Linear Korteweg-de Vries Equation Research Articles

The maximal regularity of the third‐order differential equation and its applications

In this work, we study the solvability and maximal regularity questions for the singular third‐order differential equation with unbounded coefficients and some applications. Unlike previously studied cases, the leading and intermediate coefficients of this equation can grow independently. We obtain sufficient coefficient conditions for correctness of this equation and compactness for inverse of the corresponding differential operator. We also prove the maximal regularity estimate for a generalized solution. Using these results, we obtain upper and lower estimates for the number of Kolmogorov ‐diameters of the set associated with the linear Korteweg–de Vries equation's solutions. We give an example and show that from the above inequalities follow two‐sided estimates of the Kolmogorov ‐diameters.

Some finite difference methods for solving linear fractional KdV equation

The time-fractional Korteweg de Vries equation can be viewed as a generalization of the classical KdV equation. The KdV equations can be applied in modeling tsunami propagation, coastal wave dynamics, and oceanic wave interactions. In this study, we construct two standard finite difference methods using finite difference methods with conformable and Caputo approximations to solve a time-fractional Korteweg-de Vries (KdV) equation. These two methods are named as FDMCA and FDMCO. FDMCA utilizes Caputo's derivative and a finite-forward difference approach for discretization, while FDMCO employs conformable discretization. To study the stability, we use the Von Neumann Stability Analysis for some fractional parameter values. We perform error analysis using L1 & L∞ norms and relative errors, and we present results through graphical representations and tables. Our obtained results demonstrate strong agreement between numerical and exact solutions when the fractional operator is close to 1.0 for both methods. Generally, this study enhances our comprehension of the capabilities and constraints of FDMCO and FDMCA when used to solve such types of partial differential equations laying some ground for further research.

An explicit time for the uniform null controllability of a linear Korteweg-de Vries equation

An explicit time for the uniform null controllability of a linear Korteweg-de Vries equation

Global exponential stabilization of the linearized Korteweg-de Vries equation with a state delay

Abstract In this paper, well-posedness and global boundary exponential stabilization problems are studied for the one-dimensional linearized Korteweg-de Vries equation (KdV) with state delay, which is posed in bounded interval $[0,2\pi ]$ and actuated at the left boundary by Dirichlet condition. Based on the infinite-dimensional backstepping method for the delay-free case, a linear Volterra-type integral transformation maps the system into another homogeneous target system, and an explicit feedback control law is obtained. Under this feedback, we prove the well-posedness of the considered system in an appropriate Banach space and its exponential stabilization in the topology of $L^{2}(0,2\pi )$-norm by the use of an appropriate Lyapunov–Razumikhin functional. Moreover, under the same feedback law, we get the local exponential stability for the non-linear KdV equation. A numerical example is provided to illustrate the result.

Long wavelength limit of non-isentropic Euler-Poisson system

This paper is devoted to study the long wavelength limit of Euler-Poisson system with variable temperature for ions in plasma. By the Gardner-Morikawa transformation, we formally derive the Korteweg-de Vries (KdV) equation and the linear KdV equation from the non-isentropic Euler-Poisson system. Then, we establish the rigorous justification of the KdV equation from Euler-Poisson equation in a time interval $ [0, \tau_*\varepsilon^{-\frac{3}{2}}] $ for some $ \tau_*>0 $. The long wavelength limit of Euler-Poisson equation follows from the suitable uniform bound of the remainder terms by the nonlinear energy estimates.

