Waves In Nonlinear Dispersive Media Research Articles

$\mu$-Symmetries and $\mu$-Conservation Laws for The Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation

This work discusses the $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $-symmetry and conservation law of $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $ procedure for the nonlinear dispersive modified Benjamin-Bona-Mahony equation (NDMBBME). This equation models an approximation for surface long waves in nonlinear dispersive media. It can also describe the hydromagnetic waves in a cold plasma, acoustic waves in inharmonic crystals, and acoustic gravity waves in compressible fluids. First and foremost, we offer some essential pieces of information about the $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $-symmetry and the conservation law of $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $ concepts. In light of such information, $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $-symmetries are found. Using characteristic equations, the NDMBBME is reduced to ordinary differential equations (ODEs). We obtained the exact invariant solutions by solving the nonlinear ODEs. Furthermore, employing the variational problem procedure, we get the Lagrangian and the $% %TCIMACRO{\U{3bc} }% %BeginExpansion \mu %EndExpansion $-conservation laws. The exact solutions and conservation laws are new for the NDMBBME that are not reported by the other studies. We also demonstrate the properties with figures for these solutions.

The Analytical Stochastic Solutions for the Stochastic Potential Yu–Toda–Sasa–Fukuyama Equation with Conformable Derivative Using Different Methods

We consider in this study the (3+1)-dimensional stochastic potential Yu–Toda–Sasa–Fukuyama with conformable derivative (SPYTSFE-CD) forced by white noise. For different kind of solutions of SPYTSFE-CD, including hyperbolic, rational, trigonometric and function, we use He’s semi-inverse and improved (G′/G)-expansion methods. Because it investigates solitons and nonlinear waves in dispersive media, plasma physics and fluid dynamics, the potential Yu–Toda–Sasa–Fukuyama theory may explain many intriguing scientific phenomena. We provide numerous 2D and 3D figures to address how the white noise destroys the pattern formation of the solutions and stabilizes the solutions of SPYTSFE-CD.

Analytical and numerical investigation of soliton wave solutions in the fifth-order KdV equation within the KdV-KP framework

This study investigates the fifth-order Korteweg–de Vries (fKdV) equation, which is a generalized form of the classical KdV equation governing weakly nonlinear waves in dispersive media. By employing analytical and numerical techniques, the Khater II and variational iteration methods are utilized to obtain accurate soliton wave solutions. The combination of these approaches ensures the reliability and precision of the analytical outcomes. Graphical representations, including two-dimensional, three-dimensional, and contour plots, visually illustrate the results. Additionally, the stability of the constructed solutions is assessed through the characterization of the Hamiltonian system, providing valuable insights into the soliton wave solutions’ stability properties. This research integrates analytical and numerical methods to explore the fKdV equation within the KdV-KP framework, offering novel soliton wave solutions and enhancing their interpretation through graphical representations while assessing their stability using the Hamiltonian system’s characterization.

Optical and rogue type soliton solutions of the (2+1) dimensional nonlinear Heisenberg ferromagnetic spin chains equation

In this study, the uses of unified method for finding solutions of a nonlinear Schrödinger equation that describes the nonlinear spin dynamics of (2+1) dimensional Heisenberg ferromagnetic spin chains equation. We successfully construct solutions to these equations. For each of the derived solutions, we provide the parametric requirements for the existence of a valid soliton. In order to visualize some of the discovered solutions, we plot the 2D and 3D graphics. The results of this investigation, which have been presented, might be useful in elucidating the model's physical significance. These are a highly useful tool for studying how electrical solitons, which travel as voltage waves in nonlinear dispersive media, spread out, as well as for doing various physical calculations. The study’s findings, which have been disclosed, might be useful in illuminating the models under consideration's physical significance and electrical field.

