Generalized Zakharov Equations Research Articles

Numerical Simulation for Solitary Waves of the Generalized Zakharov Equation Based on the Lattice Boltzmann Method

The generalized Zakharov equation is a widely used and crucial model in plasma physics, which helps to understand wave particle interactions and nonlinear wave propagation in plasma. The solitary wave solution of this equation provides insights into phenomena such as electron and ion acoustic waves, as well as magnetic field disturbances in plasma. The numerical simulation of solitary wave solutions to the generalized Zakharov equation is an interesting problem worth studying. This is crucial for plasma-based technology, as well as for understanding nonlinear wave propagation in plasma physics and other fields. In this study, a numerical investigation of the generalized Zakharov equation using the lattice Boltzmann method has been conducted. The lattice Boltzmann method is a new modeling and simulating method at the mesoscale. A lattice Boltzmann model was constructed by performing Taylor expansion, Chapman–Enskog expansion, and time multiscale expansion on the lattice Boltzmann equation. By defining the moments of the equilibrium distribution function appropriately, the macroscopic equation can be restored. Furthermore, the numerical experiments for the equation are carried out with the parameter lattice size m=100, time step Δt=0.001, and space step size Δx=0.4. The solitary wave solution of the equation is numerically simulated. Numerical results under different parameter values are compared, and the convergence and effectiveness of the model are numerically verified. It is obtained that the model is convergent in time and space, and the convergence orders are all 2.24881. The effectiveness of our model was also verified by comparing the numerical results of different numerical methods. The lattice Boltzmann method demonstrates advantages in both accuracy and CPU time. The results indicate that the lattice Boltzmann method is a good tool for computing the generalized Zakharov equation.

Solitary waves of the generalized Zakharov equations via integration algorithms

<abstract><p>In many applications, the investigation of traveling wave solutions is essential in obtaining an accurate description of the dynamical behavior of most physical phenomena. The exact solutions to nonlinear equations can provide more physical descriptions and insightful details for many problems of practical interest. This paper focuses on investigating the solitary wave solutions of the generalized Zakharov equations (GZEs) by using four integration algorithms, namely, the modified $ (g'/g^{2}) $-expansion method, the modified $ (g') $-expansion method, the generalized simple ($ w/g $)-expansion method, and the addendum to Kudryashov's method. The GZEs have been widely used to describe the propagation of Langmuir waves in the field of plasma physics. The efficiency and simplicity of these methods are evaluated based on their application to GZEs, which have yielded multiple new optical solitary wave solutions in the form of rational, trigonometric, and hyperbolic functions. By using a suitable wave transformation, the coupled nonlinear partial differential equations are converted into ordinary differential equations. The derived optical solutions are graphically depicted in $ 2 $D and $ 3 $D plots for some specific parameter values. The traveling wave solutions discovered in the current study constitute just one example of the desired solutions that may enable the exploration of the physical properties of many complex systems and could also contribute greatly to improving our understanding of many interesting natural phenomena that arise in different applications, including plasma physics, fluid mechanics, protein chemistry, wave propagation, and optical fibers.</p></abstract>

Soliton Solutions for Coupled Nonlinear Generalized Zakharov Equations with Anti-cubic Nonlinearity Using Various Techniques

Soliton Solutions for Coupled Nonlinear Generalized Zakharov Equations with Anti-cubic Nonlinearity Using Various Techniques

Lump-Type Solutions, Lump Solutions, and Mixed Rogue Waves for Coupled Nonlinear Generalized Zakharov Equations

This article studies diverse forms of lump-type solutions for coupled nonlinear generalized Zakharov equations (CNL-GZEs) in plasma physics through an appropriate transformation approach and bilinear equations. By utilizing the positive quadratic assumption in the bilinear equation, the lump-type solutions are derived. Similarly, by employing a single exponential transformation in the bilinear equation, the lump one-soliton solutions are derived. Furthermore, by choosing the double exponential ansatz in the bilinear equation, the lump two-soliton solutions are found. Interaction behaviors are observed and we also establish a few new solutions in various dimensions (3D and contour). Furthermore, we compute rogue-wave solutions and lump periodic solutions by employing proper hyperbolic and trigonometric functions.

Analysis of fractional‐order Schrödinger–Boussinesq and generalized Zakharov equations using efficient method

In this paper, we find the solution for an interesting family of coupled Schrödinger equations exemplifying simulating consequences. The fractional Schrödinger–Boussinesq (FSB) and fractional generalized Zakharov (FGZ) equations are analyzed in the present framework usingfractional natural decomposition method(FNDM). The hired technique is graceful amalgamations of natural transform technique with Adomian decomposition scheme. Three different cases are considered, one with FSB equations and two cases are associated with FGZ equations with different initial conditions. To validate and illustrate the proficiency of the projected solution procedure, we analyzed the projected model in terms of fractional order. Further, we captured the nature of FNDM results for different values of fractional order in terms of the plots. The considered scheme is highly effective and structured while examining nonlinear models and which can be observed and confirmed from the obtained results. Moreover, the plots show that the hired fractional operator and algorithm can help to exemplify the more fascinating properties of the complex system associated with real‐world problems.

Qualitative analysis of solutions for the generalized Zakharov equations with magnetic field in \mathbb{R}^d

Qualitative analysis of solutions for the generalized Zakharov equations with magnetic field in \mathbb{R}^d

Initial boundary value problem for generalized Zakharov equations with nonlinear function terms

In this paper, we consider the initial boundary value problem for generalized Zakharov equations. Firstly, we prove the existence and uniqueness of the global smooth solution to the problem by a priori integral estimates, the Galerkin method, and compactness theory. Furthermore, we discuss the approximation limit of the global solution when the coefficient of the strong nonlinear term tends to zero.