Perfectly Matched Layers Methods for Mixed Hyperbolic–Dispersive Equations

Absorbing boundary conditions are important when one simulates the propagation of waves on a bounded numerical domain without creating artificial reflections. In this paper, we consider various hyperbolic–dispersive equations modeling water wave propagation. A typical example is the Korteweg–de Vries equation 1 $$\begin{aligned} \displaystyle u_t+u\,u_x+\varepsilon u_{xxx}=0,\quad \forall x\in {\mathbb {R}}, \quad \forall t>0. \end{aligned}$$ In the case of linearized equations, some progress was recently done for one dimensional scalar dispersive equations using discrete transparent boundary conditions. However, a generalization of this approach to multi-dimensional setting is not obvious. In this paper, we consider the alternative perfectly matched layer (PML) approach for the linearized Korteweg–de Vries equation: 2 $$\begin{aligned} \displaystyle u_t+U\,u_x+\varepsilon u_{xxx}=0\quad \forall x\in {\mathbb {R}}, \quad \forall t>0, \end{aligned}$$ where $$U\in {\mathbb {R}}$$ denotes a reference speed. We first propose a direct perfectly matched layer approach and study the stability of the modified system. These equations are not always stable, the main obstruction being the classical condition $$v_g(k)v_\phi (k)\ge 0$$ found in the literature on PML (Bécache et al in J Comput Phys 188(2):399–433, 2003) that we recover in our analysis. Then, we introduce a hyperbolic system with a source term that is an approximation of the Korteweg–de Vries equations. In this case, the complete PML equations are not, again, completely stable. However, a version of the PML equations for this system derived without the source term is found to be stable and can absorb outgoing waves although it may create reflections as it is not perfectly matched. Finally, we consider the BBM–Boussinesq system that models bi-directional waves at the surface of an inviscid fluid layer. The dispersive properties for a subclass of physically relevant models are better suited for PML techniques since the condition $$v_g(k)v_\phi (k)\ge 0$$ is always satisfied. We show that the PML equations are always stable in this case. We illustrate numerically the absorbing and stability properties of these PML models and provide also KdV type simulation by choosing properly initial data.

Властивостi розв’язкiв лiнiйного рiвняння KdV iзφ-субгауссовими початковими умовами

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in L_2(Ω) sense) to the linear Korteweg–de Vries equation (called also the Airy equation) with random initial conditions given by φ-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over bounded domains. Also, there are presented some examples to illustrate the results of the study.

Stability analysis of a Korteweg-de Vries equation with saturated boundary feedback

The stability analysis of the linear and nonlinear Korteweg-de Vries equations in presence of a saturated feedback actuator is studied. The well-posedness is derived by using nonlinear semigroup results, Schauder's and Banach fixed point theorems. The exponential stability is shown thanks to an observability inequality, which is obtained via contradiction and compactness arguments. Finally, an alternative proof of the asymptotic stability of the linear Korteweg-de Vries equation is presented.

Exponential stabilization of a linear Korteweg-de Vries equation with input saturation

<p style='text-indent:20px;'>This article deals with the issue of the exponential stability of a linear Korteweg-de Vries equation with input saturation. It is proved that the system is well-posed and the origin is exponentially stable for the closed loop system, by using the classical argument used in this kind of problems.</p>

Classical and Multisymplectic Schemes for Linearized KdV Equation: Numerical Results and Dispersion Analysis

We construct three finite difference methods to solve a linearized Korteweg–de-Vries (KdV) equation with advective and dispersive terms and specified initial and boundary conditions. Two numerical experiments are considered; case 1 is when the coefficient of advection is greater than the coefficient of dispersion, while case 2 is when the coefficient of dispersion is greater than the coefficient of advection. The three finite difference methods constructed include classical, multisymplectic and a modified explicit scheme. We obtain the stability region and study the consistency and dispersion properties of the various finite difference methods for the two cases. This is one of the rare papers that analyse dispersive properties of methods for dispersive partial differential equations. The performance of the schemes are gauged over short and long propagation times. Absolute and relative errors are computed at a given time at the spatial nodes used.

Stability and interaction of compactons in the sublinear KdV equation

Compactons are studied in the framework of the Korteweg–de Vries (KdV) equation with the sublinear nonlinearity. Compactons represent localized bell-shaped waves of either polarity which propagate to the same direction as waves of the linear KdV equation. Their amplitude and width are inverse proportional to their speed. The energetic stability of compactons with respect to symmetric compact perturbations with the same support is proven analytically. Dynamics of compactons is studied numerically, including evolution of pulse-like disturbances and interactions of compactons of the same or opposite polarities. Compactons interact inelastically, though almost restore their shapes after collisions. Compactons play a two-fold role of the long-living soliton-like structures and of the small-scale waves which spread the wave energy.

Non-standard boundary conditions for the linearized Korteweg-de Vries equation

<p style='text-indent:20px;'>This paper aims to solve numerically the linearized Korteweg-de Vries equation. We begin by deriving suitable boundary conditions then approximate them using finite difference method. The methodology of derivation, used in this paper, yields to Non-Standard Boundary Conditions (NSBC) that perfectly absorb wave reflections at the boundary. In addition, these NSBC are exact and local in time and space for non necessarily supported initial data and source terms. We finish with numerical examples that show the absorbing quality of these boundary conditions. Further comparisons are made using standard boundary conditions like, Dirichlet, Neumann and a variant of absorbing boundary conditions called discrete artificial ones.

Singularity for solutions of linearized KdV equations

We investigate the time propagation of singularity of a solution to the linearized Korteweg-deVries equation by using the characterization of wave front sets via the wave packet transform (short time Fourier transform).

