KdV-Burgers Equation Research Articles

Traveling wave solutions of a hybrid KdV-Burgers equation with arbitrary real coefficients in relation with beam-permeated multi-ion plasma fluids

The Korteweg de Vries-Burgers (KdV-B) (1+1) equation incorporating constant (real) coefficients representing nonlinearity (a), dispersion (b) and dissipation (c), is a long known paradigm in e.g. plasma physics, where it can be derived from plasma fluid-dynamical models, so that all coefficients depend parametrically on the plasma composition. For a positive dispersion coefficient b (value), which is the general case in beam-free electron-ion plasma, this PDE possesses analytical solutions representing “shock”-shaped traveling waves with a characteristic kink (or anti-kink) soliton-like profile, for negative (or positive, respectively) values of the nonlinearity coefficient (a). In a plasma context, these excitations represent a monotonic transition between two (different) asymptotic values of the electrostatic potential , associated with a monopolar (i.e. bell-shaped) disturbance of the electric field (E). Contrary to widespread belief (based on a beam-free plasma description), an investigation of nonlinear electrostatic waves in beam-permeated plasmas reveals that the sign(s) of all (any) of the coefficients (a, b or c) may be reversed, independently from each other, depending on the beam velocity (value). In the light of this result, the analytical solutions have been reexamined in an effort to elucidate their applicability in plasma-physical scenarios (e.g., reconnection jets and other planetary plasma environments) in terms of the combined sign(s) of the various coefficients involved in the KdV-B equation. Different types of excitations are demonstrated to exist and the influence of the various coefficients on the solution’s propagation characteristics is examined.

High-Accuracy Solutions to the Time-Fractional KdV–Burgers Equation Using Rational Non-Polynomial Splines

A non-polynomial spline is a technique that utilizes information from symmetric functions to solve mathematical or physical models numerically. This paper introduces a novel non-polynomial spline construct incorporating a rational function term to develop an efficient numerical scheme for solving time-fractional differential equations. The proposed method is specifically applied to the time-fractional KdV–Burgers (TFKdV) equation. and time-fractional differential equations are crucial in physics as they provide a more accurate description of various complex processes, such as anomalous diffusion and wave propagation, by capturing memory effects and non-local interactions. Using Taylor expansion and truncation error analysis, the convergence order of the numerical scheme is derived. Stability is analyzed through the Fourier stability criterion, confirming its conditional stability. The accuracy and efficiency of the rational non-polynomial spline (RNPS) method are validated by comparing numerical results from a test example with analytical and previous solutions, using norm errors. Results are presented in 2D and 3D graphical formats, accompanied by tables highlighting performance metrics. Furthermore, the influences of time and the fractional derivative are examined through graphical analysis. Overall, the RNPS method has demonstrated to be a reliable and effective approach for solving time-fractional differential equations.

Qualitative Analysis and Novel Exact Soliton Solutions to the Compound Korteweg–De Vries–Burgers Equation

This paper deals with the exact wave results of the (1+1)-dimensional nonlinear compound Korteweg–De Vries and Burgers (KdVB) equation with a truncated M-fractional derivative. This model represents the generalization of Korteweg–De Vries-modified Korteweg–De Vries and Burgers equations. We obtained periodic, combo singular, dark–bright, and other wave results with the use of the extended sinh-Gordon equation expansion (EShGEE) and modified (G′/G2)-expansion techniques. The use of the effective fractional derivative makes our results much better than the existing results. The obtained solutions are useful as well as applicable in various fields, including mathematical physics, plasma physics, ocean engineering, optics, etc. The obtained solutions are demonstrated by 2D, 3D, and contour plots. The achieved results will be fruitful for future research on this equation. Stability analysis is used to check that the results are precise as well as exact. Modulation instability (MI) analysis is performed to find stable steady-state solutions of the abovementioned model. In the end, it is concluded that the methods used are easy and reliable.

