Nonlinear Wave Model Research Articles

Strong diffraction of nonlinear surface acoustic waves in crystals

Nonlinear distortion and shock formation in planar surface acoustic waves in anisotropic crystals have been modeled without [Hamilton et al., JASA (1999)] and with [Cormack et al., JASA 2022] piezoelectricity. Weak diffraction has been included in the paraxial approximation for nonlinear surface wave beams in isotropic solids [Shull et al., JASA (1995)]. Here, a procedure for including strong diffraction, i.e., without the paraxial approximation, in the model for nonlinear surface waves in crystals is presented, which incorporates the angular spectrum approach described by Kharusi and Farnell (JASA 1970). Anisotropy is defined by expressing the phase speed of a plane wave, and therefore, the magnitude of the corresponding wavenumber, as a function of the direction of propagation. Next, an explicit expression relating the two wavenumber components in the planar surface along which the wave propagates must be obtained as a function of direction, requiring iterative solution of a transcendental relation. The beam is then propagated incrementally away from the source, advancing the angular spectrum in k-space and including nonlinear interactions in the spatial domain to characterize the combined effects of diffraction and nonlinearity. Preliminary results are presented for converging nonlinear surface waves in crystals. [Work supported by IR&D at ARL:UT.]

Spatial estimation of unidirectional wave evolution based on ensemble data assimilation

With the limitation of the high sensitivity of nonlinear models to initial conditions, the accurate estimation of wave spatial evolution is difficult to perform at a long distance. At this stage, a helpful approach is to improve the accuracy and robustness of the model through data assimilation technique. A robust data assimilation framework is developed by coupling ensemble Kalman filtering (EnKF) with the nonlinear wave model. The spatial evolution is obtained by numerically integrating the viscous modified Nonlinear Schrödinger (MNLS) equation. The performance of the EnKF-MNLS coupled framework is tested using synthetic data and laboratory measurements. The synthetic data is generated by the MNLS simulation superposing the Gaussian noise. In the synthetic cases, the estimated wave envelopes agree well with the clean solution. The results of laboratory experiments indicate that the EnKF-MNLS framework can improve the accuracy of wave forecasts compared to noised MNLS simulations. This study aims to enhance the noise resistance of the nonlinear wave model in spatial evolution and improve the accuracy of the model forecast.

Applications of two kinds of Kudryashov methods for time fractional (2 + 1) dimensional Chaffee–Infante equation and its stability analysis

In this study, the beta time fractional (2 + 1) dimensional Chaffee–Infante equation used to describe the behavior of gas diffusion in a homogeneous medium is discussed. Generalized Kudryashov and modified Kudryashov procedures were used to discovered solitons of the equation. These methods can be easily applied and offer different solutions checked to other methods in the literature. At the same time, these two methods use symbolic calculations to better understand various nonlinear wave models and offer a powerful and effective mathematical approach. The solutions created in this article are different from those in the literature and will guide those working in the field of physics and engineering to better understand this model. Figures of the results were made values different from each other. The stability of the equations in applications has been demonstrated by testing the stability feature on some solutions obtained using the features of the Hamilton system. This work demonstrates the power and effectiveness of the methods discussed in applying many different forms of fractional-order nonlinear equations. The results obtained in this paper are original to our research and have the potential to be helpful in the fields of mathematical engineering and physics.

A dynamical behavior of the coupled Broer-Kaup-Kupershmidt equation using two efficient analytical techniques.

The aim of the present study is to identify multiple soliton solutions to the nonlinear coupled Broer-Kaup-Kupershmidt (BKK) system, including beta, conformable, local-fractional, and M-truncated derivatives. The coupled Broer-Kaup-Kupershmidt system is employed for modelling nonlinear wave evolution in mathematical models of fluid dynamics, plasmic, optical, dispersive, and nonlinear long-gravity waves. The travelling wave solutions to the above model are found using the Unified and generalised Bernoulli sub-ODE techniques. By modifying certain parameter values, we may create bright soliton, squeezed bell-shaped wave, expanded v-shaped soliton, W-shaped wave, singular soliton, and periodic solutions. The four distinct kinds of derivatives are compared quite effectively using 2D line graphs. Also, contour plots and 3D graphics are given by using Mathematica 10. Lastly, any pair of propagating wave solutions has symmetrical geometrical forms.