Global control aspects for long waves in nonlinear dispersive media

A class of models of long waves in dispersive media with coupled quadratic nonlinearities on a periodic domain T are studied. We used two distributed controls, supported in ω ⊂ T and assumed to be generated by a linear feedback law conserving the “mass” (or “volume”), to prove global control results. The first result, using spectral analysis, guarantees that the system in consideration is locally controllable in Hs(T), for s ≥ 0. After that, by certain properties of Bourgain spaces, we show a property of global exponential stability. This property together with the local exact controllability ensures for the first time in the literature that long waves in nonlinear dispersive media are globally exactly controllable in large time. Precisely, our analysis relies strongly on the bilinear estimates using the Fourier restriction spaces in two different dispersions that will guarantee a global control result for coupled systems of the Korteweg—de Vries type. This result, of independent interest in the area of control of coupled dispersive systems, provides a necessary first step for the study of global control properties to the coupled dispersive systems in periodic domains.

Bifurcation of Exact Solutions for the Space-Fractional Stochastic Modified Benjamin–Bona–Mahony Equation

This paper studies the influence of space-fractional and multiplicative noise on the exact solutions of the space-fractional stochastic dispersive modified Benjamin–Bona–Mahony equation, driven in Ito’s sense by a multiplicative Wiener process. The bifurcation of the exact solutions is investigated, and novel fractional stochastic solutions are presented. The dependence of the solutions on the initial conditions is discussed. Due to the significance of the fractional stochastic modified Benjamin–Bona–Mahony equation in describing the propagation of surface long waves in nonlinear dispersive media, the derived solutions are significantly more helpful for and influential in comprehending diverse, crucial, and challenging physical phenomena. The effect of the Wiener process and the fractional order on the exact solutions are studied.

Lie group classification of the nonlinear transmission line model and exact traveling wave solutions

A nonlinear transmission line (NLTL) model is very essential tools in understanding of propagation of electrical solitons which can propagate in the form of voltage waves in nonlinear dispersive media. These models are often formulated using nonlinear partial differential equations. One of the basic tools available to study these equations are numerical methods such as finite difference method, finite element method, etc, have been developed for nonlinear partial differential equations. These methods require a great amount of time and memory due to the discretization and usually the effect of round-off error causes loss of accuracy in the results. So in this paper, we use one of the most famous analytical methods the Lie group analysis due to Sophus Lie. One of the advantages of this approach is that requires only algebraic calculations. The main aim of this study is to explore the nonlinear transmission line model with arbitrary capacitor's voltage dependence, through the use of Lie group classification, we show that the specifying form of arbitrary capacitor's voltage are power law nonlinearity, exponential law nonlinearity and constant capacitance. The exact solutions and similarity reductions generated from the symmetries are also provided. Furthermore, translational symmetries were utilized to find a family of traveling wave solutions via the \(\tanh\)-method of the governing nonlinear problem.

On the controllability of a model system for long waves in nonlinear dispersive media

Considered here is a higher-order generalization of the classical Boussinesq system which models the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal or the propagation of long-crested waves on large lakes and oceans. Our aim is to investigate the controllability properties of this nonlinear model in terms of the values of the parameters involved in the system. We give general conditions which ensure both the well-posedness and the local exact controllability of the nonlinear problem in some well chosen Sobolev spaces.

Stability estimate for the semi-discrete linearized Benjamin-Bona-Mahony equation

In this work we study the semi-discrete linearized Benjamin-Bona-Mahony equation (BBM) which is a model for propagation of one-dimensional, unidirectional, small amplitude long waves in non-linear dispersive media. In particular, we derive a stability estimate which yields a unique continuation property. The proof is based on a Carleman estimate for a finite difference approximation of Laplace operator with boundary observation in which the large parameter is connected to the mesh size.

Symmetry Breaking Soliton, Breather, and Lump Solutions of a Nonlocal Kadomtsev–Petviashvili System

The Kadomtsev–Petviashvili equation is one of the well-studied models of nonlinear waves in dispersive media and in multicomponent plasmas. In this paper, the coupled Alice-Bob system of the Kadomtsev–Petviashvili equation is first constructed via the parity with a shift of the space variable x and time reversal with a delay. By introducing an extended Bäcklund transformation, symmetry breaking soliton, symmetry breaking breather, and symmetry breaking lump solutions for this system are presented through the established Hirota bilinear form. According to the corresponding constants in the involved ansatz function, a few fascinating symmetry breaking structures of the presented explicit solutions are shown.

