KdV Burgers Research Articles

A single bounded input‐feedback control to the generalized Korteweg–de Vries–Burgers–Kuramoto–Sivashinsky equation

The generalized Korteweg–de Vries–Burgers–Kuramoto–Sivashinsky equation (GKdVBKS) is a nonlinear partial differential equation that models propagation of waves in a thick elastic tube filled with a viscous fluid. First, the dynamics of the GKdVBKS equation that depend on the physical parameters and , where is a positive parameter and is a real number, are presented. We show that the set of constant equilibrium solutions is unstable when , Lyapunov stable when , and globally exponentially stable when . Second, we show that the controlled GKdVBKS equation has a unique strong solution. Then, we synthesize a single bounded input‐feedback controller for the GKdVBKS equation through its linearized system avoiding any spillover. The approach used is valid due to the Fréchet differentiability of the nonlinear ‐semigroup generated by the nonlinear GKdVBKS equation with its derivative being the linear ‐semigroup generated by the linearized system. Finally, numerical results for different values of the parameters of the equation are presented to illustrate the developed theory.

Finite-time boundary stabilization for Korteweg–de Vries–Burgers equations

The finite-time boundary stabilization is investigated for Korteweg–de Vries–Burgers (KdVB) equations. Firstly, a distributed controller is derived, and criteria are presented to guarantee the finite-time stability (FTS) of KdVB equations via the distributed controller. Then, considering the installation in real applications, a vogue control strategy, a boundary controller design, is adopted to ensure the FTS. In light of the FTS lemma, sufficient conditions are obtained for KdVB equations to ensure the FTS. When there exist uncertainties in the system’s coefficients, the robust finite-time stabilization is studied for uncertain KdVB equations. With the designed boundary controller, a sufficient condition is provided to ensure the FTS for the uncertain KdVB equation. Finally, the theoretical results are verified by numerical examples.

Ion temperature gradient mode linear and nonlinear structures in electron-ion plasma, with stationary charged dust grain

Ion temperature gradient mode nonlinear structures are studied with stationary charged dust. The dynamical role here is played by heavy ions. The small amplitude limit influence of various plasma parameters shows that the phase velocity of the mode modifies the overall plasma dynamics. Further, from the solution of the model equations we obtain Kortweg-de-Vries (KdV) and Korteweg-de-Vries Burger (KdVB) like equations, the solution of which gives solitary and shock wave nonlinear structures. Numerical analysis of these nonlinear structures shows that dust number density and polarity have a substantial impact on these structures. The present observation may be beneficial in space and laboratory plasma investigations.

Meromorphic solutions of autonomous ordinary differential equations without the finiteness property

We consider the problem of finding meromorphic solutions of autonomous algebraic ordinary differential equations possessing an infinite number of local solutions given by Laurent series in neighborhoods of poles. We present an algebraic-geometric method for finding all meromorphic solutions that are elliptic or rational in the exponential function. As an example, we classify such families of meromorphic solutions for fourth-order ordinary differential equations arising via integrating once traveling-wave reductions of the generalized Rosenau–Korteweg–de Vries equation and the fifth-order Korteweg–de Vries–Burgers equation. Solutions important from a physical point of view are presented.

Wind-induced changes to shoaling surface gravity wave shape

Unforced shoaling waves experience growth and changes to wave shape. Similarly, wind-forced waves on a flat-bottom likewise experience growth/decay and changes to wave shape. However, the combined effect of shoaling and wind-forcing on wave shape, particularly relevant in the near-shore environment, has not yet been investigated theoretically. Here, we consider small-amplitude, shallow-water solitary waves propagating up a gentle, planar bathymetry forced by a weak, Jeffreys-type wind-induced surface pressure. We derive a variable-coefficient Korteweg-de Vries-Burgers (vKdV-B) equation governing the wave profile's evolution and solve it numerically using a Runge-Kutta third-order finite difference solver. The simulations run until convective prebreaking -- a Froude number limit appropriate to the order of the vKdV-B equation. Offshore winds weakly enhance the ratio of prebreaking height to depth as well as prebreaking wave slope. Onshore winds have a strong impact on narrowing the wave peak, and wind also modulates the rear shelf formed behind the wave. Furthermore, wind strongly affects the width of the prebreaking zone, with larger effects for smaller beach slopes. After converting our pressure magnitudes to physically realistic wind speeds, we observe qualitative agreement with prior laboratory and numerical experiments on breakpoint location. Additionally, our numerical results have qualitatively similar temporal wave shape to shoaling and wind-forced laboratory observations, suggesting that the vKdV-B equation captures the essential aspects of wind-induced effects on shoaling wave shape. Finally, we isolate the wind's effect by comparing the wave profiles to the unforced case. This reveals that the numerical results are approximately a superposition of a solitary wave, a shoaling-induced shelf, and a wind-induced, bound, dispersive, and decaying tail.