Nonlinear supersonic flutter of a composite panel backed by an acoustic cavity with finite-amplitude sound waves

In this paper, an implicit partitioned fluid-structure-acoustic interaction method is presented for predicting nonlinear aeroelastic response of an elastic composite laminated panel backed by an acoustic cavity and subject to a supersonic flow. The method is formulated in an arbitrary Lagrangian-Eulerian framework based on the unsteady Navier-Stokes equations for the supersonic flow, a geometrical nonlinear dynamics model for the composite laminated panel, and the nonlinear Westervelt equation for the finite-amplitude sound waves in cavity. A theoretical fluid-structure-acoustic model of the supersonic panel is proposed for validating the present numerical method, in which the nonlinear structural model of the panel, the first-order piston aerodynamics model of the supersonic flow and the nonlinear sound wave model of the cavity are coupled in a monolithic manner. It is found that cavity acoustic resonance can appear in the case a vibration mode of the panel coalescing with an acoustic mode of the cavity. Several new physical features of nonlinear acoustic resonance have been discovered, such as the presence of shock wave with high pressure gradients travelling along the resonant cavity in axial direction and the presence of higher-order harmonic wave components, which emphasize a nonlinear acoustic wave model of the cavity must be used to fully characterize these phenomena. In addition, the nonlinearity of the standing wave is quantitatively related to the acoustic modes by means of a proper orthogonal decomposition analysis.

Unconditional analysis of the linearized second-order time-stepping scheme combined with a mixed element method for a nonlinear time fractional fourth-order wave equation

In this article, we present a numerical algorithm for solving a two-dimensional nonlinear fourth-order time fractional wave model, where a Galerkin mixed finite element method yielded by introducing two auxiliary variables v=ut and σ=Δu−f(u) is used in the spatial direction and a second-order θ scheme with the weighted shifted Grünwald difference (WSGD) formula is applied in the time direction. Utilizing the space-time splitting technique without limiting the relationship between spatial mesh size and temporal step size, we derive the unconditional optimal error estimate in L2-norm and the stability result. Further, for verifying the feasibility and effectiveness of our algorithm with or without starting parts, we implement numerical calculations by choosing nonsmooth solutions.

Simulating Erosive and Accretive Conditions in the Swash: Applications of a Nonlinear Wave and Morphology Evolution Model

This work presents a new model for surf and swash zone morphology evolution induced by nonlinear waves. Wave transformation in the surf and swash zones is computed by a nonlinear wave model based on the higher order Boussinesq equations for breaking and non-breaking waves. Regarding sediment transport, the model builds on previous research by the authors and incorporates the latest update of a well-founded sediment transport formula. The wave and morphology evolution model is validated against two sets of experiments on beach profile change and is afterwards used to test the performance of a widely-adopted erosion/accretion criterion. The innovation of this work is the validation of a new Boussinesq-type morphology model under both erosive and accretive conditions at the foreshore (accretion is rarely examined in similar studies), which the model reproduces very well without modification of the empirical coefficients of the sediment transport formula used; furthermore, the model confirms the empirical erosion/accretion criterion even for conditions beyond the ones it was developed for and without imposing any model constraints. The presented set of applications highlights model capabilities in simulating swash morphodynamics, as well as its suitability for coastal erosion mitigation and beach restoration design

Optimal time two-mesh mixed finite element method for a nonlinear fractional hyperbolic wave model

<abstract><p>In this article, a second-order time discrete algorithm with a shifted parameter $ \theta $ combined with a fast time two-mesh (TT-M) mixed finite element (MFE) scheme was considered to look for the numerical solution of the nonlinear fractional hyperbolic wave model. The second-order backward difference formula including a shifted parameter $ \theta $ (BDF2-$ \theta $) with the weighted and shifted Grünwald difference (WSGD) approximation for fractional derivative was used to discretize time direction at time $ t_{n-\theta} $, the $ H^1 $-Galerkin MFE method was applied to approximate the spatial direction, and the fast TT-M method was used to save computing time of the developed MFE system. A priori error estimates for the fully discrete TT-M MFE system were analyzed and proved in detail, where the second-order space-time convergence rate in both $ L^2 $-norm and $ H^1 $-norm were derived. Detailed numerical algorithms with smooth and weakly regular solutions were provided. Finally, some numerical examples were provided to illustrate the feasibility and effectiveness for our scheme.</p></abstract>

The Riccati-Bernoulli sub-optimal differential equation method for analyzing the fractional Dullin-Gottwald-Holm equation and modeling nonlinear waves in fluid mediums

