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1
Deya, Aurélien. "On a non-linear 2D fractional wave equation." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 56, no. 1 (2020): 477–501. http://dx.doi.org/10.1214/19-aihp969.
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Vivas-Cortez, Miguel, Maham Nageen, Muhammad Abbas, and Moataz Alosaimi. "Investigation of Analytical Soliton Solutions to the Non-Linear Klein–Gordon Model Using Efficient Techniques." Symmetry 16, no. 8 (2024): 1085. http://dx.doi.org/10.3390/sym16081085.
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Nonlinear distinct models have wide applications in various fields of science and engineering. The present research uses the mapping and generalized Riccati equation mapping methods to address the exact solutions for the nonlinear Klein–Gordon equation. First, the travelling wave transform is used to create an ordinary differential equation form for the nonlinear partial differential equation. This work presents the construction of novel trigonometric, hyperbolic and Jacobi elliptic functions to the nonlinear Klein–Gordon equation using the mapping and generalized Riccati equation mapping methods. In the fields of fluid motion, plasma science, and classical physics the nonlinear Klein–Gordon equation is frequently used to identify of a wide range of interesting physical occurrences. It is considered that the obtained results have not been established in prior study via these methods. To fully evaluate the wave character of the solutions, a number of typical wave profiles are presented, including bell-shaped wave, anti-bell shaped wave, W-shaped wave, continuous periodic wave, while kink wave, smooth kink wave, anti-peakon wave, V-shaped wave and flat wave solitons. Several 2D, 3D and contour plots are produced by taking precise values of parameters in order to improve the physical description of solutions. It is noteworthy that the suggested techniques for solving nonlinear partial differential equations are capable, reliable, and captivating analytical instruments.
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Flores, Jesús, Ángel García, Mihaela Negreanu, Eduardo Salete, Francisco Ureña, and Antonio M. Vargas. "Numerical Solutions to Wave Propagation and Heat Transfer Non-Linear PDEs by Using a Meshless Method." Mathematics 10, no. 3 (2022): 332. http://dx.doi.org/10.3390/math10030332.
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The applications of the Eikonal and stationary heat transfer equations in broad fields of science and engineering are the motivation to present an implementation, not only valid for structured domains but also for completely irregular domains, of the meshless Generalized Finite Difference Method (GFDM). In this paper, the fully non-linear Eikonal equation and the stationary heat transfer equation with variable thermal conductivity and source term are solved in 2D. The explicit formulae for derivatives are developed and applied to the equations in order to obtain the numerical schemes to be used. Moreover, the numerical values that approximate the functions for the considered domain are obtained. Numerous examples for both equations on irregular 2D domains are exposed to underline the effectiveness and practicality of the method.
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Elmandouh, Adel, Aqilah Aljuaidan, and Mamdouh Elbrolosy. "The Integrability and Modification to an Auxiliary Function Method for Solving the Strain Wave Equation of a Flexible Rod with a Finite Deformation." Mathematics 12, no. 3 (2024): 383. http://dx.doi.org/10.3390/math12030383.
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Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, we modify an auxiliary equation method to obtain a new systematic and effective method that can be used for a wide class of non-linear evolution equations. This method is summed up in an algorithm that explains and clarifies the ease of its applicability. The proposed method is successfully applied to construct wave solutions. The developed solutions are grouped as periodic, solitary, super periodic, kink, and unbounded solutions. A graphic representation of these solutions is presented using a 3D representation and a 2D representation, as well as a 2D contour plot.
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P., Parameshwari, and Pushpalatha G. "Application of Nonlinear Two Dimension Wave Equation Dual Reciprocity Boundary Element Method." International Journal of Trend in Scientific Research and Development 2, no. 3 (2019): 2041–42. https://doi.org/10.31142/ijtsrd11584.
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The constructive numerical implementation of the two dimensional dual boundary element method. This paper present to solve nonlinear 2 D wave equation defined over a rectangular spatial domain the boundary conditions. Two dimension wave equation is a time domain problem, with three independent variables u,v,t. The applied to 2 D wave equation satisfactory authority. P. Parameshwari | G. Pushpalatha "Application of Nonlinear Two-Dimension Wave Equation Dual Reciprocity Boundary Element Method" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: https://www.ijtsrd.com/papers/ijtsrd11584.pdf
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Макаренко, Николай Иванович, Валерий Юрьевич Ляпидевский, Данила Сергеевич Денисенко, and Дмитрий Евгеньевич Кукушкин. "Nonlinear internal wave packets in shelf zone." Вычислительные технологии, no. 2(24) (April 17, 2019): 90–98. http://dx.doi.org/10.25743/ict.2019.24.2.008.
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В рамках модели невязкой слабостратифицированной жидкости рассматривается длинноволновое приближение, описывающее нелинейные волновые пакеты типа кноидальных волн. Построены семейства асимптотических решений, одновременно описывающие периодические последовательности приповерхностных волн в форме впадин и придонных волн типа возвышений. Показано, что картины расчетных профилей качественно согласуются со структурами внутренних волн, наблюдавшихся авторами в натурных экспериментах в шельфовой зоне моря. The problem on nonlinear internal waves propagating permanently in shallow fluid is studied semi-analytically in comparison with the field data measured on the sea shelf. At present, the most studied in this context are nonlinear solitary-type waves generated due to the tidal activity over continental slope. This paper deals with periodic cnoidaltype wave packets considered in the framework of mathematical model of continuously stratified fluid. Basic model involves the Dubreil-Jacotin-Long equation for a stream function that results from stationary fully non-linear 2D Euler equations. The longwave approximate equation describing periodic non-harmonic waves is derived by means of scaling procedure using small Boussinesq parameter. This parameter characterizes slight stratification of the fluid layer with the density profile being close to the linear stratification. The fine-scale density plays important role here because it determines the non-linearity rate of model equation, so it permits to consider strongly non-linear dispersive waves of large amplitude. As a result, constructed asymptotic solutions can simulate periodic wave-trains of sub-surface depression coupled with near-bottom wavetrains of isopycnal elevation. It is demonstrated that calculated wave profiles are in good qualitative agreement with internal wave structures observed by the authors in the field experiments performed annually during 2011-2018 in expeditions on the shelf of the Japanese sea.
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Jayewardene, Indra FW, Aliasghar Golshani, and Ed Couriel. "MAXIMUM MOMENTUM FLUX FOR STABILITY ANALYSIS OF MODEL AND PROTOTYPE BREAKWATERS." Coastal Engineering Proceedings, no. 37 (September 1, 2023): 6. http://dx.doi.org/10.9753/icce.v37.papers.6.
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None of the established formulae for breakwater armour stone stability, Hudson (1958) or van der Meer (1987), explicitly account for the water depth at the toe of the structure. More recently, Hughes (2004), Melby and Hughes (2004), Melby and Kobayashi (2011) have developed equations for stability and runup utilising the concept of wave momentum flux which explicitly accounts for the water depth of the wave probe(s) in close proximity to the structure. The equations are adapted to three forms; namely (1) linear, (2) extended-linear and (3) non-linear. In the paper linearity is assessed by using the Ursell number at each probe depth. Also, due to the placement of probes at various depths in MHL’s 2D wave flume it is possible to correlate the linearity of the wave measurement for the same time series and subsequently test the appropriateness of the momentum flux equation applied for assessment of stability and runup. The stability and runup data from 43 2D physical model tests where stability was previously assessed using van der Meer’s and Hudson’s equations are assessed using the momentum flux equations and an evaluation of the results has been made. It was found that the estimation of notional permeability and selection of the use of the plunging or surging formulae was critical to obtaining a closer match between measurement and prediction. The equations were also utilised in conjunction with numerical models to evaluate the armour size for repair of two breakwater heads in South Camden Haven and Bellambi. The maximum momentum flux equations were found to perform satisfactorily at these locations where the Ursell numbers were found to be high and the waves non-linear.
