Generalized Korteweg De Vries Burgers Research Articles

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.

Structures of Classical and Special Discontinuities for the Generalized Korteweg–de Vries–Burgers Equation in the Case of a Flux Function with Four Inflection Points

Structures of Classical and Special Discontinuities for the Generalized Korteweg–de Vries–Burgers Equation in the Case of a Flux Function with Four Inflection Points

On the Instability of Monotone Traveling-Wave Solutions for a Generalized Korteweg–de Vries–Burgers Equation

On the Instability of Monotone Traveling-Wave Solutions for a Generalized Korteweg–de Vries–Burgers Equation

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.

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.

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.

Dynamics and control of the modified generalized Korteweg–de Vries–Burgers equation with periodic boundary conditions

The paper deals with the dynamics and control of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) with periodic boundary conditions. First, the dynamics of the MGKdVB equation is studied using the Fourier Galerkin and the Karhunen–Loeve (K–L) Galerkin methods. The Fourier Galerkin approach is used to generate a system of nine ordinary differential equations (ODEs) from a partial differential equation (PDE), and the K–L Galerkin method is used as a model reduction technique for nonlinear systems to derive a reduced-order system of two ODEs that imitates the dynamics of the MGKdVB equation. It is shown that the two-dimensional reduced-order ODE system based on the K–L Galerkin method is superior to the Fourier Galerkin n-dimensional ODE system for any dimension $$n<2N+1$$ , where $$N=2$$ is the number of unstable eigenvalues, and it is comparable to the traditional higher nine-dimensional Fourier Galerkin system. Then, in order to enhance the stability of the MGKdVB equation, we propose to use state feedback linearization control schemes for both systems and show that the reduced-order system based on K–L Galerkin method is less expensive to control as compared to the Fourier Galerkin system. Finally, numerical simulations of the controlled systems are shown to illustrate the developed theory.

Traveling waves and undercompressive shocks in solutions of the generalized Korteweg–de Vries–Burgers equation with a time-dependent dissipation coefficient distribution

Solutions of the generalized KdV–Burgers equation are analyzed in the case when the dissipation coefficient depends on the spatial coordinate and time. Solutions in the form of traveling waves describing the structure of discontinuities, including special discontinuities, are studied. In the frame of this problem, nonstationary solutions of the generalized KdV–Burgers equation including the special discontinuity are numerically found.

Nonlinear Adaptive Boundary Control of the Modified Generalized Korteweg–de Vries–Burgers Equation

In this paper, we study the nonlinear adaptive boundary control problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is 0,1. Four different nonlinear adaptive control laws are designed for the MGKdVB equation without assuming the nullity of the physical parameters ν, μ, γ1, and γ2 and depending whether these parameters are known or unknown. Then, using Lyapunov theory, the L2-global exponential stability of the solution is proven in each case. Finally, numerical simulations are presented to illustrate the developed control schemes.

Boundary linear stabilization of the modified generalized Korteweg\u2013de Vries\u2013Burgers equation

The linear stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) is considered when the spatial variable lies in [0,1]. First, the existence and uniqueness of global solutions are proved. Next, the exponential stability of the equation is established in L^{2} (0,1). Then, a linear adaptive boundary control is put forward. Finally, numerical simulations for both non-adaptive and adaptive problems are provided to illustrate the analytical outcomes.

Numerical methods based on radial basis function-generated finite difference (RBF-FD) for solution of GKdVB equation

The main aim of present work is to develop the two meshless collocation methods based on radial basis function-generated finite difference (RBF-FD) and global RBF(GRBF) methods to solve the non-linear generalized Korteweg-de Vries-Burgers (GKdVB) equation. For this purpose, at first, we use Taylor expansion to approximate the non-linear terms of the GKdVB equation by linear terms and discretize the temporal derivative with a finite difference scheme. Then we discretize the spatial derivatives with RBF-FD and GRBF methods which give us a linear system of equations. By solving the linear system we get numerical solutions for GKdVB equation. The efficiency and accuracy of these methods are shown with five test problems. Furthermore, the comparisons between the obtained results with spline approximation, RBF collocation, finite element (FEM), HBIM and analytical numerical solution (ANS) methods for some test problems are given. The results show that our methods are more accurate than the mentioned ones.

On the well-posedness and asymptotic behaviour of the generalized Korteweg–de Vries–Burgers equation

In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg–de Vries–Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval [1, 5). We first prove the global well-posedness in Hs(ℝ) for 0 ≤ s ≤ 3 and 1 ≤ p &lt; 2, and in H3(ℝ) when p ≥ 2. For 2 ≤ p &lt; 5, we prove the existence of global solutions in the L2-setting. Then, by using multiplier techniques and interpolation theory, the exponential stabilization is obtained with an indefinite damping term and 1 ≤ p &lt; 2. Under the effect of a localized damping term the result is obtained when 2 ≤ p &lt; 5. Combining multiplier techniques and compactness arguments, we show that the problem of exponential decay is reduced to proving the unique continuation property of weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.