Stable three-dimensional Langmuir vortex soliton

We present a numerical solution in the form of a three-dimensional vortex soliton in unmagnetized plasma in the model of the generalized Zakharov equations with saturating exponential nonlinearity. To find the solution with a high accuracy, we use a two-step numerical method combining the Petviashvili iteration procedure and the Newton–Kantorovich method. The vortex soliton with the topological charge m = 1 turns out to be stable, provided that the nonlinear frequency shift exceeds a certain critical value. The stability predictions are verified by direct simulations of the full dynamical equation.

Solution for fractional generalized Zakharov equations with Mittag-Leffler function

Abstract The pivotal aim of the present work is to find the solution for fractional generalized Zakharov (FGZ) equations using q -homotopy analysis transform method ( q -HATM). The proposed technique is graceful amalgamations of Laplace transform technique with q -homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. The fixed point hypothesis considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to illustrate and validate the efficiency of the proposed technique, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order and the numerical simulation is also demonstrated. The obtained results elucidate that, the considered algorithm is easy to implement, highly methodical as well as accurate and very effective to analyse the behaviour of coupled nonlinear differential equations of arbitrary order arisen in the connected areas of science and engineering.

Approximation to the global solution of generalized Zakharov equations in mathbf{R}^{2}

We consider the initial value problem for the two-dimensional generalized Zakharov equations which model the propagation of Langmuir waves in plasmas. It is obtained that the solutions of the two-dimensional generalized Zakharov equations converge as alphato0 to a solution of the Zakharov equations. Both weak and strong solutions are considered.

Structure of solitary wave solutions of the nonlinear complex fractional generalized Zakharov dynamical system

The analytical and solitary traveling solutions of the nonlinear complex fractional generalized Zakharov equations are investigated. The nonlinear complex fractional generalized Zakharov equations describe the interaction between dispersive and non-dispersive waves in one dimension. Analytical and solitary traveling wave solutions were obtained through applying a generalized Kudryashov and a novel (frac{G'}{G})-expansion methods. Novel solutions were the results of our investigated model, which illustrated the effectiveness and the power of the obtained methods in regards to accuracy, power, and effectiveness compared to the previously used methods.

Orbital stability of periodic traveling wave solutions to the generalized zakharov equations

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Zakharov equations {iut+uxx=uv+|u|2u,vtt-vxx=(|u|2)xx.First, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that this solution is orbitally stable by perturbations with period L. The results on the orbital stability of periodic traveling wave solutions for the generalized Zakharov equations in this paper can be regarded as a perfect extension of the results of [15, 16, 19].

New complex exact travelling wave solutions for the generalized-Zakharov equation with complex structures

In this paper, we apply the sine-Gordon expansion method which is one of the powerful methods to the generalized-Zakharov equation with complex structure. This algorithm yields new complex hyperbolic function solutions to the generalized-Zakharov equation with complex structure. Wolfram Mathematica 9 has been used throughout the paper for plotting two- and three-dimensional surface of travelling wave solutions obtained.

Retracted: Bifurcation of Travelling Wave Solutions of the Generalized Zakharov Equation

Retracted: Bifurcation of Travelling Wave Solutions of the Generalized Zakharov Equation

On global smooth solution for generalized Zakharov equations

This paper considers the existence and uniqueness of the global smooth solution for the initial value problem of generalized Zakharov equations in dimension two. By means of a priori integral estimates, Galerkin method and compactness theory, one has the existence and uniqueness for small initial data.

THE EXACT SOLUTIONS OF GENERALIZED ZAKHAROV EQUATIONS WITH HIGH ORDER SINGULAR POINTS AND ARBITRARY POWER NONLINEARITIES

In this paper, we using bifurcation theory method of dynamical systems to find the exact solutions of generalized Zakharov equations with high order singular points and arbitrary power nonlinearities. Under different parameter conditions, we obtain exact solitary wave solutions, periodic wave solutions as well as kink and anti-kink wave solutions.

Symmetry reductions, exact solutions, and conservation laws of the generalized Zakharov equations

In this paper, the generalized Zakharov equations, which describe interactions between high- and low-frequency waves in plasma physics are studied from the perspective of Lie symmetry analysis and conservation laws. Based on some subalgebras of symmetries, several reductions and numerous new exact solutions are obtained. All of these solutions represent modified traveling waves. The obtained solutions include expressions involving Airy functions, Bessel functions, Whittaker functions, and generalized hypergeometric functions. Previously unknown conservation laws are constructed for the generalized Zakharov equations using the direct method. Profiles are presented for some of these new solutions.

New Exact Travelling Wave Solutions for Generalized Zakharov Equations Using the First Integral Method

In this paper, we investigate the first integral method for solving the solutions of nonlinear Generalized Zakharov Equations. This idea can obtain some exact solutions of this equations based on the theory of Commutative algebra.

Bifurcation of Travelling Wave Solutions of the Generalized Zakharov Equation

By using bifurcation theory of planar ordinary differential equations all different bounded travelling wave solutions of the generalized Zakharov equation are classified in to different parametric regions. In each of these parametric regions the exact explicit parametric representation of all solitary, kink (antikink), and periodic wave solutions as well as their numerical simulation and their corresponding phase portraits are obtained.

New Exact Explicit Solutions of the Generalized Zakharov Equation via the First Integral Method

The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral method, in the form of trigonometric and exponential functions. The results show the first integral method is an efficient way to solve the coupled nonlinear equations and get rich explicit analytical solutions.