On the utility of C/S as an indicator of nonlinearity in finite-amplitude dispersive waves

A frequency-domain nonlinearity indicator C/S, recently introduced by Ohm et al. [Proc. Mtgs. Acoust. 29, 045003 (2016)], is a companion indicator to the well-known Morfey-Howell Q/S [AIAA J. 19, 986–992 (1981)]. The two indicators can be regarded as “two sides of the same coin,” because C/S and Q/S are, respectively, defined by the real and imaginary parts of the cross-spectrum of the pressure and squared-pressure waveforms. Despite the common origin, however, C/S and Q/S describe the pertinent nonlinear wave process in a different light. Furthermore, for nonlinear waves in media with strong dispersion only C/S may be applicable, because the use of Q/S could lead to erroneous interpretations. In this paper, we demonstrate the utility of C/S in the context of the Korteweg-de Vries (KdV) equation, which is one of the most celebrated model equations for nonlinear waves in dispersive media. The success of C/S is juxtaposed with the limitation of Q/S in describing nonlinear dispersive wave processes, especially in the case of solitons.

Solitonic Dispersive Hydrodynamics: Theory and Observation.

Ubiquitous nonlinear waves in dispersive media include localized solitons and extended hydrodynamic states such as dispersive shock waves. Despite their physical prominence and the development of thorough theoretical and experimental investigations of each separately, experiments and a unified theory of solitons and dispersive hydrodynamics are lacking. Here, a general soliton-mean field theory is introduced and used to describe the propagation of solitons in macroscopic hydrodynamic flows. Two universal adiabatic invariants of motion are identified that predict trapping or transmission of solitons by hydrodynamic states. The result of solitons incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves is the same, an effect termed hydrodynamic reciprocity. Experiments on viscous fluid conduits quantitatively confirm the soliton-mean field theory with broader implications for nonlinear optics, superfluids, geophysical fluids, and other dispersive hydrodynamic media.

Stability Analysis, Numerical and Exact Solutions of the (1+1)-Dimensional NDMBBM Equation

A newly propose mathematical approach is presented in this study. We utilize the new approach in investigating the solutions of the (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation. The new analytical technique is based on the popularly known sinh-Gordon equation and a wave transformation. In developing this new technique at each every steps involving integration, the integration constants are considered to not be zero which gives rise to new form of travelling wave solutions. The (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony is used in modelling an approximation for surface long waves in nonlinear dispersive media. We construct some new trigonometric function solution to this equation. Moreover, the finite forward difference method is utilized in investigating the numerical behavior of this equation by taking one of the obtained analytical solutions into consideration. We finally, give a comprehensive conclusions.

Exact and soliton solutions to nonlinear transmission line model

A nonlinear transmission line (NLTL) is comprised of a transmission line periodically loaded with varactors, where the capacitance nonlinearity arises from the variable depletion layer width, which depends both on the DC and AC voltages of the propagating wave. An equivalent circuit model of NLTL is discussed analytically, in this article, and different type of solutions are celebrated. The improved extended tanh-function method has been applied successfully to extract the solutions. The obtained solutions are solitary wave solutions, singular periodic solutions, singular soliton solutions, Jacobi elliptic doubly periodic type solutions and Weierstrass elliptic doubly periodic type solutions. It is a very convenient tool to study the propagation of electrical solitons which propagate in the form of voltage waves in nonlinear dispersive media.