On asymptotics of the solution to the Cauchy problem for a singularly perturbed operator-differential transport equation with weak diffusion in the case of several space variables

A formal asymptotic expansion is constructed for the solution of the Cauchy problem for a singularly perturbed operator differential transport equation with small nonlinearities and weak diffusion in the case of several space variables. Under the conditions imposed on the data of the problem, the leading asymptotic term is described by the multidimensional generalized Burgers–Korteweg–de Vries equation. Under certain conditions, the remainder is estimated with respect to the residual.

A heterogeneous continuum traffic flow model considering mixed connected and autonomous vehicles

Under the background of mixed scenarios of Connected and Autonomous Vehicles and Connected and Autonomous Electric Vehicles, a novel mixed car-following model considering the information of multiple vehicles and the difference of safety spacing in different driving scenarios is proposed. Then, the corresponding heterogeneous continuum traffic flow model is derived. Secondly, through the linear and nonlinear analysis approaches, the linear stability condition and the KdV-Burgers equation are yielded to describe the characteristics of traffic flow evolution. Finally, numerical simulations are performed to explore the density evolution of traffic flow under different initial density and permeability conditions. The conclusions can be summarized as follows: (1) The derived model can well simulate spatiotemporal phase changes and traffic flow phenomena such as stop-and-go waves and local cluster effects. (2) With the increase of the permeability of electric vehicles and speed difference sensitivity coefficient, the road density decreases from 0.15veh/m to 0.06veh/m, the traffic congestion area decreases, and the traffic congestion is effectively suppressed, which also verifies the rationality of the above theory.

Pressure wave propagation in high viscous magma containing crystals and bubbles

The presence of bubbles and crystals in highly viscous liquid (e.g., glass, magma) has been important for the purpose of both basic and practical viewpoints. In this presentation, especially, we focus on a magma, and investigate theoretically P-wave (i.e., pressure wave) propagation in the magma containing many bubbles. Based on an averaged theory of dispersed multiphase flow, we can regard the mixture of crystals (i.e., solid phase) and melt (i.e., liquid phase) as a single liquid phase, i.e., single non-Newtonian liquid, which is approximated to a Newtonian liquid by using the concept of an effective viscosity composed of various parameters (especially crystals). The system of basic equations for multiphase flow containing the effective viscosity, can be reduced to the KdV-Burgers equation as a nonlinear evolution equation via theoretical approximation based on singular perturbation method. As a result, the effect of fraction of crystals and bubbles on the acoustic characteristics (e.g., waveform evolution, nonlinearity, attenuation) is discussed.

Evolution of water wave packets by wind in shallow water

We use the Korteweg–de Vries (KdV) equation, supplemented with several forcing/friction terms, to describe the evolution of wind-driven water wave packets in shallow water. The forcing/friction terms describe wind-wave growth due to critical level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave stress in the air near the interface induced by a turbulent wind and wave decay due to a turbulent bottom boundary layer. The outcome is a modified KdV–Burgers equation that can be a stable or unstable model depending on the forcing/friction parameters. To analyse the evolution of water wave packets, we adapt the Whitham modulation theory for a slowly varying periodic wave train with an emphasis on the solitary wave train limit. The main outcome is the predicted growth and decay rates due to the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to validate the theory for various cases of initial wave amplitudes and growth and/or decay parameter ranges. The results from the modulation theory agree well with these simulations. In most cases we examined, many solitary waves are generated, suggesting the formation of a soliton gas.

Boundary controllability for the Korteweg–de Vries–Burgers equation on a finite domain

Boundary controllability for the Korteweg–de Vries–Burgers equation on a finite domain

Impact of Ion Pressure Anisotropy in Collisional Quantum Magneto-Plasma with Heavy and Light Ions

We have examined collisional degenerate plasma composed of charged state of heavy positive ion and light positive as well as negative ion. Employing the reductive perturbation method, we derived the damped Korteweg-de Vries-Burgers (dKdV-B) equation and by using its standard solution we analyze the characteristics of the solitary-shock profile under varying parameters. Furthermore, with the application of planar dynamical systems bifurcation theory, the phase portraits have been analyzed. This dynamical system analysis allowed us to extract important information on the stability of these structures as represented by the dKdV-B equation.