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Ansar, Rimsha, Muhammad Abbas, Pshtiwan Othman Mohammed, Eman Al-Sarairah, Khaled A. Gepreel, and Mohamed S. Soliman. "Dynamical Study of Coupled Riemann Wave Equation Involving Conformable, Beta, and M-Truncated Derivatives via Two Efficient Analytical Methods." Symmetry 15, no. 7 (2023): 1293. http://dx.doi.org/10.3390/sym15071293.
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In this study, the Jacobi elliptic function method (JEFM) and modified auxiliary equation method (MAEM) are used to investigate the solitary wave solutions of the nonlinear coupled Riemann wave (RW) equation. Nonlinear coupled partial differential equations (NLPDEs) can be transformed into a collection of algebraic equations by utilising a travelling wave transformation. This study’s objective is to learn more about the non-linear coupled RW equation, which accounts for tidal waves, tsunamis, and static uniform media. The variance in the governing model’s travelling wave behavior is investigated using the conformable, beta, and M-truncated derivatives (M-TD). The aforementioned methods can be used to derive solitary wave solutions for trigonometric, hyperbolic, and jacobi functions. We may produce periodic solutions, bell-form soliton, anti-bell-shape soliton, M-shaped, and W-shaped solitons by altering specific parameter values. The mathematical form of each pair of travelling wave solutions is symmetric. Lastly, in order to emphasise the impact of conformable, beta, and M-TD on the behaviour and symmetric solutions for the presented problem, the 2D and 3D representations of the analytical soliton solutions can be produced using Mathematica 10.
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Iqbal, Mujahid, Aly R. Seadawy, Dianchen Lu, and Xianwei Xia. "Construction of bright–dark solitons and ion-acoustic solitary wave solutions of dynamical system of nonlinear wave propagation." Modern Physics Letters A 34, no. 37 (2019): 1950309. http://dx.doi.org/10.1142/s0217732319503097.
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The nonlinear (2 + 1)-dimensional Zakharov–Kuznetsov (ZK) equations deal with the nonlinear behavior of waves in collision-less plasma, which contains non-isothermal cold ions and electrons. Two-dimensional dust acoustic solitary waves (DASWs) in magnetized plasma, which consist of trapped electrons and ions are leading to (2 + 1)-dim (ZK) equation by using the perturbation technique. We found the solitary wave solutions of (2 + 1)-dimensional (ZK)-equation, generalized (ZK)-equation and generalized form of modified (ZK)-equation by implementing the modified mathematical method. As a result, we obtained the bright–dark solitons, traveling wave and solitary wave solutions. The physical structure of obtained solutions is represented in 2D and 3D, graphically with the help of Mathematica.
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MAAS, LEO R. M. "WAVE ATTRACTORS: LINEAR YET NONLINEAR." International Journal of Bifurcation and Chaos 15, no. 09 (2005): 2757–82. http://dx.doi.org/10.1142/s0218127405013733.
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A number of physical mechanisms give rise to confined linear wave systems whose spatial structure is governed by a hyperbolic equation. These lack the discrete set of regular eigenmodes that are found in classical wave systems governed by an elliptic equation. In most 2D hyperbolic cases the discrete eigenmodes are replaced by a continuous spectrum of wave fields that possess a self-similar spatial structure and have a (point, line or planar) singularity in the interior. These singularities are called wave attractors because they form the attracting limit set of an iterated nonlinear map, which is employed in constructing exact solutions of this hyperbolic equation. While this is an inviscid, ideal fluid result, observations support the physical relevance of wave attractors by showing localization of wave energy onto their predicted locations. It is shown that in 3D, wave attractors may co-exist with a regular kind of trapped wave. Wave attractors are argued to be of potential relevance to fluids that are density-stratified, rotating, or subject to a magnetic field (or a combination of these) all of which apply to geophysical media.
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Fokina, K. V. "Testing of the accelerated two-dimensional model of surface potential waves." Fundamental and Applied Hydrophysics 16, no. 2 (2023): 34–43. http://dx.doi.org/10.59887/2073-6673.2023.16(2)-3.
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The paper focuses on the validation of the accelerated method for simulation of 2D-surface waves with a use of 2D-model derived by simplifications of 3D-equations for potential periodic waves at infinite depth. The separation of the velocity potential into nonlinear and linear components is used. A derivation of the equation for the total kinetic energy calculation in the surface-following coordinate system is given for the first time. The spectral characteristics of the wave field calculated with the accelerated model are compared with the results from the equivalent full 3D-model. The 3D-model is based on the numerical solution of the 3D-Poisson equation written in surface coordinates for the nonlinear component of the velocity potential. The similarity of the results obtained from the two versions of the model confirms that the new accelerated model can be used to quickly reproduce the wave field dynamics and thereby increase a speed of calculations by about two orders.
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Stepanyants, Yury. "The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns." Symmetry 12, no. 10 (2020): 1586. http://dx.doi.org/10.3390/sym12101586.
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The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe stationary moving symmetric wave patterns consisting of two plane solitary waves of equal amplitudes moving at an angle to each other. The results obtained within the approximate asymptotic theory are validated by comparison with the exact two-soliton solution of the Kadomtsev–Petviashvili equation (KP2-equation). The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin–Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.
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Arnold, Abramov, Yue Yutao, and Wang Mingming. "Non-split Perfectly Matched Layer Boundary Conditions for Numerical Solution of 2D Maxwell Equation." International Journal of Electromagnetics ( IJEL ) 3, no. 1 (2015): 1 to 9. https://doi.org/10.5281/zenodo.4563977.
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This paper developed a non - split perfectly matched layer (PML) boundary condition (BC) for Finite Difference Time Domain (FDTD) simulation of electromagnetic wave propagation in 2D structure. The point source for electric field has been exploited for propagation of electromagnetic field through 2D structures to validate developed approach. The identity of resulted field distribution to that obtained for split PML BC have been demonstrated.
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Goray, Leonid I. "Rigorous accounting diffraction on non-plane gratings irradiated by non-planar waves." Journal of Optics 24, no. 2 (2022): 025601. http://dx.doi.org/10.1088/2040-8986/ac4438.
