construct a new solution which represents Pollard-like three-dimensional nonlinear geophysical internal water waves. The Pollard-like solution includes the effects of the rotation of Earth and describes the internal water wave which exists at all latitudes across Earth and propagates above the thermocline. The solution is provided in Lagrangian coordinates. In the process we derive the appropriate dispersion relation for the internal water waves in a stable stratification and discuss the particles paths. An analysis of the dispersion relation for the constructed model identifies one mode of the internal water waves.

Stability analysis and applications of traveling wave solutions of three-dimensional nonlinear modified Zakharov–Kuznetsov equation in a magnetized plasma

Propagation of three-dimensional nonlinear solitary waves in a magnetized electron–positron plasma is analyzed. Modified extended mapping method is further modified and applied to three-dimensional nonlinear modified Zakharov–Kuznetsov equation to find traveling solitary wave solutions. As a result, electrostatic field potential, electric field, magnetic field and quantum statistical pressure are obtained with the aid of Mathematica. The new exact solitary wave solutions are obtained in different forms such as periodic, kink and anti-kink, dark soliton, bright soliton, bright and dark solitary waves, etc. The results are expressed in the forms of trigonometric, hyperbolic, rational and exponential functions. The electrostatic field potential and electric and magnetic fields are shown graphically. The soliton stability of these solitary wave solutions is analyzed. These results demonstrate the efficiency and precision of the method that can be applied to many other mathematical physical problems.

Asymptotics of a Solution of a Three-Dimensional Nonlinear Wave Equation near a Butterfly Catastrophe Point

The solution of the three-dimensional nonlinear wave equation −U″TT + U″XX + U″YY + U″ZZ = f(eT, eX, eY, eZ, U) by means of the method of matched asymptotic expansions is considered. Here e is a small positive parameter and the right-hand side is a smoothly changing source term of the equation. A formal asymptotic expansion of the solution of the equation is constructed in terms of the inner scale near a typical butterfly catastrophe point. It is assumed that there exists a standard outer asymptotic expansion of this solution suitable outside a small neighborhood of the catastrophe point. We study a nonlinear second-order ordinary differential equation (ODE) for the leading term of the inner asymptotic expansion depending on three parameters: u″xx = u5 − tu3 − zu2 − yu − x. This equation describes the appearance of a step-like contrast structure near the catastrophe point. We briefly describe the procedure for deriving this ODE. For a bounded set of values of the parameters, we obtain a uniform asymptotics at infinity of a solution of the ODE that satisfies the matching conditions. We use numerical methods to show the possibility of locating a shock layer outside a neighborhood of zero in the inner scale. The integral curves found numerically are presented.

Model Equations for Three-Dimensional Nonlinear Water Waves under Tangential Electric Field

We are concerned with gravity-capillary waves propagating on the surface of a three-dimensional electrified liquid sheet under a uniform electric field parallel to the undisturbed free surface. For simplicity, we make an assumption that the permittivity of the fluid is much larger than that of the upper-layer gas; hence, this two-layer problem is reduced to be a one-layer problem. In this paper, we propose model equations in the shallow-water regime based on the analysis of the Dirichlet-Neumann operator. The modified Benney-Luke equation and Kadomtsev-Petviashvili equation will be derived, and the truly three-dimensional fully localized traveling waves, which are known as “lumps” in the literature, are numerically computed in the Benney-Luke equation.