KdV-Burgers Equation Research Articles

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.

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.

On new computations of the time-fractional nonlinear KdV-Burgers equation with exponential memory

This paper examines the Korteweg–de Vries-Burgers (KdV-Burgers) equation with nonlocal operators using the exponential decay and Mittag-Leffler kernels. The Caputo-Fabrizio and Atangana-Baleanu operators are used in the natural transform decomposition method (NTDM). By coupling a decomposition technique with the natural transform methodology, the method provides an effective analytical solution. When the fractional order is equal to unity, the proposed approach computes a series form solution that converges to the exact values. By comparing the approximate solution to the precise values, the efficacy and trustworthiness of the proposed method are confirmed. Graphs are also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both operators outcome is examined using diagrams and numerical data. These graphs show how the approximated solution’s graph and the precise solution’s graph eventually converge as the non-integer order gets closer to 1. The outcomes demonstrate the method’s high degree of accuracy and its wide applicability to fractional nonlinear evolution equations. In order to further explain these concepts, simulations are run using a computationally packed software that helps interpret the implications of solutions. NTDM is considered the best analytical method for solving fractional-order phenomena, especially KdV-Burgers equations.

Passivity‐based boundary control for stochastic Korteweg–de Vries–Burgers equations

The passivity‐based boundary control is considered for stochastic Korteweg–de Vries–Burgers (SKdVB) equations. Both the stochastic input strictly passive (SISP) and stochastic output strictly passive (SOSP) are studied. By introducing Lyapunov functionals and Wirtinger's inequality, sufficient criteria are derived to establish SISP and SOSP for SKdVB equations with boundary disturbances. Moreover, when parameter uncertainties arise in SKdVB equations, the robust stochastic passivity is also investigated and sufficient criteria are presented to achieve the robust SISP and SOSP. Two numerical simulations are employed to show the effectiveness and advantages of our theoretical results.

How do various forces affect pressure waves in bubbly flows?

This study investigated the weakly nonlinear propagation of pressure waves in compressible, flowing water with spherical microbubbles, considering various forces. Previous theoretical studies on nonlinear pressure waves in bubbly flows did not consider the forces acting on the bubbles, although the validity of ignoring these forces has not been demonstrated. We focused on every possible force such as drag, gravity, buoyancy, and Bjerknes (acoustic radiation) forces acting on bubbles and studied their effects on pressure waves in a one-dimensional setting. Using a singular perturbation method, the Korteweg–de Vries–Burgers equation describing wave propagation was derived. The following results were obtained: (i) Bjerknes force on the bubbles enhanced the nonlinearity, dissipation, and dispersion of the waves; (ii) Drag, gravity, and buoyancy forces acting on the bubbles increased wave dissipation; (iii) Thermal conduction had the most substantial dissipation effect, followed by acoustic radiation, drag, buoyancy, and gravity. We confirmed that the dissipation due to forces on gas bubbles was quantitatively minor.

Approximate solution of KdV-Burgers equation using improved PINNs algorithm

Approximate solution of KdV-Burgers equation using improved PINNs algorithm

Undular bore theory for the modified Korteweg-de Vries-Burgers equation.

We consider nonlinear wave structures described by the modified Korteweg-de Vries equation, taking into account a small Burgers viscosity for the case of steplike initial conditions. The Whitham modulation equationsare derived, which include the small viscosity as a perturbation. It is shown that for a long enough time of evolution, this small perturbation leads to the stabilization of cnoidal bores, and their main characteristics are obtained. The applicability conditions of this approach are discussed. Analytical theory is compared with numerical solutions and good agreement is found.