Ion-acoustic waves at the night side of Titan’s ionosphere: higher-order approximation

Nonlinear solitary waves are investigated for a plasma system at the night side of Titan’s ionosphere. The plasma model consists of three positive ions, namely C2H, HCNH+, and C3H, as well as Maxwellian electrons. The basic set of fluid equations is reduced to a Korteweg de-Vries (KdV) equation and linear inhomogeneous higher order KdV (LIHO-KdV) equation. The solitary wave solutions of both equations are obtained using a renormalization method. The solitary waves’ existence region and the wave profile are investigated, and their dependences on the plasma parameters at the night side of Titan’s ionosphere are examined. The solitary waves’ phase velocities are subsonic or supersonic, and the propagating pulses are usually positive. The effect of higher-order corrections on the perturbation theory is investigated. It is found that the higher-order contribution makes the amplitude slightly taller, which is suitable for describing the solitary waves when the amplitude augments.

Output feedback stabilization of the linearized Korteweg–de Vries equation with right endpoint controllers

In this paper, we prove the output feedback stabilization for the linearized Korteweg–de Vries (KdV) equation posed on a finite domain in the case the full state of the system cannot be measured. We assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution. This allows us to design a suitable observer system whose states can be used for constructing boundary feedbacks acting at the right endpoint so that both the observer and the original plant become exponentially stable. Stabilization of the original system is proved in the L2-sense, while the convergence of the observer system to the original plant is also proved in higher order Sobolev norms. The standard backstepping approach used to construct a left endpoint controller fails and presents mathematical challenges when building right endpoint controllers due to the overdetermined nature of the related kernel models. In order to deal with this difficulty we use the method of Özsarı and Batal, (2019) which is based on using modified target systems involving extra trace terms. In addition, we show that the number of controllers and boundary measurements can be reduced to one, with the cost of a slightly lower exponential rate of decay. We provide numerical simulations illustrating the efficacy of our controllers.

Exact Controllability of a Linear Korteweg--de Vries Equation by the Flatness Approach

We consider a linear Korteweg-de Vries equation on a bounded domain with a left Dirichlet boundary control. The controllability to the trajectories of such a system was proved in the last decade by using Carleman estimates. Here, we go a step further by establishing the exact controllability in a space of analytic functions with the aid of the flatness approach.

Parameter estimation and output feedback stabilization for the linear Korteweg-de Vries equation with disturbed boundary measurement

This paper is concerned with the parameter estimation and boundary feedback stabilization for the linear Korteweg-de Vries equation posed on a finite interval with the boundary observation at the right end and the non-collocated control at the left end. The boundary observation suffers from some unknown disturbance. An adaptive observer is designed and the adaptive laws of the parameters are obtained by the Lyapunov method. The resulted closed-loop system is proved to be well-posed and asymptotically stable in case that the length of the interval is not critical. Moreover, it is shown that the estimated parameter converges to the unknown parameter. As a by-product, a hidden regularity result is proved.

Null Controllability of a Linearized Korteweg--de Vries Equation by Backstepping Approach

This paper deals with the controllability problem of a linearized Korteweg-de Vries equation on bounded interval. The system has a homogeneous Dirichlet boundary condition and a homogeneous Neumann boundary condition at the right end-points of the interval, a non homogeneous Dirichlet boundary condition at the left end-point which is the control. We prove the null controllability by using a backstepping approach, a method usually used to handle stabilization problems.

Points of constancy of the periodic linearized Korteweg–deVries equation

We investigate the points of constancy in the piecewise constant solution profiles of the periodic linearized Korteweg–deVries equation with step function initial data at rational times. The solution formulae are given by certain Weyl sums, and we employ number theoretic techniques, including Kummer sums, in our analysis. These results constitute an initial attempt to understand the complementary phenomenon of ‘fractalization’ at irrational times.

Stability Analysis of Dissipative Systems Subject to Nonlinear Damping via Lyapunov Techniques

In this paper, we provide a general strategy based on Lyapunov functionals to analyze global asymptotic stability of linear infinite-dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymptotically stable with a linear damping . To do so, we use the fact that for any linear infinite-dimensional system that is globally exponentially stable, there exists a Lyapunov functional. Then, we derive a Lyapunov functional for the nonlinear system, which is the sum of a Lyapunov functional coming from the linear system and another term that compensates the nonlinearity. Our results are then applied to the linearized Korteweg–de Vries equation and some wave equations.