KdV Burgers Research Articles

A continuous traffic flow model considering predictive headway variation and preceding vehicle’s taillight effect

The car taillights in the deceleration process of preceding vehicle may greatly affect the acceleration and deceleration processes of the following vehicle. Additionally, the driver can predict the headway at the next time period from surrounding traffic information, and adjust the vehicle acceleration based on the difference between the predicted headway and the current headway information. To analyze these combined effects, we propose a new continuous traffic flow model taking into account the predictive headway variation and preceding vehicle’s taillight. The stability condition and KdV–Burgers equation of the continuum model are derived in linear and nonlinear stability analysis. The density wave solution obtained by solving the KdV–Burgers equation can be used to describe the formation and propagation mechanism of traffic jams near stable conditions. The complex traffic phenomena such as shock waves, rarefaction waves and local cluster effects are reproduced by simulation. Results show that the taillight information of preceding vehicle and the driver’s prediction time step both contribute greatly to the stability of the traffic flow and energy consumption.

Nonlinear and bifurcation analysis for a novel heterogeneous continuum model and numerical tests

In this paper, an improved heterogeneous continuum model is presented accounting for multiple optimal velocity functions with probabilities, in which self-stabilizing effect and velocity uncertainty are considered simultaneously. The linear stability criterion of this new model is obtained by applying linear stability theory. By means of nonlinear analysis method, the Korteweg-de Vries-Burgers (KdV-Burgers) equation is deduced to discuss the evolution process of traffic flow density wave. Through bifurcation analysis, the conditions and stability of the Hopf bifurcation for the new model are derived in detail. Numerical simulations are performed to study the influences of above mentioned factors on the stability of traffic flow. Fuel consumption and emissions are also explored in this paper. It is notable that when the Hopf bifurcation point is selected as the initial point of density evolution, significant periodic oscillations will occur in the traffic flow. The results of numerical simulations coincide well with the theoretical analysis.

Cylindrical and Spherical Dust–Ion-Acoustic Shock Solitary Waves by Korteweg–de Vries–Burgers Equation

Cylindrical and Spherical Dust–Ion-Acoustic Shock Solitary Waves by Korteweg–de Vries–Burgers Equation

Ion-acoustic stable oscillations, solitary, periodic and shock waves in a quantum magnetized electron–positron–ion plasma

Abstract Nonlinear stable oscillations, solitary, periodic and shock waves in electron–positron–ion (EPI) quantum plasma in the presence of an external static magnetic field are reported. The Korteweg-de Vries-Burgers (KdVB) equation is derived by the reductive perturbation technique (RPT). The wave solution gives shock waves depending on various parameters as quantum diffraction parameter (β), electron and positron Fermi temperatures, and densities of the system species. Amplitude, polarity, speed, and width of wave solutions are remarkably modified by species densities, kinematic viscosity, and the Bohm potential. Existence of stable oscillation of ion-acoustic waves (IAWs) is shown by using the concept of phase plane analysis. Stability of wave solution is analysed by examining the Bohm potential effect. In the absence of dissipation, phase plane of the considered plasma system is analysed to discuss the existence of periodic wave solution. The results of this study could be helpful for comprehension of the wave features in dense quantum plasmas, like white dwarfs, laboratory plasma as interaction experiments of intense laser-solid matter and microelectronic devices.

Numerical Study on Weakly Nonlinear Evolution of Pressure Waves in Water Flows Containing Many Translational Bubbles Acting a Drag Force

Weakly nonlinear (i.e., finite but small amplitude) propagation of plane progressive pressure waves in compressible water flow uniformly containing many spherical gas bubbles is numerically investigated with a special attention to a drag force acting bubbles and translation of bubbles. The gas and liquid phases are flowing with initially independent velocities. Drag force and virtual mass force are introduced as interfacial momentum transports. Translation and spherically symmetric oscillations are considered as bubble dynamics. In this paper, under these assumptions, we numerically solve the KdVB (Korteweg-de Vries-Burgers) equation previously derived by ourselves (Yatabe et al., Phys. Fluids, 33 (2021), 033315) from basic equations based on a two-fluid model. The main results are summarized as follows: (i) The drag force acting on bubbles increases a dissipation effect of waves and drastically changes the phase and amplitude of waves. (ii) Although the translation of bubbles increases the nonlinear effect of waves, its contribution to waveform is quantitatively small. (iii) The effect of the drag force decreases with decreasing the initial void fraction and with increasing the initial bubble radius. That of the translation decreases with decreasing the initial void fraction, and is almost independent of the initial bubble radius. (iv) The spatiotemporal evolution of two type of dissipation effects (i.e., dissipation terms) due to the acoustic radiation and to the drag force is different tendency.