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Abstract The modified boundary integral equation method (MIM) is considered a rigorous theoretical application for the diffraction of cylindrical waves by arbitrary profiled plane gratings, as well as for the diffraction of plane/non-planar waves by concave/convex gratings. This study investigates 2D diffraction problems of the filiform source electromagnetic field scattered by a plane lamellar grating and of plane waves scattered by a similar cylindrical-shaped grating. Unlike the problem of plane wave diffraction by a plane grating, the field of a localised source does not satisfy the quasi-periodicity requirement. Fourier transform is used to reduce the solution of the problem of localised source diffraction by the grating in the whole region to the solution of the problem of diffraction inside one Floquet channel. By considering the periodicity of the geometry structure, the problem of Floquet terms for the image can be formulated so that it enables the application of the MIM developed for plane wave diffraction problems. Accounting of the local structure of an incident field enables both the prediction of the corresponding efficiencies and the specification of the bounds within which the approximation of the incident field with plane waves is correct. For 2D diffraction problems of the high-conductive plane grating irradiated by cylindrical waves and the cylindrical high-conductive grating irradiated by plane waves, decompositions in sets of plane waves/sections are investigated. The application of such decomposition, including the dependence on the number of plane waves/sections and radii of the grating and wave front shape, was demonstrated for lamellar, sinusoidal and saw-tooth grating examples in the 0th and −1st orders as well as in the transverse electric and transverse magnetic polarisations. The primary effects of plane wave/section partitions of non-planar wave fronts and curved grating shapes on the exact solutions for 2D and 3D (conical) diffraction problems are discussed.
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Farag, Neveen G. A., Ahmed H. Eltanboly, Magdi S. El-Azab, and Salah S. A. Obayya. "Numerical Solutions of the (2+1)-Dimensional Nonlinear and Linear Time-Dependent Schrödinger Equations using Three Efficient Approximate Schemes." Fractal and Fractional 7, no. 2 (2023): 188. http://dx.doi.org/10.3390/fractalfract7020188.
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In this paper, the (2+1)-dimensional nonlinear Schrödinger equation (2D NLSE) abreast of the (2+1)-dimensional linear time-dependent Schrödinger equation (2D TDSE) are thoroughly investigated. For the first time, these two notable 2D equations are attempted to be solved using three compelling pseudo-spectral/finite difference approaches, namely the split-step Fourier transform (SSFT), Fourier pseudo-spectral method (FPSM), and the hopscotch method (HSM). A bright 1-soliton solution is considered for the 2D NLSE, whereas a Gaussian wave solution is determined for the 2D TDSE. Although the analytical solutions of these partial differential equations can sometimes be reached, they are either limited to a specific set of initial conditions or even perplexing to find. Therefore, our suggested approximate solutions are of tremendous significance, not only for our proposed equations, but also to apply to other equations. Finally, systematic comparisons of the three suggested approaches are conducted to corroborate the accuracy and reliability of these numerical techniques. In addition, each scheme’s error and convergence analysis is numerically exhibited. Based on the MATLAB findings, the novelty of this work is that the SSFT has proven to be an invaluable tool for the presented 2D simulations from the speed, accuracy, and convergence perspectives, especially when compared to the other suggested schemes.
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Tsai, Ching-Piao, Hong-Bin Chen, and John R. C. Hsu. "Second-Order Time-Dependent Mild-Slope Equation for Wave Transformation." Mathematical Problems in Engineering 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/341385.
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This study is to propose a wave model with both wave dispersivity and nonlinearity for the wave field without water depth restriction. A narrow-banded sea state centred around a certain dominant wave frequency is considered for applications in coastal engineering. A system of fully nonlinear governing equations is first derived by depth integration of the incompressible Navier-Stokes equation in conservative form. A set of second-order nonlinear time-dependent mild-slope equations is then developed by a perturbation scheme. The present nonlinear equations can be simplified to the linear time-dependent mild-slope equation, nonlinear long wave equation, and traditional Boussinesq wave equation, respectively. A finite volume method with the fourth-order Adams-Moulton predictor-corrector numerical scheme is adopted to directly compute the wave transformation. The validity of the present model is demonstrated by the simulation of the Stokes wave, cnoidal wave, and solitary wave on uniform depth, nonlinear wave shoaling on a sloping beach, and wave propagation over an elliptic shoal. The nearshore wave transformation across the surf zone is simulated for 1D wave on a uniform slope and on a composite bar profile and 2D wave field around a jetty. These computed wave height distributions show very good agreement with the experimental results available.
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MURAKI, DAVID J. "Large-amplitude topographic waves in 2D stratified flow." Journal of Fluid Mechanics 681 (June 16, 2011): 173–92. http://dx.doi.org/10.1017/jfm.2011.187.
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Our fundamental understanding of steady, stratified flow over two-dimensional (2D) topography rests on the pioneering works of G. Lyra and R. Long. Within linear theory, Lyra established the far-field radiation conditions that determine the downstream pattern of buoyancy waves. Soon after, Long discovered that the steady, nonlinear streamfunction for special cases of stratified, 2D flow could satisfy the same equations as linear theory, subject to an exact topographic boundary condition. Fourier methods are currently used to compute solutions to Long's theory for arbitrary topography in the near-hydrostatic or small-amplitude topographic parameter regimes. It is not generally appreciated however, that these methods encounter difficulties for flows that are both strongly non-hydrostatic and beyond linear amplitudes. By recasting Long's theory into a linear integral equation, this difficulty is shown to be a computational barrier associated with an ill-conditioning of the Fourier method. The problem is overcome through the development of a boundary integral computation which relies on some lesser known solutions from Lyra's original analysis. This method is well-conditioned for strongly non-hydrostatic flows, and is used to extend the exploration of critical overturning flows over Gaussian and bell-shaped ridges. These results indicate that the critical value of the non-dimensional height () asymptotes to a finite value with increasing non-hydrostatic parameter ().
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Siddique, Imran, Khush Bukht Mehdi, Sayed M. Eldin, and Asim Zafar. "Diverse optical solitons solutions of the fractional complex Ginzburg-Landau equation via two altered methods." AIMS Mathematics 8, no. 5 (2023): 11480–97. http://dx.doi.org/10.3934/math.2023581.
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<abstract> <p>This work evaluates the fractional complex Ginzburg-Landau equation in the sense of truncated M- fractional derivative and analyzes its soliton solutions and other new solutions in the appearance of a detuning factor in non-linear optics. The multiple, bright, and bright-dark soliton solutions of this equation are obtained using the modified $\left({{{G'} / {{G^2}}}} \right)$ and $\left({{1 / {G'}}} \right) - $expansion methods. The equation is evaluated with Kerr law, quadratic –cubic law and parabolic law non-linear fibers. To shed light on the behavior of solitons, the graphical illustrations in the form of 2D and 3D of the obtained solutions are represented for different values of various parameters. All of the solutions have been verified by substitution into their corresponding equations with the aid of a symbolic software package. The various forms of solutions to the aforementioned nonlinear equation that arises in fluid dynamics and nonlinear processes are presented. Moreover, we guarantee that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory.</p> </abstract>
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Lai, Yong G., and Han Sang Kim. "A Near-Shore Linear Wave Model with the Mixed Finite Volume and Finite Difference Unstructured Mesh Method." Fluids 5, no. 4 (2020): 199. http://dx.doi.org/10.3390/fluids5040199.