Nonstationary solutions of a generalized Korteweg-de Vries-Burgers equation

Nonstationary solutions of the Cauchy problem are found for a model equation that includes complicated nonlinearity, dispersion, and dissipation terms and can describe the propagation of nonlinear longitudinal waves in rods. Earlier, within this model, complex behavior of traveling waves has been revealed; it can be regarded as discontinuity structures in solutions of the same equation that ignores dissipation and dispersion. As a result, for standard self-similar problems whose solutions are constructed from a sequence of Riemann waves and shock waves with stationary structure, these solutions become multivalued. The interaction of counterpropagating (or copropagating) nonlinear waves is studied in the case when the corresponding self-similar problems on the collision of discontinuities have a nonunique solution. In addition, situations are considered when the interaction of waves for large times gives rise to asymptotics containing discontinuities with nonstationary periodic oscillating structure.

Adaptive control of the generalized Korteweg-de Vries-Burgers equation

In this paper, the adaptive and non-adaptive stabilization of the generalized Korteweg-de Vries (KdV)-Burgers equation by nonlinear boundary control are analyzed. This is motivated by the method proposed by Smaoui (2004). We use this method to resolve the adaptive control of the generalized KdV-Burgers equation. For the non-adaptive case, using Lyapunov method, we show that the controlled system is exponentially stable in . As for the adaptive case, we show the regulation of the solution of the generalized KdV-Burgers equation. Key words: Adaptive control, KdV-Burgers equation, stabilization.

Adaptive boundary control of the unforced generalized Korteweg–de Vries–Burgers equation

This paper deals with the adaptive control problem of the unforced generalized Korteweg–de Vries–Burgers (GKdVB) equation when the spatial domain is [0,1]. Three adaptive control laws are designed for the GKdVB equation when either the kinematic viscosity ν or the dynamic viscosity μ is unknown, or when both viscosities ν and μ are unknowns. Using the Lyapunov theory, the L2-global exponential stability of the solutions of this equation is shown for each of the proposed control laws. Also, numerical simulations based on the Finite Element method (FEM) are given to illustrate the analytical results.

Pullback attractors for generalized Korteweg–de Vries–Burgers equations

In this paper, we study the long-time behavior of solutions to the generalized Korteweg–de Vries–Burgers equation of the formut+uxxx+(g(u))x+γu−νuxx=f(t),where ν⩾0,γ>0. We prove the existence of pullback attractors Aν in H1(R) for this equation in both cases ν=0 and ν>0. Moreover, the upper semicontinuity of the pullback attractors at ν=0 is also investigated.

Examples of the absence of a traveling wave for the generalized Korteweg-de Vries-Burgers equation

An example of convex function f(u) for which the generalized Korteweg-de Vries-Burgers equation ut + (f(u))x + auxxx − buxx = 0 has no solutions in the form of a traveling wave with specified limits at infinity is constructed. This example demonstrates the difficulties in analyzing asymptotic behavior of the Cauchy problem for the Korteweg-de Vries-Burgers equation that is not inherent in the type of equation for the conservation law, the Burgers-type equation, and its finite difference analog.

Nonlinear boundary control of the unforced generalized Korteweg–de Vries–Burgers equation

In this paper, we consider the boundary con- trol problem of the unforced generalized Korteweg- de Vries-Burgers (GKdVB) equation when the spatial domain is (0, 1). Three control laws are derived for this equation and the L 2 -global exponential stability of the solution is proved analytically. Numerical results us- ing the finite element method (FEM) are presented to illustrate the developed control schemes.

Asymptotic behavior of the generalized Korteweg–de Vries–Burgers equation

Cauchy’s problem for a generalization of the KdV–Burgers equation is considered in Sobolev spaces \({H^1(\mathbb{R})}\) and \({H^2(\mathbb{R})}\). We study its local and global solvability and the asymptotic behavior of solutions (in terms of the global attractors). The parabolic regularization technique is used in this paper which allows us to extend the strong regularity properties and estimates of solutions of the fourth order parabolic approximations onto their third order limit—the generalized Korteweg–de Vries–Burgers (KdVB) equation. For initial data in \({H^2(\mathbb{R})}\) we study the notion of viscosity solutions to KdVB, while for the larger \({H^1(\mathbb{R})}\) phase space we introduce weak solutions to that problem. Finally, thanks to our general assumptions on the nonlinear term f guaranteeing that the global attractor is usually nontrivial (i.e., not reduced to a single stationary solution), we study an upper semicontinuity property of the family of global attractors corresponding to parabolic regularizations when the regularization parameter \({\epsilon}\) tends to 0+ (which corresponds the passage to the KdVB equation).

Adaptive Boundary Control of the Forced Generalized Kortewegde Vries-Burgers Equation

This paper considers the adaptive control problem of the forced generalized Korteweg-de Vries-Burgers (GKdVB) equation when the spatial domain is [0,1]. Three different adaptive control laws are designed for the forced GKdVB equation when either the kinematic viscosity ν or the dynamic viscosity μ is unknown, or when both viscosities ν and μ are unknowns. The L2 -global exponential stability of the solutions of these equations is shown for each of the proposed control laws by using the Lyapunov theory. Numerical simulations based on the Finite Element method (FEM) are presented to validate the analytical developments.