Shock Waves in Weakly Relativistic e-p-i Plasma: Application to Plasma Created by Ultraintense Short Pulse Laser

In this article, the role of the positrons shock waves generation in weakly relativistic laser-created plasma is investigated. The plasma considered is generated on the rear face of a target, irradiated by a high intensity short pulse laser, in the framework of target normal sheath acceleration (TNSA) mechanism. This plasma is composed of Boltzmann distributed positrons, hot and cold super-thermal electrons, and dynamic ions. Kinematic viscosity among the plasma constituents has been also considered. With the purpose of studying the behavior of nonlinear ion acoustic shock waves (IASHWs), the reductive perturbation method (RPM) has been employed and Korteweg-de Vries Burgers equation has been derived. It was observed that different plasma parameters (namely, electrons super-thermality, number of positrons, ion temperature, kinematic viscosity, etc.) have a real impact on the propagation of IASHWs. As electron–positron plasmas with ion possessing relativistic velocities are often detected in laboratory experiments, astrophysical, and space environments, the present work will help to explain the different types of collective processes and instabilities in regions where e-p-i plasmas can exist.

Plasma Shock Wave in Gamma-Ray Bursts: Nonlinear Phenomena and Radiative Process

Since gamma-ray bursts (GRBs) are related to extreme explosions in the universe, they provide a unique setup to study various space plasma phenomena. The GRB spectra supply a wide range of information regarding the various nonlinear interactions and associated structures. In this work, we study the generation of shock fronts and their properties in a plasma. Such a plasma shock is the aftereffect of an explosion of which the prompt emission of GRBs are the signatures. The shock fronts are studied analytically by deriving the KdV–Burger’s equation. The solution of the KdV–Burgers equation via the “Cole-Hoff transformation” is studied alongside the usual “hyperbolic-tangent method.” The dynamical properties of the shock solitons are studied further to second our findings and find the stability and any possibility of chaos. The results will help to interpret the stellar plasma properties from data. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PACS</i> —95.30.Qd, 52.65.-y, 52.35.-g, 52.35.Tc, 52.27.Lw

Asymptotics of solutions to periodic problem for the Korteweg–de Vries–Burgers equation

AbstractWe consider the periodic problem for the Korteweg–de Vries–Burgers (KdVB) equation with pumping where , . We study the solutions, which satisfy the periodic boundary conditions for all and , with the 2π—periodic initial data . Our aim is to find large time asymptotic profile of solutions. The large time asymptotic behavior of solutions to the Cauchy problem for the KdVB equations with different dissipative terms was extensively studied. In the present paper we find the asymptotic profile of solutions for large time. We develop the approach started in Refs. [1, 2, 3]. We prove the following asymptotics for the solutions: as uniformly with respect to , where , , , , , Λ is a constant.

Magnetosonic Shocks and Solitons in Fermi Plasma With Quasiperiodic Perturbation

In this article, we consider a dense stellar plasma consisting of predominantly electrons, positrons, and ions under the action of the magnetic field of the star. We derived the Korteweg–de Vries Burgers (KdV-Burgers) equations by using the reductive perturbation technique and obtained shock and solitary profiles for magnetoacoustic waves. We further studied the self-interaction of such stationary formations and the amplitude modulated envelope solitons. The possibility of a rogue wave-like structure is also discussed. To understand the physical problem in greater detail, we studied the Hamiltonian formulation to study stability. We studied how modulational instability can lead to the formation of rogue waves by investigating the nonlinear Schrodinger equation. In addition, drift-like instability showing the density and charge separation effects in plasma has been studied. The results will be helpful to interpret magnetoacoustic wave formations in solar corona or other stellar entities and can help in the study of inhomogeneous plasmas in laboratory and fusion reactors. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PACS</i> —52.35.Fp, 03.50.De, 41.20.-q, 52.30.Cv, 47.85.Dh, 52.35.Hr, 52.27.Ep, 52.30.-q, 52.35.Tc, and 52.35.Sb.

