Linearized KdV Equation Research Articles

Stability of KdV equation on a network with bounded and unbounded lengths

In this work, we studied the exponential stability of the nonlinear KdV equation posed on a finite star shaped network with finite number of branches. On each branch of the network we define a KdV equation posed on a finite domain [[EQUATION]] or the half-line [[EQUATION]]. We start by proving well-posedness and some regularity results. Then, we state the exponential stability of the linear KdV equation by acting with a damping term on some branches. The main idea is to prove a suitable observability inequality. In the nonlinear case, we obtain two kinds of results: The first result holds for small amplitude solutions, and is proved using a perturbation argument from the linear case but without acting on all edges. The second result is a semiglobal stability result, and it is obtained by proving an observability inequality directly for the nonlinear system, but we need to act with damping terms on all the branches. In this case, we are able to prove the stabilization in weighted spaces.

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.

Fourier analysis of the local discontinuous Galerkin method for the linearized KdV equation

Fourier analysis of the local discontinuous Galerkin method for the linearized KdV equation

Non-homogeneous boundary value problems of the Kawahara equation posed on a finite interval

Considered in this paper is the initial boundary value problem (IBVP) of the Kawahara equation, a class of the fifth order KdV equation, posed on a finite interval, (0.1)ut+ux+βuxxx+uxxxxx+uux=0,0<x<L,t>0,u(x,0)=ϕ(x), subject to the non-homogeneous boundary conditions (0.2)Bju=hj(t),j=1,2,3,4,5t>0where Bju=∑k=04(ajk∂xku(0,t)+bjk∂xku(L,t)),j=1,2,3,4,5and ajk,bjk(k=0,1,2,3,4andj=1,2,3,4,5) are real constants. Under some general assumptions imposed on the coefficients ajk,bjk, the IBVP (0.1)–(0.2) is shown to be locally well posed in the space Hs(0,L) for any s≥0 with naturally compatible ϕ∈Hs(0,L) and boundary values hj,j=1,2,3,4,5 belonging to some appropriate spaces with optimal regularity. The sharp Kato smoothing properties (due to Kenig, Ponce and Vega (Kenig et al., 1991; Kenig et al., 1993)) of the pure initial value problem (IVP) of the linear inhomogeneous fifth order KdV equation posed on the whole line R, vt+βvxxx+vxxxxx=g(x,t),v(x,0)=ψ(x),x,t∈R,have played an important role in establishing the well-posedness of the IBVP (0.1)–(0.2).

Exponential decay for the KdV equation on ℝ with new localized dampings

In this paper, we prove several results on the exponential decay in $L^{2}$ norm of the KdV equation on the real line with localized dampings. First, for the linear KdV equation, the exponential decay holds if and only if the averages of the damping coefficient on all intervals of a fixed length have a positive lower bound. Moreover, under the same damping condition, the exponential decay holds for the (nonlinear) KdV equation with small initial data. Finally, with the aid of certain properties of propagation of regularity in Bourgain spaces for solutions of the associated linear system and the unique continuation property, the exponential decay for the KdV equation with large data holds if the damping coefficient has a positive lower bound on $E$, where $E$ is equidistributed over the real line and the complement $E^{c}$ has a finite Lebesgue measure.

The Solutions of Initial (-Boundary) Value Problems for Sharma-Tasso-Olver Equation

A nonlinear transformation from the solution of linear KdV equation to the solution of Sharma-Tasso-Olver (STO) equation is derived out by using simplified homogeneous balance (SHB) method. According to the nonlinear transformation derived here, the exact explicit solution of initial (-boundary) value problem for STO equation can be constructed in terms of the solution of initial (-boundary) value problem for the linear KdV equation. The exact solution of the latter problem is obtained by using Fourier transformation.

New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method

Symmetry performs an essential function in finding the correct techniques for solutions to time space fractional differential equations (TSFDEs). In this article, we present the Novel Analytic Method (NAM) for approximate solutions of the linear and non-linear KdV equation for TSFDs. To enunciate the non-integer derivative for the aforementioned equation, the Caputo operator is manipulated. Furthermore, the formula implemented is a numerical way that is postulated from Taylor’s series, which confirms an analytical answer in the form of a convergent series. For delineation of the efficiency and functionality of the method in question, four applications are exemplified along with graphical interpretation and numerical solutions to finitely illustrate the behavior of the solution to this equation. Moreover, the 3D graphs of some of these numerical examples are plotted with specific values. Observing the effectiveness of this process, we can easily decide that this process can be implemented to other TSFDEs applied in the mathematical modeling of a real-world aspect.

Observability inequality at two time points for the KdV equation from measurable sets

We prove two observability inequalities at two time points for the linear KdV equation on the real line. In the two inequalities, the observable region in space is the complement of (i) a measurable set with finite Lebesgue measure, and (ii) of a measurable set contained in a half line with some density conditions, whose Lebesgue measure may be infinite.

Conservative Galerkin methods for dispersive Hamiltonian problems

An energy conservative discontinuous Galerkin scheme for a generalised third order KdV type equation is designed. Based on the conservation principle, we propose techniques that allow for the derivation of optimal a priori bounds for the linear KdV equation and a posteriori bounds for the linear and modified KdV equation. Extensive numerical experiments showcasing the good long time behaviour of the scheme are summarised which are in agreement with the analysis proposed.