QUADRATIC B-SPLINE FINITE ELEMENT METHOD FOR THE BENJAMIN-BONA-MAHONY-BURGERS EQUATION

Abstract. A quadratic B-spline nite element method for the spatialvariable combined with a Newton method for the time variable is proposedto approximate a solution of Benjamin-Bona-Mahony-Burgers (BBMB)equation. Two examples were considered to show the eciency of theproposed scheme. The numerical solutions obtained for various viscositywere compared with the exact solutions. The numerical results show thatthe scheme is ecient and feasible. 1. IntroductionThe mathematical model of propagation of small amplitude long wavesin nonlinear dispersive media is described by the following Benjamin-Bona-Mahony-Burgers equation[1]:8> :u t u xxt u xx + u x + uu = f in [0;L] [0;T];u(0;t) = u(L;t) = 0 on [0;T];u(x;0) = u 0 (x) in [0;L];(1)where >0; are constants, fis a given forcing term. In the physical case, thedispersive e ect of (1) is the same as the Benjamin-Bona-Mahony (BBM) equa-tion, while the dissipative e ect is the same as the Burgers equation, and whichis an alternative model for the Korteweg-de Vries-Burgers (KdVB) equation [2].Numerical methods based on either nite elements[3]-[7], nte di erences[8]-[10],or Adomian decomposition scheme[11, 12]. Quadratic B-spline nite ele-ment method for approximating the solution of Burgers equation can be foundin [13, 14]. Cubic B-spline collocation method for numerical solution of theBBMB equation can be found in [15]. In this paper, we apply the quadraticB-spline nite element method to convert BBMB equation to a nite set of

Painlevé Analysis, Lie Symmetries and Exact Solutions for Variable Coefficients Benjamin—Bona—Mahony—Burger (BBMB) Equation

In this paper, a variable-coefficient Benjamin—Bona—Mahony—Burger (BBMB) equation arising as a mathematical model of propagation of small-amplitude long waves in nonlinear dispersive media is investigated. The integrability of such an equation is studied with Painlevé analysis. The Lie symmetry method is performed for the BBMB equation and then similarity reductions and exact solutions are obtained based on the optimal system method. Furthermore different types of solitary, periodic and kink waves can be seen with the change of variable coefficients.

Few-cycle optical solitary waves in nonlinear dispersive media

We study the propagation of few-cycle optical solitons in nonlinear media with an anomalous, but otherwise arbitrary, dispersion and a cubic nonlinearity. Our approach does not derive from the slowly varying envelope approximation. The optical field is derived directly from Maxwell's equations under the assumption that generation of the third harmonic is a nonresonant process or at least cannot destroy the pulse prior to inevitable linear damping. The solitary wave solutions are obtained numerically up to nearly single-cycle duration using the spectral renormalization method originally developed for the envelope solitons. The theory explicitly distinguishes contributions between the essential physical effects such as higher-order dispersion, self-steepening, and backscattering, as well as quantifies their influence on ultrashort optical solitons.

Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves

Super rogue waves with an amplitude of up to 5 times the background value are observed in a water-wave tank for the first time. Nonlinear focusing of the local wave amplitude occurs according to the higher-order breather solution of the nonlinear wave equation. The present result shows that rogue waves can also develop from very calm and apparently safe sea states. We expect the result to have a significant impact on studies of extreme ocean waves and to initiate related studies in other disciplines concerned with waves in nonlinear dispersive media, such as optics, plasma physics, and superfluidity.Received 17 January 2012DOI:https://doi.org/10.1103/PhysRevX.2.011015This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical Society

Dark Solitons, Dispersive Shock Waves, and Transverse Instabilities

The nature of transverse instabilities of dark solitons for the (2+1)-dimensional defocusing nonlinear Schrödinger/Gross–Pitaevski (NLS/GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev–Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS/GP equation. The dispersion relation for shallow solitons is obtained asymptotically beyond the KP limit. This yields (1) the maximal growth rate and associated wavenumber of unstable perturbations and (2) the separatrix between convective and absolute instabilities. The instability properties of the dark soliton are directly related to those of oblique dispersive shock wave (DSW) solutions. Stationary and nonstationary oblique DSWs are constructed analytically and investigated numerically by direct simulations of the NLS/GP equation. It is found that stationary and nonstationary oblique DSWs have the same jump conditions in the shallow and hypersonic regimes. These results have application to controlling nonlinear waves in dispersive media.

Analysis for a two-step dispersion scheme for coupled non-linear Schrödinger equations

A model for the evolution of envelope waves in non-linear dispersive media is analyzed. With the help of coupled Non-Linear Schrödinger Equations (NLSE), the Modulation Instability (MI) of electromagnetic waves with periodic dispersion is investigated. A mathematical formulation is also derived for the MI gain. The analytical predictions are confirmed with numerical simulations of coupled NLSE with periodic constants.