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The near-shore and estuary environment is characterized by complex natural processes. A prominent feature is the wind-generated waves, which transfer energy and lead to various phenomena not observed where the hydrodynamics is dictated only by currents. Over the past several decades, numerical models have been developed to predict the wave and current state and their interactions. Most models, however, have relied on the two-model approach in which the wave model is developed independently of the current model and the two are coupled together through a separate steering module. In this study, a new wave model is developed and embedded in an existing two-dimensional (2D) depth-integrated current model, SRH-2D. The work leads to a new wave–current model based on the one-model approach. The physical processes of the new wave model are based on the latest third-generation formulation in which the spectral wave action balance equation is solved so that the spectrum shape is not pre-imposed and the non-linear effects are not parameterized. New contributions of the present study lie primarily in the numerical method adopted, which include: (a) a new operator-splitting method that allows an implicit solution of the wave action equation in the geographical space; (b) mixed finite volume and finite difference method; (c) unstructured polygonal mesh in the geographical space; and (d) a single mesh for both the wave and current models that paves the way for the use of the one-model approach. An advantage of the present model is that the propagation of waves from deep water to shallow water in near-shore and the interaction between waves and river inflows may be carried out seamlessly. Tedious interpolations and the so-called multi-model steering operation adopted by many existing models are avoided. As a result, the underlying interpolation errors and information loss due to matching between two meshes are avoided, leading to an increased computational efficiency and accuracy. The new wave model is developed and verified using a number of cases. The verified near-shore wave processes include wave shoaling, refraction, wave breaking and diffraction. The predicted model results compare well with the analytical solution or measured data for all cases.
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Tao, Sixing. "Breathers, resonant multiple waves and complexiton solutions of a (2+1)-dimensional nonlinear evolution equation." AIMS Mathematics 8, no. 5 (2023): 11651–65. http://dx.doi.org/10.3934/math.2023590.
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<abstract><p>Based on the Hirota bilinear form of a (2+1)-dimensional equation, breathers and resonant multiple waves as well as complexiton solutions are considered in this paper. First, the breather waves are constructed via employing the extend homoclinic test method. By calculation, two kinds of solutions are obtained. Through analysis, three pairs of breathers consisting of hyperbolic functions and trigonometric functions are derived. Furthermore, a rouge wave solution is deduced by applying the Taylor expansion method to a obtained breather wave. In addition, related figures are plotted to illustrate the dynamical features of these obtained solutions. Then, two types of the resonant multi-soliton solutions are obtained by applying the linear superposition principle to the the Hirota bilinear form. At the same time, 3D profiles and 2D density plots are presented to depict the intersection progression of wave motion. Finally, the complexiton solutions are constructed according to the yielded resonant multi-soliton solutions by further utilizing the linear superposition principle. By considering different domain fields, several types of complexiton solutions including the positive ones are derived. Moreover, related 3D and 2D figures are plotted for the obtained results in order to vividly exhibit their dynamics properties.</p></abstract>
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Cimpoiasu, Rodica. "Multiple explicit solutions of the 2D variable coefficients Chafee–Infante model via a generalized expansion method." Modern Physics Letters B 35, no. 19 (2021): 2150312. http://dx.doi.org/10.1142/s0217984921503127.
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In this work, we do apply a generalized expansion method to the realistic two-dimensional (2D) Chafee–Infante model with time-variable coefficients which is encountered in physical sciences.The key ideas of this method consist in: (i) to choose a nonlinear wave variable in respect to time-variable into the general finite series solution of the governing model; (ii) to take a full advantage from the general elliptic equation introduced as an auxiliary equation which can degenerate into sub-equations such as Riccati equation, the Jacobian elliptic equations, the generalized Riccati equation. Based upon this powerful technique, we successfully construct for the first time several types of non-autonomous solitary waves as well as some non-autonomous triangular solutions, rational or doubly periodic type ones. We investigate the propagation of non-autonomous solitons and we emphasize as well upon the influence of the variable coefficients. We are providing and analyzing a few graphical representations of some specific solutions. The results of this paper will be valuable in the study of nonlinear physical phenomena. The above- mentioned method could be employed to solve other partial differential equations with variable coefficients which describe various complicated natural phenomena.
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Aljber, Morhaf, Kaho Nogami, Jae-Soon Jeong, Jonathan Salar Cabrera, and Han Soo Lee. "TSUNAMI MODELLING IN A BUILT-IN COASTAL ENVIRONMENT WITH ADAPTIVE MESH RIFINEMENT AND VARYING BOTTOM FRICTION." Coastal Engineering Proceedings, no. 37 (September 1, 2023): 40. http://dx.doi.org/10.9753/icce.v37.management.40.
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The great Japan earthquake on March/11/2011provoked an extreme tsunami wave toward the northeastern Japanese coast. The tsunami inundation covered a wide range of territories stretching over 500 km2 and caused devastating influences that claimed more than 15,000 lives (Mori et al. (2011)). Miyagi Prefecture, the closest mainland to the epicentre, received massive tsunami waves that reached over 40 m in run-up height. Prasetyo et al. (2019) aimed to reproduce the tsunami wave inundation in Onagawa Town of Miyagi Prefecture by building a physical model using Hybrid Tsunami Open Flume in Ujigawa Open Laboratory (HyTOFU) at Kyoto University to study the inundation process by measuring the wave height and arrival time. For that purpose, two wave types were used, the (Pump-type) hydraulic bore, and the (Piston-type) solitary wave. Also, 2D and quasi-3D (Q3D) models were tested for their robustness and applied to emulate tsunami propagation in the complex coastal city model. In this study, we adopted adaptive mesh refinement (AMR) for its efficiency in tsunami modelling with Basilisk open-source flow solver for non-linear shallow water (Saint-Venant) equations and fully non-linear Boussinesq (Green-Naghdi) equations (Popinet (2015)). The numerical experiment was conducted to validate the model by comparing the wave height with the physical model experiment, the 2D, and Q3D model results by Prasetyo et al. (2019).
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Yang, Renhu, Weijian Mao, and Xu Chang. "An efficient seismic modeling in viscoelastic isotropic media." GEOPHYSICS 80, no. 1 (2015): T63—T81. http://dx.doi.org/10.1190/geo2013-0439.1.
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Energy is absorbed and attenuated when seismic waves propagate in real earth media. Hence, the viscoelastic medium needs to be considered. There are many ways to construct the viscoelastic body, in which the generalized standard linear viscoelastic body is the most representative one. For viscoelastic wave propagation and imaging, it is very important to obtain a compact and efficient viscoelastic equation. Because of this, we derived a set of simplified viscoelastic equations in isotropic media on the basis of the standard linear solid body and the constitutive relation for a linear viscoelastic isotropic solid. The simplified equations were composed of the linear equations of momentum conservation, the stress-strain relations, and the memory variable equations. During the derivation of the equations, the Lamé differentiation matrix, which has a similar form to the stiffness matrix and indicates the relations between viscoelastic and elastic stiffness matrices, was introduced to simplify the memory variable equations. Analogous to the elastic equations, the simplified equations have symmetrically compact forms and are very useful for efficient viscoelastic modeling, migration, and inversion. Applied to a 2D simple model and the 2D SEG/EAGE salt model, the results show that our simplified equations are more efficient in computation than Carcione’s equations.
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KHAWAR, JAWAD, ANWAR UL-HAQUE, and SAJID RAZA CHAUDHRY. "VALIDATION OF 2D MULTI-BLOCK HIGH-SPEED COMPRESSIBLE TURBULENT FLOW SOLVER." International Journal of Computational Methods 04, no. 01 (2007): 33–57. http://dx.doi.org/10.1142/s0219876207000947.