Theoretical analysis of electron acoustic shock waves in magnetized superthermal plasma with electron beam

AbstractWe have analysed the small‐amplitude non‐linear electron acoustic shock waves by taking into account the effects of electron beam in magnetized plasma. Satellite observations in different regions of the Earth's magnetosphere have shown that the electrostatic solitary waves are generally associated with electron or/and ion beams. The nonlinear Korteweg‐de‐Vries Burgers (KdVB) equation has been derived by considering the basic fluid equations and dissipation effects. The nonlinear coefficient of KdVB equation comes out to be negative. Only dip‐shaped potential structures are reported here. For the parameters discussed in this paper, we did not find positive polarity shocks. This could be due to the restrictions on the plasma parameters since we are using the fixed densities of the cold, hot, and beam electrons as observed by the Viking satellite in the auroral region. In this paper, the importance of the cold electron to hot electron temperature in conjunction with the beam speed is pointed out. Increase in beam density, kinematic viscosity, and magnetic field results in increase in the amplitude while the increase in hot electron concentration and superthermality leads to decrease in potential. The numerical analysis is presented for the parameters corresponding to the observation of burst b event by Viking satellite in the dayside auroral zone of the earth's magnetosphere.

Observer‐based H∞ control of a stochastic Korteweg–de Vries–Burgers equation

AbstractThis article mainly deals with observer‐based H∞ control problem for a stochastic Korteweg–de Vries–Burgers equation under point or averaged measurements. Due to the nonlinearity of the stochastic partial differential equations, special emphases are given to computation complexity. By constructing an appropriate Lyapunov functional, we derive sufficient conditions in terms of linear matrix inequalities to guarantee the internal exponential stability and H∞ performance of the perturbed closed‐loop system by means of the Lyapunov approach. Consistent simulation results that support the proposed theoretical statements are provided. Finally, we have made important instructions for future research directions.

Stability analysis of traveling wave solutions of a generalized Korteweg–de Vries–Burgers equation with variable dissipation parameter

We consider traveling wave solutions of a generalized Korteweg–de Vries–Burgers equation. The dissipation coefficient depends on coordinate and time and has a smoothed step-like form at every instant of time. Small-scale processes of dissipation and dispersion determine the solution of the traveling wave problem in the high-gradient region. The flux function is non-convex and has two inflection points. It is shown that traveling wave solutions with the same wave speed can converge to the three different limiting values behind the wave. The Evans function technique is used to conduct linear stability analysis. We demonstrate that traveling wave solutions can be linearly stable for each of the three possible cases. Thus, we found linearly stable solutions in the form of a traveling wave, which correspond to admissible discontinuities in hyperbolic model. These discontinuities have the same speed but different states behind them.

Weakly nonlinear theory on ultrasound propagation in liquids containing many microbubbles encapsulated by visco-elastic shell

Aimed towards an application of ultrasound diagnosis using contrast agents, the dynamics of encapsulated bubbles has been theoretically investigated under the restriction of a single bubble. In this paper, we extend the theory for single bubble or some bubbles to that for many bubbles, and theoretically investigate weakly nonlinear propagation of ultrasound in an initially quiescent incompressible liquid, uniformly containing many microbubbles encapsulated by the shell as a viscoelastic body (Kelvin–Voigt model). As a result, we derived the Korteweg–de Vries–Burgers equation for a low-frequency long wave and clarified that the shell affects the advection, nonlinear, and dissipation (not dispersion) effects of ultrasound propagation. In particular, shell rigidity, surface tension, and shell viscosity increased the advection, nonlinear, and dissipation effects, respectively.

