Wave Solutions Of Nonlinear Equations Research Articles

Rescaling method for blow‐up solutions of nonlinear wave equations

Rescaling method for blow‐up solutions of nonlinear wave equations

Traveling Wave Solutions of Nonlinear Wave Equation Based on Functional Analysis and Multiple Variables as Applied to IoT

Abstract In applied mathematics, physics, and mechanics, many mathematical models are established in the form of nonlinear wave equations. Therefore, a method for studying traveling wave solutions of nonlinear wave equations based on functional analysis is proposed. The boundary value stability of the nonlinear second order nonlinear wave equation is modified by using functional analysis of multiple variables. In this paper, a new integral term is introduced into the organization mapping functional, the concept and construction of the nonlinear wave equation are analyzed, and its continuity is proved. According to the growth order of the traveling wave solution of nonlinear wave equation, the method of finding the traveling wave solution and its stability are studied.

Persistence and asymptotic analysis of solutions of nonlinear wave equations

Persistence and asymptotic analysis of solutions of nonlinear wave equations

Existence and asymptotic behavior of solutions for non-linear wave equations of Kirchhoff type with viscoelasticity

"In this paper we discuss the global existence and the asymptotic behavior of solutions of an initial boundary value problem of a non-linear wave equation of Kirchhoff type. "

Numerical Simulation of the Korteweg–de Vries Equation with Machine Learning

A machine learning procedure is proposed to create numerical schemes for solutions of nonlinear wave equations on coarse grids. This method trains stencil weights of a discretization of the equation, with the truncation error of the scheme as the objective function for training. The method uses centered finite differences to initialize the optimization routine and a second-order implicit-explicit time solver as a framework. Symmetry conditions are enforced on the learned operator to ensure a stable method. The procedure is applied to the Korteweg–de Vries equation. It is observed to be more accurate than finite difference or spectral methods on coarse grids when the initial data is near enough to the training set.

Novel Exact Traveling Wave Solutions for Nonlinear Wave Equations with Beta-Derivatives via the sine-Gordon Expansion Method

The main objectives of this research are to use the sine-Gordon expansion method (SGEM) along with the use of appropriate traveling transformations to extract new exact solitary wave solutions of the (2 + 1)- dimensional breaking soliton equation and the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) system equipped with beta partial derivatives. Using the chain rule, we convert the proposed nonlinear problems into nonlinear ordinary differential equations with integer orders. There is then no further demand for any normalization or discretization in the calculation process. The exact explicit solutions to the problems obtained with the SGEM are written in terms of hyperbolic functions. The exact solutions are new and published here for the first time. The effects of varying the fractional order of the beta-derivatives are studied through numerical simulations. 3D, 2D, and contour plots of solutions are shown for a range of values of fractional orders. As parameter values are changed, we can identify a kink-type solution, a bell-shaped solitary wave solution, and an anti-bell shaped soliton solution. All of the solutions have been carefully checked for correctness and could be very important for understanding nonlinear phenomena in beta partial differential equation models for systems involving the interaction of a Riemann wave with a long wave and interactions of two long waves with distinct dispersion relations.

Travelling Wave Solutions of the Non-Linear Wave Equations

Abstract This article focuses on the exact periodic solutions of nonlinear wave equations using the well-known Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. The periodic solutions are found using this method which contains both solitary wave and shock wave solutions. In this paper, the new results are computed using the closed-form solution including solitary or shock wave solutions which are obtained using Jacobi elliptic function method. The corresponding solitary or shock wave solutions are compared with the actual results. The results are visualised and the periodic behaviour of the solution is described in detail. The shock waves are found to break with time, whereas, solitary waves are found to be improved continuously with time.

Discontinuous Galerkin method for blow-up solutions of nonlinear wave equations

e develop and study an explicit time-space discrete discontinuous Galerkin finite element method to approximate the solution of one-dimensional nonlinear wave equations. We show that the numerical scheme is stable if a nonuniform time mesh is considered. We also investigate the blow-up phenomena and we prove that under weak convergence assumptions, the numerical blow-up time tends toward the theoretical one. The validity of our results is confirmed throughout several examples and benchmarks.

Blow-up of solutions for nonlinear wave equations on locally finite graphs

<abstract><p>Let $ G = (V, E) $ be a local finite connected weighted graph, $ \Omega $ be a finite subset of $ V $ satisfying $ \Omega^\circ\neq\emptyset $. In this paper, we study the nonexistence of the nonlinear wave equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial^2_t u = \Delta u + f(u) $\end{document} </tex-math></disp-formula></p> <p>on $ G $. Under the appropriate conditions of initial values and nonlinear term, we prove that the solution for nonlinear wave equation blows up in a finite time. Furthermore, a numerical simulation is given to verify our results.</p></abstract>

Travelling wave solutions of the third-order KdV equation using Jacobi elliptic function method

For the purpose of constructing the exact periodic solutions of nonlinear wave equations, it has been proposed to use a method known as the Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. It has been demonstrated that the periodic solutions obtained using this method contain both solitary wave solutions and shock wave solutions in some instances.