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A 2D multi-block high-speed compressible turbulent flow solver CFD2D based on the Jones and Launders two-equation k –ε turbulence model is developed. Method of solution employed is Finite Volume Method. Its basic algorithm is based on the approximate Riemann solver with the three-step Runge–Kutta time integration. Its additional feature includes Wilcox model for compressibility correction of k–ε turbulence model, Girmaji algebraic Reynolds stress (non-linear stress) model and linear stress model for evaluation of turbulent stresses. For validation purpose, code is applied to a 2D diamond aerofoil and a wedge ramp attached to a flat plate. CFD-predicted results are compared to the experimental results for shock wave and shock wave boundary layer interaction on the trailing edge of the fin. Contour plots are also compared to the Schlieren photographs. Flow simulation shows ability of the code to capture the physics of the flow both qualitatively and quantitatively.
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Wang, Xiaofeng, Weizhong Dai, and Shuangbing Guo. "A conservative linear difference scheme for the 2D regularized long-wave equation." Applied Mathematics and Computation 342 (February 2019): 55–70. http://dx.doi.org/10.1016/j.amc.2018.09.029.
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Torii, André Jacomel, Roberto Dalledone Machado, and Marcos Arndt. "GFEM for modal analysis of 2D wave equation." Engineering Computations 32, no. 6 (2015): 1779–801. http://dx.doi.org/10.1108/ec-07-2014-0144.
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Purpose – The purpose of this paper is to present an application of the Generalized Finite Element Method (GFEM) for modal analysis of 2D wave equation. Design/methodology/approach – The GFEM can be viewed as an extension of the standard Finite Element Method (FEM) that allows non-polynomial enrichment of the approximation space. In this paper the authors enrich the approximation space with sine e cosine functions, since these functions frequently appear in the analytical solution of the problem under study. The results are compared with the ones obtained with the polynomial FEM using higher order elements. Findings – The results indicate that the proposed approach is able to obtain more accurate results for higher vibration modes than standard polynomial FEM. Originality/value – The examples studied in this paper indicate a strong potential of the GFEM for the approximation of higher vibration modes of structures, analysis of structures subject to high frequency excitations and other problems that concern high frequency oscillatory phenomena.
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Wu, Reng-Lai, Ye-Jun Long, Hong-Jie Xue, Yabin Yu, and Hui-Fang Hu. "Plasmon dispersions in ultrathin metallic films." International Journal of Modern Physics B 28, no. 27 (2014): 1450189. http://dx.doi.org/10.1142/s0217979214501896.
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We present an eigen-equation for plasmon of ultrathin films based on the self-consistent linear response approximation (SCLRA). The calculations for plasmon dispersion in both single and multilayer systems are reported. There are two types of plasmon in the plasmon spectrum, two-dimensional (2D) and bulk-like (BL) modes. The plasmon energy of the 2D mode is zero in the long wave limit, while the one of BL mode is nonzero in the long-wave limit. Given a surface electron density, with the decrease of the wave vector the dispersions of the 2D plasmon of different layer systems become equal to each other, and approach results of the pure 2D system.
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Alsafri, Naher Mohammed A., and Hamad Zogan. "Probing the diversity of kink solitons in nonlinear generalised Zakharov-Kuznetsov-Benjamin-Bona-Mahony dynamical model." AIMS Mathematics 9, no. 12 (2024): 34886–905. https://doi.org/10.3934/math.20241661.
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<p>This investigation offers an innovative analytical strategy, namely the Riccati modified extended simple equation method (RMESEM), to establish and analyze soliton results of the (2+1)-dimensional dynamical generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation (GZK-BBME) in plasma physics. This equation models the physical phenomena of long waves with small and finite amplitude in magnetic plasma. Using a wave transformation, the employed transformative technique first converts GZK-BBME to a nonlinear ordinary differential equation (NODE). With the incorporation of the Riccati equation, a close-form solution is then assumed for the resultant NODE by RMESEM, which converts the NODE into a set of algebraic equations. The fresh plethora of soliton results in the form of rational, exponential, rational-hyperbolic and periodic functional cases are obtained by addressing this set of equations. Several contour, 3D, and 2D graphs are also employed to visualizes the dynamics of these constructed soliton results. These graphs demonstrate that the acquired solitons adopts the type of diverse kink solitons, including cuspon, dark, bright, lump-type, and dark-bright kinks. In addition, our proposed RMESEM shows the applications of the model by producing different traveling soliton results, providing qualitative information on the GZK-BBMEs and possible applications in dealing with other similar kinds of non-linear models.</p>
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Alekseev, Gennady, Olga Dyakonova, Alexey Lobanov, and Andrey Yu. Fershalov. "2D Electromagnetic Wave Scattering Problem for Cylindrical Cloak Incorporating PEMC-Layer." Key Engineering Materials 685 (February 2016): 75–79. http://dx.doi.org/10.4028/www.scientific.net/kem.685.75.
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The 2D scattering problem is studied for the case of a two-dimensional cylindrical invisibility cloak, incorporating a tiny layer of perfect electromagnetic conductor (PEMC). The solution of the corresponding Helmholtz equation with variable coefficients is seeked as cylindrical functions Fourier series. The coefficients of these series are determined by solving the system of four linear algebraic equations with respect to four unknown coefficients with badly-conditioned matrix. The Singular Value Decomposition (SVD) method is applied for solving the system. Based on this idea the efficient numerical method is developed for solving the cloaking problem under study. The properties of this algorithm are studied and some results of numerical experiments are discussed.
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Suarez, Leandro, Rodrigo Cienfuegos, Eric Barthélemy, Hervé Michallet, and Cristian Escauriaza. "LAGRANGIAN DRIFTER MODELLING OF AN EXPERIMENTAL RIP CURRENT." Coastal Engineering Proceedings 1, no. 33 (2012): 35. http://dx.doi.org/10.9753/icce.v33.currents.35.
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A non-uniform alongshore wave forcing on an experimental uneven mobile bathymetry create mean circulation on a rip channel. A 2D numerical hydrodynamic model that integrates the non-linear shallow-water equations in a shock-capturing finite-volume framework is used to validate the nearshore circulation, and drifters displacement.
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Sulaiman, Tukur Abdulkadir, Canan Unlu, and Hasan Bulut. "On the wave solutions to the TRLW equation." ITM Web of Conferences 22 (2018): 01033. http://dx.doi.org/10.1051/itmconf/20182201033.
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In this study, a nonlinear model is investigated, namely; the time regularized long wave equation. Various solitary wave solutions are constructed such as the non-topological, compound topological-non-topological bell-type, singular and compound singular soliton solutions. Under the choice of suitable parameters values, the 2D and 3D graphs to all the obtained solutions are plotted. The reported results in this study may be helpful in explaining the physical meanings of some important nonlinear models arising in the field of nonlinear science.
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An, Xinliang, Haoyang Chen, and Silu Yin. "Low regularity ill-posedness for elastic waves driven by shock formation." American Journal of Mathematics 145, no. 4 (2023): 1111–81. http://dx.doi.org/10.1353/ajm.2023.a902956.