Boundary Stabilization of the Korteweg-de Vries-Burgers Equation with an Infinite Memory-Type Control and Applications: A Qualitative and Numerical Analysis

Boundary Stabilization of the Korteweg-de Vries-Burgers Equation with an Infinite Memory-Type Control and Applications: A Qualitative and Numerical Analysis

Magnetosonic shock waves in degenerate electron–positron–ion plasma with separated spin densities

This study investigates magnetosonic shock waves in a spin-polarized three-component quantum plasma using the quantum magnetic hydrodynamic model. We explore the influence of spin effects, specifically spin magnetization current and spin pressure, on shock wave behavior. Numerical analysis of the linear dispersion relation under varying parameters such as positron imbalance, spin polarization ratio, plasma beta, quantum diffraction, and magnetic diffusivity reveals differential impacts, with diffusion exerting significant influence on the plasma frequency. Our findings highlight the sensitivity discrepancy between the real and imaginary parts of the dispersion relation. Furthermore, nonlinear behavior of magnetosonic shock waves is examined via the Korteweg–de Vries–Burgers equation, showcasing transitions between oscillatory and monotonic wave patterns based on changes in dimensionless parameters. Notably, we observe the combined effects of spin-up and spin-down positrons with spin-up and spin-down electrons on shock wave dynamics, contributing to a deeper understanding of spin-plasma interactions with implications across various fields.

Exploring dust-ion acoustic shocks in a plasma in the light of phase plane analysis

This paper presents a comprehensive exploration of shock waves in a dusty plasma, incorporating the dynamics of inertial dust and ions with Maxwellian distributed electrons. By deriving the Korteweg-de Vries-Burgers (KdV-B) equation and by using its standard solution we analyze the characteristics of the shock front under varying parameters. Furthermore, we conduct a phase plane analysis to elucidate the system's behavior and dynamics by tuning in the parameters that provide various types of perturbation in the system. The investigation sheds light on the intricate behavior of shock waves in dusty plasmas and contributes valuable insights to the understanding of plasma dynamics.

Convergence rate toward shock wave under periodic perturbation for generalized Korteweg–de Vries–Burgers equation

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg–de Vries–Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.

Comparative Study of Solutions of Fractional Order Mixed KdV Burger’s Equation

Objective: The aim of this study is to obtain numerical solutions of fractional order mixed KdV Burger’s equation using iterative methods. Methodology: The study is centered on a time fractional mixed KdV Burger’s equation in the Caputo sense. The approximate solutions of this equation are obtained using Adomian decomposition method and Homotopy Perturbation method. Also, a Python code is developed to obtained numerical solutions and simulate graphically. Finding: The obtained solutions are compared with the exact solution and presented graphically using Python program for better analysis. Novelty: These methods shall be used to obtain approximate solution of such type of non-linear fractional order differential equations. Keywords: Fractional Differential Equation, Caputo Derivative, ADM, HPM, KdV Burger equation, Python

Exact solutions of stochastic Burgers–Korteweg de Vries type equation with variable coefficients

We will present exact solutions for three variations of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equation featuring variable coefficients. In each variant, white noise exhibits spatial uniformity, and the three categories include additive, multiplicative, and advection noise. Across all cases, the coefficients are time-dependent functions. Our discovery indicates that solving certain deterministic counterparts of KdV–Burgers equations and composing the solution with a solution of stochastic differential equations leads to the exact solution of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equations.