An exhaustive theoretical analysis of thermal effect inside bubbles for weakly nonlinear pressure waves in bubbly liquids

Weakly nonlinear propagation of pressure waves in initially quiescent compressible liquids uniformly containing many spherical microbubbles is theoretically studied based on the derivation of the Korteweg–de Vries–Burgers (KdVB) equation. In particular, the energy equation at the bubble–liquid interface [Prosperetti, J. Fluid Mech. 222, 587 (1991)] and the effective polytropic exponent are introduced into our model [Kanagawa et al., J. Fluid Sci. Technol. 6, 838 (2011)] to clarify the influence of thermal effect inside the bubbles on wave dissipation. Thermal conduction is investigated in detail using some temperature-gradient models. The main results are summarized as follows: (i) Two types of dissipation terms appeared; one was a well-known second-order derivative comprising the effect of viscosity and liquid compressibility (acoustic radiation) and the other was a newly discovered term without differentiation comprising the effect of thermal conduction. (ii) The coefficients of the KdVB equation depended more on the initial bubble radius rather than on the initial void fraction. (iii) The thermal effect contributed to not only the dissipation effect but also to the nonlinear effect, and nonlinearity increased compared with that observed by Kanagawa et al. (2011). (iv) There were no significant differences among the four temperature-gradient models for milliscale bubbles. However, thermal dissipation increased in the four models for microscale bubbles. (v) The thermal dissipation effect observed in this study was comparable with that in a KdVB equation derived by Prosperetti (1991), although the forms of dissipation terms describing the effect of thermal conduction differed. (vi) The thermal dissipation effect was significantly larger than the dissipation effect due to viscosity and compressibility.

The numerical solution of fractional Korteweg‐de Vries and Burgers' equations via Haar wavelet

In this article, Haar wavelet collocation technique is adapted to acquire the approximate solution of fractional Korteweg‐de Vries (KdV), Burgers', and KdV–Burgers' equations. The fractional order derivatives involved are described using the Caputo definition. In the proposed technique, the given nonlinear fractional differential equation is discretized with the help of Haar wavelet and reduced to the nonlinear system of equations, which are solved with Newton's or Broyden's method. The proposed method is semi‐analytic as it involves exact integration of Caputo derivative. The proposed technique is widely applicable and robust. The technique is tested upon many test problems. The results are computed and presented in the form of maximum absolute errors which show the accuracy, efficiency, and simple applicability of the proposed method.

SYMMETRY ANALYSIS, ANALYTICAL SOLUTIONS AND CONSERVATION LAWS OF A GENERALIZED KdV–BURGERS–KURAMOTO EQUATION AND ITS FRACTIONAL VERSION

Wave motions play an important role in fluid dynamics and engineering issues. In this paper, a systematic study to the generalized KdV–Burgers–Kuramoto equation is presented resort to symmetry method. First, based on the Lie point symmetries of the generalized KdV–Burgers–Kuramoto equation, we get invariants and invariant solutions. In particular, we obtain series solutions of generalized KdV–Burgers–Kuramoto equation. Meanwhile, we find that this equation just exists in Lie point symmetries. Then, we present a conservation law, and derive reciprocal Bäcklund transformations of conservation law for the first time. Furthermore, Bäcklund transformation of KdV–Burgers–Kuramoto equation associated with truncated Painlevé expansion is studied for the first time. Subsequently, mCK method is employed to study KdV–Burgers–Kuramoto equation. Lastly, we investigate symmetries of generalized time fractional KdV–Burgers–Kuramoto equation, and derive a conservation law. The obtained results illustrate that symmetry method is a very powerful method to deal with nonlinear partial differential equations. The results provide theoretical support for explaining complex nonlinear phenomena.

Weakly nonlinear theory on pressure waves in bubbly liquids with a weak polydispersity

Weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid uniformly containing many spherical microbubbles is theoretically investigated, especially focusing on an initial small polydispersity of both the bubble radius and the number density of bubbles (i.e., void fraction), which appears in a field far from the sound source. Nonlinear waves in polydispersed bubbly liquids are classified into a form of three cases of nonlinear wave equation describing long-range propagation of waves. Using the method of multiple scales with perturbation expansions and the scaling relations of some nondimensional ratios, from the set of basic equations based on a two-fluid model, (i) for a low-frequency long wave, the Korteweg–de Vries–Burgers (KdVB) equation is derived and (ii) for a moderately high-frequency short wave, (ii-a) the NLS (nonlinear Schrödinger)-I (or LG (Landau–Ginzburg)-I) equation for a weak polydisperse medium and (ii-b) the NLS-II (or LG-II) equation in a strong polydisperse medium are derived in a unified manner. For all cases, polydispersity contributes to the advection effect of waves and induces variable coefficients into the KdVB, NLS-I, and NLS-II equations. Furthermore, the KdVB equation includes an inhomogeneous term owing to the polydispersity and the NLS-II equation includes second-order nonlinearity of polydispersity. The polydisperse effect is finally clarified quantitatively by focusing on the advection coefficients with a help of numerical examples.