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abstract: In this paper, we construct counterexamples to the local existence of low-regularity solutions to elastic wave equations in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad's classic results on the scalar wave equation by showing that the Cauchy problem for 3D elastic waves, a physical system with multiple wave-speeds, are ill-posed in $H^3(\Bbb{R}^3)$. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties of the 3D case come from the multiple wave-speeds and its associated non-strict hyperbolicity. We obtain the desired results by designing and combining a geometric approach and an algebraic approach, equipped with detailed studies and calculations of the structures and coefficients of the corresponding non-strictly hyperbolic system. Moreover, the ill-posedness we depict also applies to 2D elastic waves, which corresponds to a strictly hyperbolic case.
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Ali Shehab Shams Eldeen, Ahmed M. R. El-Baz, and Abdalla Mostafa Elmarhomy. "CFD Modeling of Regular and Irregular Waves Generated by Flap Type Wave Maker." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 85, no. 2 (2021): 128–44. http://dx.doi.org/10.37934/arfmts.85.2.128144.
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The improvement of wave generation in numerical tanks represents the key factor in ocean engineering development to save time and effort in research concerned with wave energy conversion. For this purpose, this paper introduces a numerical simulation method to generate both regular and irregular waves using Flap-Type wave maker. A 2D numerical wave tank model is constructed with a numerical beach technique, the independence of the numerical beach slope is tested to reduce the wave reflections. The different governing parameters of the Flap type wave maker were studied such as periodic time dependency and length of the flap stroke. The linear wave generated was validated against the wave maker theory WMT, the numerical results agreed with WMT. The Pierson-Moskowitz model is used to generate irregular waves with different frequencies and amplitudes. The numerical model succeeded to generate irregular waves which was validated against published experimental data and with Pierson-Moskowitz spectrum model using Fourier expansion theory in the frequency domain. Useful results are presented in this paper based on the numerical simulation to understand the characteristics of the waves. This paper produces a full guide to generate both regular and irregular waves numerically using ANSYS-CFX approach to solve the 2D Unsteady Reynolds Averaged Navier-Stokes Equation (URANS).
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Operto, Stéphane, Jean Virieux, A. Ribodetti, and J. E. Anderson. "Finite-difference frequency-domain modeling of viscoacoustic wave propagation in 2D tilted transversely isotropic (TTI) media." GEOPHYSICS 74, no. 5 (2009): T75—T95. http://dx.doi.org/10.1190/1.3157243.
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A 2D finite-difference, frequency-domain method was developed for modeling viscoacoustic seismic waves in transversely isotropic media with a tilted symmetry axis. The medium is parameterized by the P-wave velocity on the symmetry axis, the density, the attenuation factor, Thomsen’s anisotropic parameters [Formula: see text] and [Formula: see text], and the tilt angle. The finite-difference discretization relies on a parsimonious mixed-grid approach that designs accurate yet spatially compact stencils. The system of linear equations resulting from discretizing the time-harmonic wave equation is solved with a parallel direct solver that computes monochromatic wavefields efficiently for many sources. Dispersion analysis shows that four grid points per P-wavelength provide sufficiently accurate solutions in homogeneous media. The absorbing boundary conditions are perfectly matched layers (PMLs). The kinematic and dynamic accuracy of the method wasassessed with several synthetic examples which illustrate the propagation of S-waves excited at the source or at seismic discontinuities when [Formula: see text]. In frequency-domain modeling with absorbing boundary conditions, the unstable S-wave mode is not excited when [Formula: see text], allowing stable simulations of the P-wave mode for such anisotropic media. Some S-wave instabilities are seen in the PMLs when the symmetry axis is tilted and [Formula: see text]. These instabilities are consistent with previous theoretical analyses of PMLs in anisotropic media; they are removed if the grid interval is matched to the P-wavelength that leads to dispersive S-waves. Comparisons between seismograms computed with the frequency-domain acoustic TTI method and a finite-difference, time-domain method for the vertical transversely isotropic elastic equation show good agreement for weak to moderate anisotropy. This suggests the method can be used as a forward problem for viscoacoustic anisotropic full-waveform inversion.
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Procesi, Michela. "Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D." Discrete & Continuous Dynamical Systems - A 13, no. 3 (2005): 541–52. http://dx.doi.org/10.3934/dcds.2005.13.541.
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Altybay, Arshyn, Dauren Darkenbayev, and Nurbapa Mekebayev. "Parallel numerical simulation of the 2D acoustic wave equation." International Journal of Electrical and Computer Engineering (IJECE) 14, no. 6 (2024): 6519. http://dx.doi.org/10.11591/ijece.v14i6.pp6519-6525.
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Mathematical simulation has significantly broadened with the advancement of parallel computing, particularly in its capacity to comprehend physical phenomena across extensive temporal and spatial dimensions. High-performance parallel computing finds extensive application across diverse domains of technology and science, including the realm of acoustics. This research investigates the numerical modeling and parallel processing of the two-dimensional acoustic wave equation in both uniform and non-uniform media. Our approach employs implicit difference schemes, with the cyclic reduction algorithm used to obtain an approximate solution. We then adapt the sequential algorithm for parallel execution on a graphics processing unit (GPU). Ultimately, our findings demonstrate the effectiveness of the parallel approach in yielding favorable results.
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Claes, N., and R. Keppens. "Thermal stability of magnetohydrodynamic modes in homogeneous plasmas." Astronomy & Astrophysics 624 (April 2019): A96. http://dx.doi.org/10.1051/0004-6361/201834699.
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Context. Thermal instabilities give rise to condensations in the solar corona, and are the most probable scenario for coronal rain and prominence formation. We revisit the original theoretical treatment done by Field (1965, ApJ, 142, 531) in a homogeneous plasma with heat-loss effects and combine this with state-of-the-art numerical simulations to verify growth-rate predictions and address the long-term non-linear regime. We especially investigate interaction between multiple magnetohydrodynamic (MHD) wave modes and how they in turn trigger thermal mode development. Aims. We assess how well the numerical MHD simulations retrieve the analytically predicted growth rates. We complete the original theory with quantifications of the eigenfunctions, calculated to consistently excite each wave mode. Thermal growth rates are fitted also in the non-linear regime of multiple wave–wave interaction setups, at the onset of thermal instability formation. Methods. We performed 2D numerical MHD simulations, including an additional (radiative) heat-loss term and a constant heating term to the energy equation. We mainly focus on the thermal (i.e. entropy) and slow MHD wave modes and use the wave amplitude as a function of time to make a comparison to predicted growth rates. Results. It is shown that the numerical MHD simulations retrieve analytically predicted growth rates for all modes, of thermal and slow or fast MHD type. In typical coronal conditions, the latter are damped due to radiative losses, but their interaction can cause slowly changing equilibrium conditions which ultimately trigger thermal mode development. Even in these non-linear wave-wave interaction setups, the growth rate of the thermal instability agrees with the exponential profile predicted by linear theory. The non-linear evolutions show systematic field-guided motions of condensations with fairly complex morphologies, resulting from thermal modes excited through damped slow MHD waves. These results are of direct interest to the study of solar coronal rain and prominence fine structure. Our wave–wave interaction setups are relevant for coronal loop sections which are known to host slow wave modes, and hence provide a new route to explain the sudden onset of coronal condensation.