Obliquely propagating dust acoustic shock waves in magnetized plasma under the influence of polarization force and nonadiabaticity of dust charge variation

In this work, we have analyzed the impact of polarization force, angle of obliqueness under the influence of nonadiabatic dust charge fluctuation on shock waves formation in magneto rotating plasma. The present plasma model is consisting of negatively charged dust grains, Maxwellian electrons, non-Maxwellian hybrid distributed ions. The dissipation is introduced in the system via nonadiabaticity, a new mechanism for the formation of shock waves. Using the standard reductive perturbation method, nonlinear equation, namely, Korteweg-de Vries-Burgers (KdVB) equation is derived and solution is obtained using the Tanh method. It is shown that dust charge fluctuation is the main source of dissipation. We have studied the various parameteric influences on such shock structure and also showed how the gradual variations of these parameters affect the generation and structure of the shocks in their respective domain. Much of the experiments on dusty plasma with nonadiabatic dust charge fluctuation will benefit from the parametric study.

Global stability of traveling wave solutions of generalized Korteveg–de Vries–Burgers equation with non-constant dissipation parameter

We consider traveling wave solutions of generalized Korteweg–de Vries–Burgers equation when the flux function has two inflection points. The dissipation coefficient μ depends only on the spatial coordinate in some moving coordinate system and increases monotonically from μ1 to μ2 in the narrow spatial region. Some external influence causes the change in the dissipation coefficient. The set of admissible shocks is defined. In order to determine which discontinuities are admissible, we study the nonlinear (global) stability of traveling wave solutions. Scenarios of the evolution of linearly unstable traveling waves are described, and asymptotics of unstable solutions are found. We find that a stable traveling wave solution and a solution with a time-dependent structure can correspond to the same admissible shock.

Bifurcation analysis for dust ion acoustic waves in magnetized four-component plasmas with superthermal electrons

In collisionless, dissipative, and magnetized dusty plasma, the propagation of nonlinear dust-ion-acoustic (DIA) solitary and shock waves has been studied, whose constituents are cold positive ions, negative ions obeying the Boltzmann distribution, superthermally distributed electrons and positively charged stationary dust particles. The Korteweg-de Vries–Burgers (KdVB) equation is derived using the reductive perturbation method. This equation is solved using the tangent hyperbolic method, yielding a shock wave solution. Also, other interesting physical solutions and the corresponding electric field are determined. By applying the bifurcation theory to the planar dynamical system of the derived KdVB equation under two conditions and , the presence of solitary wave solutions and periodic traveling wave solutions has been established. According to both numerical and analytical investigations, The basic properties of dust ion acoustic waves are significantly modified by the influence of superthermal electrons, viscosity coefficients, and the existence of an external magnetic field. The current results could help in a full understanding and studying of plasma waves in both space and astrophysical environments (such as the magnetospheres of Saturn and pulsars) and also in the laboratory.

Nonlinear and dissipation effects of pressure waves in water flows containing translational bubbles with a drag force

Abstract Weakly nonlinear (i.e., finite but small amplitude) propagation of plane progressive pressure waves in compressible water flows uniformly containing many spherical bubbles is theoretically studied. Drag force acting bubbles and translation of bubbles are newly considered by introducing in momentum conservation equations in a two fluid model and the bubble dynamics equation for volumetric oscillations, respectively. Although these assumptions are the same as our previous paper, in this study, the energy conservation equation for each bubble describing a thermal conduction inside bubble is introduced. By using the method of multiple scales, the Korteweg–de Vries–Burgers equation for low-frequency long wave was derived from the set of basic equations in the two-fluid model. As a result, the dissipation effect was described by two types of terms, i.e., one was the second-order partial derivative owing to the liquid compressibility and the other was the term without differentiation owing to the drag force and the thermal conduction. Finally, we clarified that the dissipation owing to the drag force was smaller than that owing to the thermal conduction.

Boundary control of stochastic Korteweg-de Vries-Burgers equations

The boundary control problem is considered for stochastic Korteweg-de Vries-Burgers equations. First, a boundary controller is proposed, and a criterion is obtained for mean square exponential stability by using the Lyapunov functional method and inequality techniques. Then, when there exist uncertainties in the system parameters, the robust mean square exponential stability is considered, and a sufficient criterion is obtained. Furthermore, if there are also additive noises in the considered system, the H-infinity performance is investigated and a sufficient condition is obtained to ensure the mean square H-infinity performance. Numerical examples illustrate the validity of the theoretical results.