On the analytical and numerical solutions of the damped nonplanar Shamel Korteweg–de Vries Burgers equation for modeling nonlinear structures in strongly coupled dusty plasmas: Multistage homotopy perturbation method

The dissipative cylindrical and spherical (nonplanar) electrostatic low-frequency dust-acoustic waves (DAWs) including solitary and shock waves in a collisional and unmagnetized strongly coupled dusty plasma are investigated analytically and numerically. The present plasma model consists of inertialess particles including thermal elections and vortex-like positive ions distribution as well as inertial strongly coupled negatively charged dust grains. In the hydrodynamic regime, the fluid governed equations of the present model are reduced to the damped nonplanar Shamel Korteweg–de Vries Burgers (SKVB) equation using the reductive perturbation technique. In the absence of the dissipative effect, the damped nonplanar Shamel Korteweg–de Vries (SKdV) equation is recovered and solved analytically for the first time using a novel analytical approach in order to describe the dynamical mechanism of the dissipative nonplanar dust-acoustic solitary waves. Also, the damped nonplanar SKdV equation is solved numerically using the homotopy perturbation method (HPM) and the hybrid homotopy perturbation method with the moving boundary method which is called multistage HPM (MsHPM). Furthermore, in the presence of the dissipative effect, the damped nonplanar SKdVB equation is solved numerically via the HPM and MsHPM for studying the characteristics of the dissipative nonplanar dust-acoustic solitary and shock waves. For checking the accuracy of the obtained solutions, the maximum global residual error is estimated. Moreover, a comparison between the approximate analytical and numerical solutions is reported. Furthermore, the dependence of dissipative nonplanar structures (solitons and shocks) characteristics on various plasma parameters is examined.

Weak dust acoustic shock wave in strongly coupled two-dimensional complex plasma

By using the equation of the state of P=αd+βdTd, we studied a shock wave propagation in a strongly coupled complex plasma. A Korteweg-de Vries–Burgers equation is obtained to describe a shock wave in strongly coupled complex plasma. Dependence of the shock wave speed on the piston velocity and the plasma parameters such as the screening parameter of the strongly coupled complex plasma are given analytically for a weak shock wave. It is in good agreement with previous results. It shows in the present paper that there are density oscillations in the postshock region which is similar to that found in the previous works. Dependence of the length between two first peaks of the density oscillation on the piston velocity and the plasma parameter are also given analytically which is also in agreement qualitatively with previous results.

Theoretical elucidation of effect of drag force and translation of bubble on weakly nonlinear pressure waves in bubbly flows

Theoretical investigation of the effects of a translation of bubbles and a drag force acting on bubbles on the wave propagation in bubbly flows has long been lacking. In this study, we theoretically and numerically investigate the weakly nonlinear (i.e., finite but small amplitude) propagation of plane progressive pressure waves in compressible water flows that contain uniformly distributed spherical gas bubbles with translation and drag forces. First, we assume that the gas and liquid phases flow at independent velocities. Then, the drag force and virtual mass force are introduced in an interfacial transport across the bubble–liquid interface in the momentum conservation equations. Furthermore, we consider the translation and spherically symmetric oscillations as bubble dynamics and deploy a two-fluid model to introduce the translation and drag forces. Bubbles do not coalesce, break up, extinct, or appear. For simplicity, the gas viscosity, thermal conductivities of the gas and liquid, and phase change and mass transport across the bubble–liquid interface are ignored. The following results are then obtained: (i) Using the method of multiple scales, two types of Korteweg–de Vries–Burgers equations with a correction term due to the drag force are derived. (ii) The translation of bubbles enhances the nonlinear effect of waves, and the drag force acting on bubbles contributes the nonlinear and dissipation effects of waves. (iii) The results of long-period numerical analysis verify that the temporal evolution of the wave (not flow) dissipation due to the drag force differs from that caused by the acoustic radiation.