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Ali, Karmina K., R. Yilmazer, H. Bulut, Tolga Aktürk, and M. S. Osman. "Abundant exact solutions to the strain wave equation in micro-structured solids." Modern Physics Letters B 35, no. 26 (2021): 2150439. http://dx.doi.org/10.1142/s021798492150439x.
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In this study, the strain wave equation in micro-structured solids which take an important place in solid physics is presented for consideration. The generalized exponential rational function method is used for this purpose which is one of the most powerful methods of constructing abundantly distinct, exact solutions of nonlinear partial differential equations. In micro-structured solids, wave propagation is based on the structure of the strain wave equation. As a consequence, we successfully received many different exact solutions, including non-topological solutions, periodic singular solutions, topological solutions, singular solutions, like periodic lump solutions. Furthermore, in order to better understand their physical interpretation, 2D, 3D, and counter plots are graphed for each of the solutions acquired.
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Rao, Wan-Ting, Jing-Dong Tang, and Jin-Fang Xing. "Factors affecting the shear wave elastic quantitative measurement of penile tissue in rats." Asian Biomedicine 17, no. 1 (2023): 22–29. http://dx.doi.org/10.2478/abm-2023-0040.
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Abstract Background: As a new ultrasound technology, 2-dimensional shear wave elastography (2D-SWE) can evaluate the elastic characteristics of penile tissue. However, no studies have reported the main factors affecting the shear wave elastic quantitative measurement (SWQ) in penile tissue. Objectives: To analyze the main factors affecting the SWQ reflecting the elastic characteristics of penile tissue by 2D-SWE. Methods: Twenty healthy male Sprague–Dawley rats (5–60 weeks old) were selected for this study. We performed the 2D-SWE examination on the penis using the Aixplorer ultrasound system, with SWQ as the measurement index. We performed penile immunohistochemistry analysis with the positive area proportion (PAP) of alpha-smooth muscle actin (PAPS) and type III collagen fiber (PAPC) as the outcomes. Then, we conducted multiple linear regression analysis to explore the correlation of SWQ with PAPS and PAPC and established the regression equation. Results: The multiple linear regression analysis showed that the linear regression equation (SWQ = 10.376 – 0.05 PAPS – 0.07 PAPC) was statistically significant (F = 21.153, P < 0.001). The content of smooth muscle cells (SMCs) and collagen fibers was negatively correlated with SWQ, affecting 42.6% of the total variation in SWQ (R 2 = 0.426). Conclusions: SMCs and collagen fibers are the main factors affecting the SWQ value of penile tissue and the primary tissue components determining the SWQ when using 2D-SWE to quantitatively evaluate the elastic characteristics of penile tissue.
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Seadawy, Aly, Asghar Ali, and Noufe Aljahdaly. "The nonlinear integro-differential Ito dynamical equation via three modified mathematical methods and its analytical solutions." Open Physics 18, no. 1 (2020): 24–32. http://dx.doi.org/10.1515/phys-2020-0004.
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AbstractIn this work, we construct traveling wave solutions of (1+1) - dimensional Ito integro-differential equation via three analytical modified mathematical methods. We have also compared our achieved results with other different articles. Portrayed of some 2D and 3D figures via Mathematica software demonstrates to understand the physical phenomena of the nonlinear wave model. These methods are powerful mathematical tools for obtaining exact solutions of nonlinear evolution equations and can be also applied to non-integrable equations as well as integrable ones. Hence worked-out results ascertained suggested that employed techniques best to deal NLEEs.
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Ozdemir, Neslihan. "M-truncated soliton solutions of the fractional (4+1)-dimensional Fokas equation." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 13, no. 1 (2023): 123–29. http://dx.doi.org/10.11121/ijocta.2023.1321.
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This article aims to examine M-truncated soliton solutions of the fractional (4+1)-dimensional Fokas equation (FE), which is a generalization of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations. The fractional (4+1)$-dimensional Fokas equation with the M-truncated derivatives is also studied first time in this study. The generalized projective Riccati equations method (GPREM) is successfully implemented. In the application of the presented method, a suitable fractional wave transformation is chosen to convert the proposed model into a nonlinear ordinary differential equation. Then, a linear equation system is acquired utilizing the GPREM, the system is solved, and the suitable solution sets are obtained. Dark and singular soliton solutions are successfully derived. Under the selection of appropriate values of the parameters, 2D, 3D, and contour plots are also displayed for some solutions.
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Riyanti, C. D., Y. A. Erlangga, R. E. Plessix, W. A. Mulder, C. Vuik, and C. Oosterlee. "A new iterative solver for the time-harmonic wave equation." GEOPHYSICS 71, no. 5 (2006): E57—E63. http://dx.doi.org/10.1190/1.2231109.
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The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can be solved efficiently by a direct method. In three dimensions, direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large. Iterative methods are an alternative. These methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence. The iterative solution of the time-harmonic wave equation has long been a notoriously difficult problem in numerical analysis. Recently, a new preconditioner based on a strongly damped wave equation has heralded a breakthrough. The solution of the linear system associated with the preconditioner is approximated by another iterative method, the multigrid method. The multigrid method fails for the original wave equation but performs well on the damped version. The performance of the new iterative solver is investigated on a number of 2D test problems. The results suggest that the number of required iterations increases linearly with frequency, even for a strongly heterogeneous model where earlier iterative schemes fail to converge. Complexity analysis shows that the new iterative solver is still slower than a time-domain solver to generate a full time series. We compare the time-domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively. The new iterative solver can be applied straightforwardly to 3D problems.
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Wang, Xiaoming, Shehbaz Ahmad Javed, Abdul Majeed, Mohsin Kamran, and Muhammad Abbas. "Investigation of Exact Solutions of Nonlinear Evolution Equations Using Unified Method." Mathematics 10, no. 16 (2022): 2996. http://dx.doi.org/10.3390/math10162996.
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In this article, an analytical technique based on unified method is applied to investigate the exact solutions of non-linear homogeneous evolution partial differential equations. These partial differential equations are reduced to ordinary differential equations using different traveling wave transformations, and exact solutions in rational and polynomial forms are obtained. The obtained solutions are presented in the form of 2D and 3D graphics to study the behavior of the analytical solution by setting out the values of suitable parameters. The acquired results reveal that the unified method is a suitable approach for handling non-linear homogeneous evolution equations.
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Ullah, Naeem, Muhammad Imran Asjad, Jan Awrejcewicz, Taseer Muhammad, and Dumitru Baleanu. "On soliton solutions of fractional-order nonlinear model appears in physical sciences." AIMS Mathematics 7, no. 5 (2022): 7421–40. http://dx.doi.org/10.3934/math.2022415.
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<abstract><p>In wave theory, the higher dimensional non-linear models are very important to define the physical phenomena of waves. Herein study we have built the various solitons solutions of (4+1)-dimensional fractional-order Fokas equation by using two analytical techniques that is, the Sardar-subequation method and new extended hyperbolic function method. Different types of novel solitons are attained such as, singular soliton, bright soliton, dark soliton, and periodic soliton. To understand the physical behavior, we have plotted 2D and 3D graphs of some selected solutions. From results we concluded that the proposed methods are straightforward, simple, and efficient. Moreover, this paper offers a hint, how we can convert the fractional-order PDE into an ODE to acquire the exact solutions. Also, the proposed methods and results can be help to examine the advance fractional-order models which seem in optics, hydrodynamics, plasma and wave theory etc.</p></abstract>
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Brühl, Markus, and Matthias Becker. "Analysis of Subaerial Landslide Data Using Nonlinear Fourier Transform Based on Korteweg-deVries Equation (KdV-NLFT)." Journal of Earthquake and Tsunami 12, no. 02 (2018): 1840002. http://dx.doi.org/10.1142/s179343111840002x.