Exponential decay for the linear KdV with a rough localized damping

We show the exponential decay of the linear damped KdV equation on the real line, whenever the damping coefficient satisfies some locally integrable assumptions and has a positive lower bound on translates of a set with positive measure.

Cost for a controlled linear KdV equation

The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [L. Rosier, ESAIM: COCV 2 (1997) 33-55.]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathematical theories are built on this totally unknown constant. We introduce a constructive method that gives the quantitative value of this constant.

Dispersive Quantization of 2D Linear Dispersive Equations

The dispersive quantization of the 2D linear KdV equation and the 2D linear Schrödinger equation were studied over a bounded rectangle domain in the plane. The research shows that, for the KdV equation, if the period ratio is a rational number, at the rational moments, the solution to the periodic initial boundary value problem will be the linear combination of the initial value conditions; whereas, at the irrational moments, the solution will be continuous and nondifferentiable, and exhibit a fractallike profile. The same is true for the 2D linear Schrödinger equation.

Observability Inequality at Two Time Points for KdV Equations

We prove an observability inequality from two time points for the linear KdV equation with an explicit constant. The proof relies on a quantitative analytic estimate for the linear KdV with compact...

Non-homogeneous boundary value problems for some KdV-type equations on a finite interval: A numerical approach

This paper addresses the approximation of solutions to some non-homogeneous boundary value problems associated with the nonlinear Korteweg-de Vries equation (KdV) and a system of two coupled KdV-type equations derived by Gear and Grimshaw posed on a bounded interval. An efficient Galerkin scheme that combines a finite element strategy for space discretization with a second-order implicit scheme for time-stepping is employed to approximate time dynamics of model equations studied. Several numerical experiments, including boundary controllability problems for nonlinear KdV and GG equations, are presented for different final states to show the performance of the numerical strategies proposed. The numerical results with nonlinear models agree with previous analytic theory and show the persistence of the behavior not uniform in time of the control functions computed already observed by Rosier [22] in the case of the linear KdV equation.

Non-homogeneous initial-boundary-value problem of the fifth-order Korteweg-de Vries equation with a nonlinear dispersive term

We study the initial-boundary-value problem of the fifth-order KdV equation∂tu−∂x5u=c1u∂xu+c2u2∂xu+b1∂xu∂x2u+b2u∂x3u,b2≠0 with initial data u0∈Hs(0,L). The main analysis difficulty of this model is caused by the nonlinear dispersive term u∂x3u since the Kato smoothing effect of the solution to the linear fifth-order KdV equation with free source term can only improve two orders concerning the initial data. The Cauchy problem of this model has recently been proved to be globally well-posed in H−1(R) by Bringmann-Killip-Visan (2019). Under appropriate non-homogeneous boundary data, we prove that it is locally well-posed in Hs(0,L) with s∈[0,32) and admits a unique local solution as s≥32. As far as we know, this is the latest well-posedness result of the fifth-order KdV type equation with the nonlinear dispersive term when posed in any finite domain. The main innovation in this paper is that we construct a source term space which makes that the solution to the linear fifth-order KdV equation improves three orders on the source term.

L2-properties for linearized KdV equation around small solutions

L2-properties for linearized KdV equation around small solutions

Analysis of local discontinuous Galerkin methods with generalized numerical fluxes for linearized KdV equations

Analysis of local discontinuous Galerkin methods with generalized numerical fluxes for linearized KdV equations

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).

Optimal Error Estimate of the Extended-WKB Approximation to the High Frequency Wave-Type Equation in the Semi-classical Regime

For the Cauchy problem of the high frequency wave-type equation with Wentzel–Kramers–Brillouin (WKB) type initial data, the extended Wentzel–Kramers–Brillouin (E-WKB) ansatz is an asymptotically valid solution. This ansatz is globally defined and formulated as an integral of coherent states over the displaced Lagrangian submanifold. This paper proves the optimal first order error estimate of the proposed E-WKB ansatz in $$L^2$$ norm for the wave-type equation in the semi-classical regime. The key ingredients in the proof are the moving frame technique developed in Zheng (Commun Math Sci 11:105–140, 2013) and the deep relations between the E-WKB analysis and the classical WKB analysis. Numerical results on the linear KdV equation verify the theoretical analysis.

Rapid exponential stabilisation of linear-KdV equation with long input delay on the left boundary

In this paper, we consider the one-dimensional linear-KdV (Korteweg–de Vries) equation posed in bounded interval with delay boundary control acting on the left boundary by Dirichlet condition where the delay is constant and known, but it can be of arbitrary length. We reformulate the considered problem as an undelayed coupled KdV-Transport system and we use the so-called infinite dimensional backstepping method to derive an explicit feedback control law that transforms system under consideration to a target system. For this target system, we prove its well posedness through the use of semigroup theory and its rapid exponential stability in a suitable functional space by a series of propositions based on the Young's inequality, Wirtinger's inequality and Agmon's inequality. Then, by invertibility of such design, we prove the well posedness and the rapid exponential stabilisation of the original plant.