The Effect of Weak Mutual Diffusion on Transport Processes in a Multiphase Medium

The Cauchy problem for a singularly perturbed system of equations describing a transport process with diffusion in a multiphase medium is considered. A formal asymptotic expansion of its solution is constructed in the case when exchange between the phases proceeds much more rapidly than the transport and diffusion processes. The case when the diffusion fluxes of the components have a mutual effect on each other is considered. Under the assumptions imposed on the data of the problem, the leading term of the asymptotics is described by a multidimensional generalized Burgers–Korteweg–de Vries equation. Under some additional conditions, the remainder is estimated by means of the residual.

Traveling fronts in dissipative granular chains and nonlinear lattices

We consider an infinite chain of particles with nonlinear elastic and dissipative nearest neighbors interactions. Assuming the existence of a traveling front between uniformly compressed (or stretched) states, we obtain jump conditions relating the wave speed and limiting particle velocities to the relative displacements at infinity. Using this result, we characterise compression fronts in chains of touching beads, for viscoelastic contact laws that include a nonlinear elastic force (generalised Hertz contact) and viscous dissipation. We compute compression fronts numerically for the generalised Kuwabara–Kono model in which the viscous contact force is proportional to the derivative of the elastic force, without precompression of the chain. Steady fronts are obtained both as the end result of the compression of one end of the chain and using a shooting method which provides numerically exact traveling waves. Depending on the magnitudes of contact damping and strain applied downstream, we obtain either overdamped (monotonic) or underdamped (oscillatory) compression fronts. To explain this transition and approximate the front profiles, we consider a continuum limit valid when the exponent of the contact nonlinearity is close to (and above) unity. Using multiscale expansions, we formally derive two different amplitude equations for long waves, a Burgers equation with logarithmic nonlinearity, and a logarithmic Korteweg–de Vries (KdV)–Burgers equation for small contact damping. Both models possess traveling front solutions that are in good agreement with the front profiles computed numerically in the granular chain. The analysis of the logarithmic KdV–Burgers equation allows one to approximate the critical damping corresponding to the transition from underdamped to overdamped fronts.

On the Multistage Differential Transformation Method for Analyzing Damping Duffing Oscillator and Its Applications to Plasma Physics

The multistage differential transformation method (MSDTM) is used to find an approximate solution to the forced damping Duffing equation (FDDE). In this paper, we prove that the MSDTM can predict the solution in the long domain as compared to differential transformation method (DTM) and more accurately than the modified differential transformation method (MDTM). In addition, the maximum residual errors for DTM and its modification methods (MSDTM and MDTM) are estimated. As a real application to the obtained solution, we investigate the oscillations in a complex unmagnetized plasma. To do that, the fluid govern equations of plasma species is reduced to the modified Korteweg–de Vries–Burgers (mKdVB) equation. After that, by using a suitable transformation, the mKdVB equation is transformed into the forced damping Duffing equation.

Bifurcation analysis for a novel heterogeneous continuum model considering electronic throttle angle changes with memory

Heterogeneous traffic flow model such as mixes of trucks and cars or connected and regular vehicles is a central issue in traffic flow modeling. However, few researches focus on the bifurcation phenomena of heterogeneous traffic flow. Due to this point, a novel heterogeneous continuum model accounting for different ratios of multiple optimal velocity functions is proposed, in which electronic throttle angle changes with memory is also considered. A simple formula for the linear stability criterion of this improved model is theoretically deduced, and the Korteweg-de Vries-Burgers (KdV-Burgers) equation is derived to discuss the characteristics of density evolution through nonlinear analysis method. The conditions and stability of the Hopf bifurcation for the modified model are then proved minutely. Numerical simulations are executed to investigate how multiple maximum velocities, multiple safety spacings and electronic throttle angle changes with memory affect the density fluctuations, in which numerical experiments on fuel consumptions and emissions are also included. To address the problems mentioned above, we set the Hopf bifurcation as the initial point of the heterogeneous continuum traffic flow to analyze the evolution of bifurcation behavior in the heterogeneous traffic flow.