Linear Korteweg-de Vries Research Articles

Stabilization of linear KdV equation with boundary time-delay feedback and internal saturation

This paper investigates the question of stabilization of linear Korteweg-de Vries equation with time-delay on the boundary feedback in presence of saturated source term. Under some conditions the well-posedness is proved. The exponential stability result is demonstrated, using an appropriate Lyapunov functional.

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.

Stabilization of the coupled ODE-linearized KdV system

This paper considers the exponential stabilization of coupled ordinary differential equation (ODE)-linearized Korteweg-de Vries (KdV) equation system coupled at right boundary point with left boundary control. Firstly, we transfer the original system into an exponentially stable target system by backstepping transformation. Secondly, we show the existence of the kernels in forward and backward transformation. Finally, we prove the exponential stability of the closed-loop system.

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.

Evolution of the Atmospheric Pressure Signal from Tonga Volcano when Moving Away from It

A model of atmospheric pressure signal propagation from the eruption of the Hunga–Tonga—Hunga–Haʻapai volcano (hereafter abbreviated as Tonga) is proposed. The model is used to explain some peculiarities in the changes in the wave form of the observed signal with increasing distance from the volcano. The model is based on the solution of the linearized Korteweg de Vries (KDV) equation, which describes the change in the wave form of the Lamb wave as a function of distance from the source. We compare the observed and model signals obtained as a superposition of the Lamb wave and the acoustic modes calculated for three infrasound stations (IS22, IS24, and IS30). The energy of the volcanic eruption is estimated from the pressure amplitude and characteristic duration of the signal recorded at one of the infrasound stations closest to the volcano (IS24).

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.

Output Feedback Control of a Cascade System of Linear Korteweg--de Vries Equations

This paper is about the stabilization of a cascade system of n linear Korteweg-de Vries equations in a bounded interval. It considers an output feedback control placed at the left endpoint of the last equation, while the output involves only the solution to the first equation. The boundary control problems investigated include two cases: a classical control on the Dirichlet boundary condition and a less standard one on its second-order derivative. The feedback control law utilizes the estimated solutions of a high-gain observer system and the output feedback control leads to stabilization for any n for the first boundary conditions case and for n = 2 for the second one.

Exponential stabilization of an ODE–linear KdV cascaded system with boundary input delay

Abstract This paper considers the well posedness and the exponential stabilization problems of a cascaded ordinary differential equation (ODE)–partial differential equation (PDE) system. The considered system is governed by a linear ODE and the one-dimensional linear Korteweg–de Vries (KdV) equation posed on a bounded interval. For the whole system, a control input delay acts on the left boundary of the KdV domain by Dirichlet condition. Whereas, the KdV acts back on the ODE by Dirichlet interconnection on the right boundary. Firstly, we reformulate the system in question as an undelayed ODE-coupled KdV-transport system. Secondly, we use the so-called infinite dimensional backstepping method to derive an explicit feedback control law that transforms system under consideration to a well-posed and exponentially stable target system. Finally, by invertibility of such design, we use semigroup theory and Lyapunov analysis to prove the well posedness and the exponential stabilization in a suitable functional space of the original plant, respectively.

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.

The Korteweg-de Vries equation on an interval

The initial-boundary value problem (IBVP) for the Korteweg-de Vries (KdV) equation on an interval is studied by extending a novel approach recently developed for the well-posedness of the KdV on the half-line, which is based on the solution formula produced via Fokas’ unified transform method for the associated forced linear IBVP. Replacing in this formula the forcing by the nonlinearity and using data in Sobolev spaces suggested by the space-time regularity of the Cauchy problem of the linear KdV gives an iteration map for the IBVP which is shown to be a contraction in an appropriately chosen solution space. The proof relies on key linear estimates and a bilinear estimate similar to the one used for the KdV Cauchy problem by Kenig, Ponce, and Vega.

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.