Instability of periodic waves for the Korteweg–de Vries–Burgers equation with monostable source

In this paper, it is proved that the KdV–Burgers equation with a monostable source term of Fisher–KPP type has small-amplitude periodic traveling wave solutions with finite fundamental period. These solutions emerge from a subcritical local Hopf bifurcation around a critical value of the wave speed. Moreover, it is shown that these periodic waves are spectrally unstable as solutions to the PDE, that is, the Floquet (continuous) spectrum of the linearization around each periodic wave intersects the unstable half plane of complex values with positive real part. To that end, classical perturbation theory for linear operators is applied in order to prove that the spectrum of the linearized operator around the wave can be approximated by that of a constant coefficient operator around the zero solution, which intersects the unstable complex half plane.

Bifurcation analysis of electron-acoustic waves in electron beam-plasma with suprathermal electrons

It has investigated the properties of nonlinear electron-acoustic waves propagating in a four-component collisionless, unmagnetized plasma. The system consists of cool inertial background electrons of temperature T c , the cool inertial electron beam of temperature T b , hot inertialess suprathermal electrons modeled by a κ-distribution of temperature T h , and ions. The nonlinear modified Korteweg-de Vries-Burgers (mKdV-Burgers) equation that characterizes the electron acoustic waves in such a plasma model is derived. The wave solutions of the resultant mKdV-Burgers equation are investigated using planar dynamical system bifurcation theory. The unperturbed hot, cool, and beam electron densities and the superthermal electron parameter are revealed to have a considerable influence on the nonlinear electron-acoustic traveling waves and the associated electric fields. The system evolves also monotonic shock waves when the dissipation term dominates over the dispersion term. The current model might be useful for understanding the generation of nonlinear electron-acoustic waves in a variety of astrophysical plasma settings, including the Earth's magnetosphere.

Magnetoacoustic waves in spin-1/2 dense quantum degenerate plasma: nonlinear dynamics and dissipative effects

Abstract Within the confines of a two-fluid quantum magnetohydrodynamic model, the investigation of magnetoacoustic shock and solitary waves is conducted in an electron-ion magnetoplasma that considers electrons of spin 1/2. When the plasma system is nonlinearly investigated using the reductive perturbation approach, the Korteweg de Vries-Burgers (KdVB) equation is produced. Sagdeev’s potential is created, revealing the presence of solitary solutions. However, when dissipative terms are included, intriguing physical solutions can be obtained. The KdVB equation is further investigated using the phase plane theory of a planar dynamical system to demonstrate the existence of periodic and solitary wave solutions. Predicting several classes of traveling wave solutions is advantageous due to various phase orbits, which manifest as soliton-shock waves, and oscillatory shock waves. The presence of a magnetic field, the density of electrons and ions, and the kinematic viscosity significantly alter the properties of magnetoacoustic solitary and shock waves. Additionally, electric fields have been identified. The outcomes obtained here can be applied to studying the nature of magnetoacoustic waves that are observed in compact astrophysical environments, where the influence of quantum spin phenomena remains significant, and also in controlled laboratory plasma experiments.

Passivity-based boundary control for Korteweg-de Vries-Burgers equations

This article considers the passivity-based boundary control for Korteweg-de Vries-Burgers (KdVB) equations. Both the input strict passivity (ISP) and the output strict passivity (OSP) are studied. By the Lyapunov functional method and Wirtinger's inequality, sufficient criteria are derived to establish ISP and OSP for KdVB equations with boundary disturbances. Moreover, when parameter uncertainties arise in KdVB equations, the robust ISP and OSP are also investigated and sufficient criteria are proposed to achieve the robust ISP and OSP. Simulation examples are also given to verify the effectiveness of theoretical results.

Fractional Spectral Approaches for Some Fractional Partial Differential Equations

The multi-order linear and nonlinear partial differential equations of fractional order have been solved numerically. A computational strategy based on fractional spectral operational matrices (OM) and generalized fractional Lageurre (GFL), generalized fractional shifted Legendre (GFSL), generalized fractional modified Bernstein (GFMB) are provided. Our fractional spectral methods have been used to solve nonlinear fractional partial differential equations and fractional Korteweg-de Vries-Burgers (KdVB) equation. Making use of a variety of test and application examples, the effectiveness of the numerical solution have been satisfied.