The Application of the Functional Variable Method for Solving the Loaded Non-linear Evaluation Equations

In this article, we construct exact traveling wave solutions of the loaded Korteweg-de Vries, the loaded modified Korteweg-de Vries, and the loaded Gardner equation by the functional variable method. The performance of this method is reliable and effective and gives the exact solitary and periodic wave solutions. All solutions to these equations have been examined and 3D graphics of the obtained solutions have been drawn by using the Matlab program. We get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, and solitary wave solutions of kink type. Our results reveal that the method is a very effective and straightforward way of formulating the exact traveling wave solutions of non-linear wave equations arising in mathematical physics and engineering.

Wave profile analysis of a couple of (3+1)-dimensional nonlinear evolution equations by sine-Gordon expansion approach

The (3+1)-dimensional Kadomtsev-Petviashvili and the modified KdV-Zakharov-Kuznetsov equations have a significant impact in modern science for their widespread applications in the theory of long-wave propagation, dynamics of shallow water wave, plasma fluid model, chemical kinematics, chemical engineering, geochemistry, and many other topics. In this article, we have assessed the effects of wave speed and physical parameters on the wave contours and confirmed that waveform changes with the variety of the free factors in it. As a result, wave solutions are extensively analyzed by using the balancing condition on the linear and nonlinear terms of the highest order and extracted different standard wave configurations, containing kink, breather soliton, bell-shaped soliton, and periodic waves. To extract the soliton solutions of the high-dimensional nonlinear evolution equations, a recently developed approach of the sine-Gordon expansion method is used to derive the wave solutions directly. The sine-Gordon expansion approach is a potent and strategic mathematical tool for instituting ample of new traveling wave solutions of nonlinear equations. This study established the efficiency of the described method in solving evolution equations which are nonlinear and with higher dimension (HNEEs). Closed-form solutions are carefully illustrated and discussed through diagrams.

Study of reflection/transmission behavior of a planar nonlinear dielectric–NID interface

The manuscript is devoted to study the behavior of reflection and transmission from a planar interface of second order nonlinear–linear NID medium. Slowly varying amplitude approximation is used to simplify the solution of nonlinear wave equation. Analysis is conducted by considering imperfect matching case of permittivity for the nonlinear medium. It is noted that behavior of the reflection coefficient changes drastically when the angle of reflection for nonlinear polarization becomes complex. It is also observed that dimension of the NID medium is an effective parameter to control the behavior of coefficients.

Variational methods for breather solutions of nonlinear wave equations

We construct infinitely many real-valued, time-periodic breather solutions of the nonlinear wave equation with suitable N ⩾ 2, p > 2 and localized nonnegative Q. These solutions are obtained from critical points of a dual functional and they are weakly localized in space. Our abstract framework allows to find similar existence results for the nonlinear Klein–Gordon equation and biharmonic wave equations.

Nonlinear Fourier Analysis: Rogue Waves in Numerical Modeling and Data Analysis

Nonlinear Fourier Analysis (NLFA) as developed herein begins with the nonlinear Schrödinger equation in two-space and one-time dimensions (the 2+1 NLS equation). The integrability of the simpler nonlinear Schrödinger equation in one-space and one-time dimensions (1+1 NLS) is an important tool in this analysis. We demonstrate that small-time asymptotic spectral solutions of the 2+1 NLS equation can be constructed as the nonlinear superposition of many 1+1 NLS equations, each corresponding to a particular radial direction in the directional spectrum of the waves. The radial 1+1 NLS equations interact nonlinearly with one another. We determine practical asymptotic spectral solutions of the 2+1 NLS equation that are formed from the ratio of two phase-lagged Riemann theta functions: Surprisingly this construction can be written in terms of generalizations of periodic Fourier series called (1) quasiperiodic Fourier (QPF) series and (2) almost periodic Fourier (APF) series (with appropriate limits in space and time). To simplify the discourse with regard to QPF and APF Fourier series, we call them NLF series herein. The NLF series are the solutions or approximate solutions of the nonlinear dynamics of water waves. These series are indistinguishable in many ways from the linear superposition of sine waves introduced theoretically by Paley and Weiner, and exploited experimentally and theoretically by Barber and Longuet-Higgins assuming random phases. Generally speaking NLF series do not have random phases, but instead employ phase locking. We construct the asymptotic NLF series spectral solutions of 2+1 NLS as a linear superposition of sine waves, with particular amplitudes, frequencies and phases. Because of the phase locking the NLF basis functions consist not only of sine waves, but also of Stokes waves, breather trains, and superbreathers, all of which undergo complex pair-wise nonlinear interactions. Breather trains are known to be associated with rogue waves in solutions of nonlinear wave equations. It is remarkable that complex nonlinear dynamics can be represented as a generalized, linear superposition of sine waves. NLF series that solve nonlinear wave equations offer a significant advantage over traditional periodic Fourier series. We show how NLFA can be applied to numerically model nonlinear wave motions and to analyze experimentally measured wave data. Applications to the analysis of SINTEF wave tank data, measurements from Currituck Sound, North Carolina and to shipboard radar data taken by the U. S. Navy are discussed. The ubiquitous presence of coherent breather packets in many data sets, as analyzed by NLFA methods, has recently led to the discovery of breather turbulence in the ocean: In this case, nonlinear Fourier components occur as strongly interacting, phase locked, densely packed breather modes, in contrast to the previously held incorrect belief that ocean waves are weakly interacting sine waves.

Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency

In this paper, one dimensional nonlinear wave equation \begin{document}$ u_{tt}-u_{xx} +mu +\varepsilon f(\omega t,x,u;\xi) = 0 $\end{document} with Dirichlet boundary condition is considered, where \begin{document}$ \varepsilon $\end{document} is small positive parameter, \begin{document}$ \omega = \xi \bar{\omega}, $\end{document} \begin{document}$ \bar{\omega} $\end{document} is weak Liouvillean frequency. It is proved that there are many quasi-periodic solutions with Liouvillean frequency for the above equation. The proof is based on an infinite dimensional KAM Theorem.

Experimental revealing of asynchronous transient-soliton buildup dynamics

The buildup process of coherent structures and patterns from the composite balance between conservative and dissipative effects is a universal phenomenon that occurs in various areas of physics, ranging from quantum mechanics to astrophysics. Dissipative solitons are highly coherent solutions of nonlinear wave equations, and provide an excellent research platform for ultrafast transient phenomena. Herein, by taking advantage of the fast detection technique provided by the dispersive Fourier transform, we experimentally observe the spectral broadening and breathing behavior of transient dissipative structures produced asynchronously during the buildup process of dissipative solitons. These observations unveil a novel dynamics of dissipative soliton generation, which is accompanied by energy quantization, self-phase modulation induced spectral broadening, structural dissipative soliton formation, and Raman soliton self-frequency shifting, thus providing a new insight in transient ultrafast laser dynamics.

Self-focusing of multi-Gaussian laser beams in nonlinear optical media as a Kepler’s central force problem

In this paper self-focusing of multi-Gaussian laser beam in nonlinear optical media has been investigated theoretically. Saturation of the optical nonlinearity has been incorporated through cubic-quintic model. Moment theory approach in W.K.B approximation has been invoked to find the numerical solution of nonlinear wave equation for the field of incident laser beam. Particularly the dynamical evolutions of the beam width of the laser beam with distance of propagation have been investigated in detail. It has been shown that the self-focusing of the laser beam resembles to Kepler’s central force problem.

Homogenous balance method for solving exact solutions of the nonlinear Benny -luke equation and Vakhnenko-Parkes equation

      In this article, We apply the homogeneous balance method to construct the many families of exact solution of travelling wave solution of nonlinear equations, the Benny -Luke equation and vakhnenko-parkes equation. As the result, many solitary wave solution are obtained from the solution by hyperbolic function when the parameters were taken at special values and we compared the result with the solution of F-expansion method and -expansion method, we obtained the same results by certain hypotheses. Also we drew a 3D graph of exact solution  for the special Benny -luke equation and vakhnenko-parkes equation by help of the maple.  

Quasi-periodic solutions for nonlinear wave equation with singular Legendre potential

In this paper, the nonlinear wave equation with singular Legendre potential \t\t\tutt−uxx+VL(x)u+mu+perturbation=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{tt}-u_{xx}+V_{L}(x)u + mu + \\mathit{perturbation}=0 $$\\end{document} subject to certain boundary conditions is considered, where m is a positive real number and V_{L}(x)=-frac{1}{2}-frac{1}{4}tan ^{2} {x}, xin (-frac{pi }{2},frac{pi }{2}). By means of the partial Birkhoff normal form technique and infinite-dimensional Kolmogorov–Arnold–Moser theory, it is proved that, for every min mathbb{R}_{+}setminus {frac{1}{4}}, the above equation admits plenty of quasi-periodic solutions with three frequencies.