<abstract><p>The present study investigates the fractional Dullin-Gottwald-Holm equation by using the Riccati-Bernoulli sub-optimal differential equation method with the Bäcklund transformation. By employing a well-established criterion, the present study reveals novel cusp soliton solutions that resemble peakons and offers valuable insights into their dynamic behaviors and mysterious phenomena. The solution family encompasses various analytical solutions, such as peakons, periodic, and kink-wave solutions. Furthermore, the impact of both the time- and space-fractional parameters on all derived solutions' profiles is examined. This investigation's significance lies in its contribution to understanding intricate dynamics inside physical systems, offering valuable insights into various domains like fluid mechanics and nonlinear phenomena across different physical models. The computational technique's straightforward, effective, and concise nature is demonstrated through introduction of some graphical representations in two- and three-dimensional plots generated by adjusting the related parameters. The findings underscore the versatility of this methodology and demonstrate its applicability as a tool to solve more complicated nonlinear problems as well as its ability to explain many mysterious phenomena.</p></abstract>

$ L^1 $ local stability to a nonlinear shallow water wave model

$ L^1 $ local stability to a nonlinear shallow water wave model

Optimal Dirichlet boundary control by Fourier neural operators applied to nonlinear optics

We present an approach for solving optimal boundary control problems of nonlinear optics by using deep learning. For computing high resolution approximations of the solution to the nonlinear wave model, we propose higher order space-time finite element methods in combination with collocation techniques. Thereby, Cl-regularity in time of the global discrete solution is ensured. The simulation data is used to train solution operators that effectively leverage the higher regularity of the data. The solution operator is represented by Fourier Neural Operators and can be used as the forward solver in the optimal Dirichlet boundary control problem.The proposed algorithm is implemented and tested on high-performance computing platforms, with a focus on efficiency and scalability. The effectiveness of the approach is demonstrated on the problem of generating Terahertz radiation in periodically poled Lithium Niobate. The neural network is used as the solver in the optimal control setting to optimize the parametrization of the optical input pulse and maximize the yield of 0.3THz-frequency radiation.We exploit the periodic layering of the crystal to design the neural networks. The networks are trained to learn the propagation through one period of the layers. The recursive application of the network onto itself yields an approximation to the full problem. Our results indicate that the proposed method can achieve a speedup by a factor of 360 compared to classical methods. A comparison of our results to experimental data shows the potential to revolutionize the way we approach optimization problems in nonlinear optics.

Exact Solutions Modelling Nonlinear Atmospheric Gravity Waves

Exact solutions to the governing equations for atmospheric motion are derived which model nonlinear gravity wave propagation superimposed on atmospheric currents. Solutions are explicitly prescribed in terms of a Lagrangian formulation, which enables a detailed exposition of intricate flow characteristics. It is shown that our solutions are well-suited to modelling two distinct forms of mountain waves, namely: trapped lee waves in the Equatorial f-plane, and vertically propagating mountain waves at general latitudes.

Stability of constrained solitary waves for the Ostrovsky–Vakhnenko model in the coastal zone

In this paper, we consider a nonlinear water wave model describing the evolution of the internal tides in the coastal zone. First, we prove the existence of minimizers for the corresponding energy functional on the L2-norm constrained manifold by the concentrated compactness principle, and thus we obtain a ground state traveling wave solution of the model. Moreover, we show that the solution has exponential decay in the far field. Finally, we show that the traveling wave solution is non-degenerate, and thus the spectral stability of the solution can be obtained by using the instability index theory.

The explicit power series solution formation and computationof Lie point infinitesimals generators: Lie symmetry approach

This work scrutinizes the well-known nonlinear non-classical Sobolev-type wave model which addresses the fluid flow via fractured rock, thermodynamics and many other fields of modern sciences. The nonlinear non-classical Sobolev-type wave model provides a more comprehensive and accurate description of wave phenomena in a wide range of fields. By incorporating both nonlinearity and the complexities of dispersive waves, these models enhance our understanding of natural phenomena and enable more precise predictions and applications in various scientific and engineering disciplines. Therefore, this study is investigating it. Prior to this study, no previous research has performed Lie symmetry analysis and achieved invariant solutions of this kind. The symmetry generators are taking into account the Lie invariance criteria. The suggested approach produces the three dimensional Lie algebra, where translation symmetries in space and time are associated with mass conservation and conservation of energy, respectively and the other symmetries are scaling or dilation. The nonlinear non-classical Sobolev-type wave partial differential equation is transformed into a system of highly nonlinear ordinary differential equations by employing appropriate similarity transformations through using Lie group methodology. The power series technique is used to generate exact wave solutions because the inverse scattering transform cannot solve the Cauchy problem for this equation. The graphical behaviour of certain solutions is demonstrated in 3-D and 2-D for particular quantities of the physical factors in the investigated equation.