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Subaerial and underwater landslides, rock falls and glacier calvings can generate impulse waves in lakes, fjords and the open sea. Experiments with subaerial landslides have shown that, depending on the slide characteristics, different wave types (Stokes, cnoidal or bore-like waves) are generated. Each of these wave types shows different wave height decay with increasing distance from the impact position. Furthermore, in very shallow water, the first impulse wave shows characteristic properties of a solitary wave. The nonlinear Fourier transform based on the Korteweg–deVries equation (KdV-NLFT) is a frequency-domain analysis method that decomposes shallow-water free-surface data into nonlinear cnoidal waves instead of linear sinusoidal waves. This method explicitly identifies solitons as spectral components within the given data. In this study, we apply the KdV-NLFT for the very first time to available 2D and 3D landslide-test data. The objective of the nonlinear decomposition is to identify the hidden nonlinear spectral structure of the impulse waves, including solitons. Furthermore, we analyze the determined solitons at different downstream positions from the impact point with respect to soliton propagation and modification. Finally, we draw conclusions for the prediction of the expected landslide-generated downstream solitons in the far-field.
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Bakaleinikov, L. A., and A. Gordon. "Dynamic peculiarities of interphase boundaries and domain walls for non-spin domains." Canadian Journal of Physics 100, no. 4 (2022): 247–53. http://dx.doi.org/10.1139/cjp-2021-0338.
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The analysis of magnetization oscillations (de Haas – van Alphen effect; dHvA effect) and features of the Fermi surface shape in three (3D)- and two-dimensional (2D) metals shows that the dynamic phenomena — the dynamics of domain walls and interphase boundaries during diamagnetic phase transitions — are described by a new nonlinear partial differential equation. The equation is a result of the inclusion of the case of multiple extremal cross sections of the Fermi surface in these metals. Our consideration indicates the possibility of the appearance of metastable non-spin domains (Condon domains) and first-order phase transitions to the ordered phase in the regime of dHvA oscillations for the two-frequency case. Sine–Gordon, Klein–Gordon, double sine–Gordon, time-dependent Ginzburg–Landau equations, and the telegraph equation are limiting cases of the derived equation. We show that particular moving kink-soliton solutions of the equation describe traveling wave fronts being moving domain walls and interphase boundaries in the Condon domain phase.
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Nasreen, Naila, Aly R. Seadawy, and Dianchen Lu. "Construction of soliton solutions for modified Kawahara equation arising in shallow water waves using novel techniques." International Journal of Modern Physics B 34, no. 07 (2020): 2050045. http://dx.doi.org/10.1142/s0217979220500459.
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The modified Kawahara equation also called Korteweg-de Vries (KdV) equation of fifth-order arises in shallow water wave and capillary gravity water waves. This study is based on the generalized Riccati equation mapping and modified the F-expansion methods. Several types of solitons such as Bright soliton, Dark-lump soliton, combined bright dark solitary waves, have been derived for the modified Kawahara equation. The obtained solutions have significant applications in applied physics and engineering. Moreover, stability of the problem is presented after being examined through linear stability analysis that justify that all solutions are stable. We also present some solution graphically in 3D and 2D that gives easy understanding about physical explanation of the modified Kawahara equation. The calculated work and achieved outcomes depict the power of the present methods. Furthermore, we can solve various other nonlinear problems with the help of simple and effective techniques.
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Chalikov, D. V., K. Yu Bulgakov, and К. V. Fokina. "Interpretation of the spectral wave forecast model results using the phase-resolving model." Fundamental and Applied Hydrophysics 16, no. 2 (2023): 21–33. http://dx.doi.org/10.59887/2073-6673.2023.16(2)-2.
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The paper presents an interpretation of the results of spectral wave-forecast model using the phase-resolving model. Spectral models provide the information on the evolution of the potential energy distribution in terms of angle and frequency though the information about the geometry and statistical wave characteristics in such models are not available. This information has to be extracted through the additional, often unsubstantiated, hypotheses. The proposed computational procedure transforms spectral information into a two-dimensional wave field which consists of a set of linear modes with randomly distributed phases is proposed. The wave field is not realistic since it does not have non-linear properties, for example, various asymmetry properties such as increased kurtosis. Afterwards the linear wave field reproduced on the basis of the wave spectrum is set as the initial condition for the exact phase-resolving model. The exact models formally suitable for such calculations are cumbersome and inefficient and that practically restricts their broad and regular application. This restriction can be overcome by using a new type of 3D wave simulation based on 2D equations. The 2D model reproduces the statistical characteristics of the wave field similar to the results of the 3D exact model and runs several times faster. The examples of using the developed method of interpretation of the spectral wave forecast in the Baltic Sea are demonstrated.
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Chu, Chunlei, and Paul L. Stoffa. "Implicit finite-difference simulations of seismic wave propagation." GEOPHYSICS 77, no. 2 (2012): T57—T67. http://dx.doi.org/10.1190/geo2011-0180.1.
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We propose a new finite-difference modeling method, implicit both in space and in time, for the scalar wave equation. We use a three-level implicit splitting time integration method for the temporal derivative and implicit finite-difference operators of arbitrary order for the spatial derivatives. Both the implicit splitting time integration method and the implicit spatial finite-difference operators require solving systems of linear equations. We show that it is possible to merge these two sets of linear systems, one from implicit temporal discretizations and the other from implicit spatial discretizations, to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm. We give the complete derivations of the implicit splitting time integration method and the implicit spatial finite-difference operators, and present the resulting discretized formulas for the scalar wave equation. We conduct a thorough numerical analysis on grid dispersions of this new implicit modeling method. We show that implicit spatial finite-difference operators greatly improve the accuracy of the implicit splitting time integration simulation results with only a slight increase in computational time, compared with explicit spatial finite-difference operators. We further verify this conclusion by both 2D and 3D numerical examples.
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Quintero, Jose R., and Alex M. Montes. "A center manifold application: existence of periodic travelling waves for the 2D abcd-Boussinesq system." Advanced Studies in Theoretical Physics 11, no. 12 (2017): 641–67. http://dx.doi.org/10.12988/astp.2017.7837.
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In this paper we study the existence of periodic travelling waves for the 2D abcd Boussinesq type system related with the three-dimensional water-wave dynamics in the weakly nonlinear long-wave regime. Small solutions that are periodic in the direction of translation form an infinite-dimensional family, by characterizing them using a center manifold reduction of infinite dimension and codimension due to the fact that at the linear level we are dealing with an ill-posed mixed-type initial-value problem. As happens for the Benney-Luke model and the KP II model for wave speed large enough and large surface tension, we show that a unique global solution exists for arbitrary small initial data for the two-component bottom velocity, specified along a single line in the direction of translation (or orthogonal to it). As a consequence of this fact, we show that the spatial evolution of bottom velocity is governed by a dispersive, nonlocal, nonlinear wave equation.