Nonlinear analysis of the ion-acoustic solitary and shock wave solutions for non-extensive dusty plasma in the framework of modified Korteweg–de Vries–Burgers equation

Nonlinear analysis of the ion-acoustic solitary and shock wave solutions for non-extensive dusty plasma in the framework of modified Korteweg–de Vries–Burgers equation

Equilibrium states of Burgers and Korteweg-de Vries equations.

We simulate Korteweg-de Vries (KdV) and dissipationless Burgers equationsusing δ-correlated random noise as an initial condition. We observe that the energy fluxes of the two equationsremain zero throughout, thus, indicating their equilibrium nature. We characterize the equilibrium states using Gaussian probability distribution for the real space field, and using Boltzmann distribution for the modal energy. We show that the single soliton of the KdV equationtoo exhibits zero energy flux, hence, it is in equilibrium. We argue that the energy flux is a good measure for ascertaining whether a system is in equilibrium or not.

Shocks and solitons in collisional dense laser produced plasmas

The characteristics of nonlinear electron-acoustic waves such as shocks and solitons, are investigated in a three component, dense laser produced plasma consisting of ions and two distinct groups of electrons, using the quantum hydrodynamic model and the standard reductive perturbation method. The modified Korteweg-deVries (mKdV) and Korteweg-deVries-Burgers (KdVB) equations have been derived for the electron-acoustic waves in the plasma. The dependence of both shocks and solitons on various parameters has been extensively studied. It is observed that whenever the density crosses the limit from the classical to the quantum range, the effective potential remains invariant for the solitary profiles; but shows a slight variation for the shock profiles. The collisional effect plays a significant role in the dissipation of solitary waves and the dissipation is larger for higher values of collision frequencies. The results obtained could prove helpful for understanding the parametric dependence of nonlinear waves in highly intense laser plasma interactions.

Evolution of cylindrical/spherical shock formation in a dusty plasma with nonadiabatic dust charge variation

Cylindrical/spherical dust acoustic shock waves are studied in an unmagnetized plasma consisting of Maxwellian electrons, superthermal ions under the effect of nonadiabatic dust charge fluctuation. The dissipation is introduced in the system via nonadiabaticity, a new mechanism for the formation of shock waves. Using standard reductive perturbation method, nonplanar equation namely Korteweg–de Vries Burgers (KdVB) equation is derived and solution is obtained using the Weighted residual method. By use of weighted residual method, a progressive wave solution to nonplanar modified KdV Burgers equation is presented. It is shown that dust charge fluctuation produces dissipation which is responsible for shock formation. The solution shows that as one goes towards the axis of the cylinder or center of the sphere, the wave attenuates faster and the effects are shown in some 2D figures. We have studied various parametric influences on such shock structure and also showed how the gradual variations of these parameter affect the generation and structure of the shocks in their respective domain. Our results should be applicable to the formation of nonlinear DIA shock wave structures in regions in which dust grain is embedded in a Kappa distributed plasma, such as Saturn's magnetosphere.

Shock formation in magnetized plasma under the influence of polarization force and nonadiabaticity of dust charge variation

In this work, we have analyzed the impact of polarization force, angle of obliqueness under the influence of nonadiabatic dust charge fluctuation on shock waves formation in magneto rotating plasma. The present plasma model is consisting of negatively charged dust grains, Maxwellian electrons, nonextensive ions. The dissipation is introduced in the system via nonadiabaticity, a new mechanism for the formation of shock waves. Using standard reductive perturbation method, nonlinear equation namely Korteweg–de Vries Burgers equation is derived and solution is obtained using the Tanh method. It is shown that dust charge fluctuation is the main source of dissipation. We have studied the various parameteric influences on such shock structure and also showed how the gradual variations of these parameter affect the generation and structure of the shocks in their respective domain. Much of experiments on dusty plasma with nonadiabatic dust charge fluctuation will benefit from the parametric study.

Well-posedness and stability results for the Korteweg–de Vries–Burgers and Kuramoto–Sivashinsky equations with infinite memory: A history approach

The main concern of the present paper is to study the well-posedness and stability problem of two different dispersive systems subject to the effect of a distributed infinite memory term. The two systems are respectively governed by the one-dimensional Korteweg–de Vries–Burgers and Kuramoto–Sivashinsky equations in a bounded domain [0,1]. In order to deal with the presence of the memory term, we adopt the history approach. First, we show that both problems are well-posed in appropriate functional spaces by means of the Fixed-Point Theorem provided that the initial condition is sufficiently small. Then, the energy method enables us to provide a decay estimate of the systems’ energy according to the assumptions satisfied by the physical parameters and the memory kernel.