Advanced computational techniques for solving the modified KdV-KP equation and modeling nonlinear waves

Advanced computational techniques for solving the modified KdV-KP equation and modeling nonlinear waves

Lump Waves in a Spatial Symmetric Nonlinear Dispersive Wave Model in (2+1)-Dimensions

This paper aims to search for lump waves in a spatial symmetric (2+1)-dimensional dispersive wave model. Through an ansatz on positive quadratic functions, we conduct symbolic computations with Maple to generate lump waves for the proposed nonlinear model. A line of critical points of the lump waves is computed, whose two spatial coordinates travel at constant speeds. The corresponding maximum and minimum values are evaluated in terms of the wave numbers, and interestingly, all those extreme values do not change with time, either. The last section is the conclusion.

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.

The dynamical study and analysis of diverse bright-dark and breathers wave solutions of nonlinear evolution equations and their applications

In this work, we establish novel solitary wave solutions of three nonlinear evolution dynamical models, namely, Boussinesq model, [Formula: see text]-dimensional generalized shallow water wave (SWW) and [Formula: see text]-dimensional Fokas dynamical models by applying two-variable [Formula: see text]- expansion technique. The SWW equations are usually appropriate when the fluid is in a state where the vertical length scale is significantly smaller than the length of horizontal scale. In computer simulations for ocean engineering, Boussinesq-type models are frequently used to represent water waves in harbors and shallow seas. In order to understand the physical processes of waves inside and on the surface of water, the Fokas dynamical model plays a crucial part in wave theory. Analytical solutions in different forms such as solitons, solitary waves, trigonometric, hyperbolic, rational function solutions, breathers-type waves and more wave solutions are devised through using the proposed method. The exact solutions are also presented in graphical form having applications in engineering and other areas of applied sciences. The obtained results show that the given technique is universal and efficient. In addition, this technique can also be applied on lots of other nonlinear dynamical wave models occurring in many scientific real-world application domains, including engineering sciences, mathematics physics, and many more.

On the Propagation Model of Two-Component Nonlinear Optical Waves

Currently, two-component integrable nonlinear equations from the hierarchies of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrödinger equation are being actively investigated. In this paper, we propose a new hierarchy of two-component integrable nonlinear equations, which have an important difference from the already known equations. To construct the hierarchical equations, we use the monodromy matrix method, as first proposed by B.A. Dubrovin. The method we use consists of solving the following sequence of problems. First, using the Lax operator, we find the monodromy matrix, which is a polynomial in the spectral parameter. More precisely, we find a sequence of monodromy matrices dependent on the degree of this polynomial. Each Lax operator has its own sequence of monodromy matrices. Then, using the terms from the decomposition of the monodromy matrix, we construct a sequence of second operators from a Lax pair. A hierarchy of evolutionary integrable nonlinear equations follows from the conditions of compatibility of the sequence of Lax pairs. Also, knowledge of the monodromy matrix allows us to find stationary equations that are analogs of the Novikov equations for the Korteweg–de Vries equation. In addition, the characteristic equation of the monodromy matrix corresponds to the spectral curve equation of the relevant multiphase solution for the integrable nonlinear equation. Since the coefficients of the spectral curve equation are integrals of the hierarchical equations, they can be utilized to find the simplest solutions of the constructed integrable nonlinear equations. In this paper, we demonstrate the operation of this method, starting with the assignment of the Lax operator and ending with the construction of the simplest solutions.

Optical soliton for (2+1)-dimensional coupled integrable NLSE using Sardar-subequation method

The [Formula: see text]-dimensional coupled nonlinear Schrödinger equation has versatile applications for modeling nonlinear waves in different areas such as optics, atmospheric science, fluid dynamics, and plasma physics. This study focuses on the propagation of optical solitons and their interaction within various mediums such as multi-mode fiber and fiber arrays. The Sardar-subequation method is applied to achieve the bright, dark, combined bright–dark, periodic, combined dark-singular and singular optical solitons solutions. The 3D, contour and 2D profiles for some of the assimilation solutions are also plotted to show the physical behavior of the solutions. The obtained results demonstrate the effectiveness of the adopted methodology in obtaining traveling wave solutions and have many applications in applied mathematics and engineering. The novelty of our work lies in the successful application of Sardar-subequation method in obtaining diverse soliton solutions and exploring their physical behavior. This study goes beyond previous efforts in the literature by presenting a comprehensive analysis of soliton dynamics. This ability to describe and predict the behavior of optical solitons in diverse mediums opens up new opportunities for investigating the existing systems.