Linear Korteweg-de Vries Research Articles

On the solution of a korteweg-de vries like equation by the decomposition method

A new approach to a linear Korteweg-de Vries like equation is implemented by the Adomian decomposition method. The approach is based on the choice of a suitable differential operator which may be ordinary or partial, linear or nonlinear, deterministic or stochastic [1–4]. It allows to obtain a decomposition series analytic solution of the equation which is calculated in the form of a convergent power series with easily computable components. The inhomogeneous problem is quickly solved by observing the self-canceling “noise” terms where the sum of components vanishes in the limit. Many test modeling problems from mathematical physics, linear and nonlinear, are discussed to illustrate the effectiveness and the performance of the decomposition method. This paper is particularly concerned with the accuracy for the modeling of various linear Korteweg-de Vries like equations by the Adomian decomposition method. Its remarkable accuracy is finally demonstrated in the study of several test problems.

The secular solutions of the linearized Korteweg–de Vries equation

We study the inhomogeneous linearized Korteweg–de Vries (KdV) equation. It is solved by the inverse scattering transform method. The secular-producing terms on the right-hand side (rhs) are characterized in several ways: first we give a mathematical characterization as resonant terms. Second, the secular-producing terms are interpreted as conserved densities of the KdV equation. Third, it is checked that the removal of all linear terms from the rhs, polynomial in the solution of KdV, ensures the boundness of the solution of the linearized equation. Fourth, considering this solution itself as the rhs, we determine which part of it is secular producing, and which part is not.

The solution of certain initial boundary-value problems for the linearized Korteweg—deVries equation

Several initial boundary value problems for the linearized KdV equation are solved. These problems are formulated in the quarter plane or in a wedge shaped domain and they involve Dirichlet, Neumann as well as changing type boundary conditions. The solution, given in terms of an explicit integral representation, is obtained by using the new spectral method introduced in Fokas (1997). The extension of these results to the KdV equation is also briefly discussed.

Inversion of the linearized Korteweg-de Vries equation at the multisoliton solutions

Uniform estimates for the decay structure of the $n$-soliton solution of the Korteweg-deVries equation are obtained. The KdV equation, linearized at the $n$-soliton solution is investigated in a class $\WW$ consisting of sums of travelling waves plus an exponentially decaying residual term. An analog of the kernel of the time-independent equation is proposed, leading to solvability conditions on the inhomogeneous term. Estimates on the inversion of the linearized KdV equation at the $n$-soliton are obtained.

Analyticity of Solutions of the Generalized Korteweg–de Vries Equation with Respect to Their Initial Values

The initial value problem (IVP) of the generalized Korteweg–de Vries (KdV) equation \[ \partial _t u + \partial _x (a(u)) + \partial _x^3 u = 0,\quad u(x,0) = \phi (x)\] is well posed in the classical Sobolev space $H^s (R)$ with $s > {3 / 4}$, which establishes a nonlinear map K from $H^s (R)$ to $C ( [ - T,T];H^s (R)$. It is shown that (i) if $a = a(x)$ is a $C^\infty $ function on R to R, then K is infinitely many times Frechet differentiable; (ii) if $a = a(x)$ is a polynomial, then K is analytic, i.e., for any $\phi \in H^s (R)$, K has a Taylor series expansion \[ K(\phi + h) = \sum_{n = 0}^\infty {\frac{1}{{n!}}} K^{(n)} (\phi )[h^n ].\] Each term $y_n = K^{(n)} (\phi )[h^n ]$ in the series solves a linearized KdV equation. Consequently, any “small” perturbation $K(\phi + h)$ of $K(\phi )$ can be obtained by solving a series of linear problems. The proof of these results relies on various smoothing properties of the associated linear KdV equation.

An analysis of numerical dispersion in approximations of the linear korteweg-de vries equation

We consider a class of purely dispersive difference schemes for the linear Korteweg-deVries equation and analyze the pointwise behavior of the approximations when the initial data is a step function. Using a normal form analysis, we give a complete description of the approximation in a large region surrounding the leading front of the solution. We show in particular that in a fixed region around the front, numerical dispersion does not have a lasting effect on the approximation as the mesh size tends to zero. This contrasts sharply with the behavior of dispersive approximations of hyperbolic problems.

Unsteady flow in a nonuniform channel: A model for wave generation

It is known that a propagating wave train can be generated in unsteady flow in a nonuniform channel. A small amplitude inviscid theory is derived in which the forced displacement of the streamlines is governed by a linearized Korteweg–de Vries equation with time-dependent coefficients. Solutions of this equation for a channel with an asymmetric expansion show a standing wave with a propagating wave front, the amplitude of which is modified by a forced oscillation and by wave packets which travel downstream at a velocity greater than the group velocity for the main wave. An analytical model, based on the assumption of a standing wave, successfully predicts the wavelength of the main wave, the magnitude of the forced oscillation, and the velocity of the wave packets. Predictions from the theory are also compared with numerical solutions of the full Navier–Stokes equations for flow in nonuniform channels; there is agreement on some but not all properties of the wave.

Dispersive Blowup of Solutions of Generalized Korteweg-de Vries Equations

The strong effect of dispersion on short-wavelength disturbances featured by the Korteweg-de Vries equation and some of its generalizations is exploited to provide solutions of these equations that correspond to infinitely smooth initial data, which exhibit a specified loss of spatial smoothness at particular times. The points in space-time at which smoothness is lost may even comprise an arbitrary discrete subset of the upper half-plane ) { (x, t): x ∈ R, t ≥ 0}. Our results are related to recent work on smoothing of solutions of such equations, some of which are sharpened here, and they show that in certain aspects these earlier results are not far from being optimal. The theory makes use of new results concerning well-posedness of such equations in weighted Sobolov spaces and some detailed analysis of the linear Korteweg-de Vries equation.

Causal Theories of Evolution and Wave Propagation in Mathematical Physics

There are still many phenomena, especially in continuum physics, that are described by means of parabolic partial differential equations whose solution are not compatible with the causality principle. Compatibility with this principle is required also by the theory of relativity. A general form of hyperbolic operators for the most frequently occurring linear governing equations in mathematical physics is written down. It is then easy to convert any given parabolic equation to the hyperbolic form without necessarily entering into the cause of the inadequacy of the governing equation. The method is verified on the well-known example of Timoshenko’s correction of the Bernoulli–Euler–Rayleigh beam equation for flexural motion. The “Love–Rayleigh” fourth-order differential equations for the longitudinal and torsional wave propagation in the rod is generalized with this method. The hyperbolic version (not to mention others) of the linear Korteweg–de Vries equation and of the “telegraph” equation governing electromagnetic wave propagation through relaxing material are given. Lagrangians of all the equations studied are listed. For all the reasons given we believe the hyperbolic governing equations to be physically and mathematically more realistic and adequate.

Flow along a channel with a time-dependent indentation in one wall: the generation of vorticity waves

We describe flow-visualization experiments and theory on the two-dimensional unsteady flow of an incompressible fluid in a channel with a time-dependent indentation in one wall. There is steady Poiseuille flow far upstream, and the indentation moves in and out sinusoidally, its retracted position being flush with the wall. The governing parameters are Reynolds number Re, Strouhal number (frequency parameter) St and amplitude parameter ε (the maximum fraction of the channel width occupied by the indentation); most of the experiments were performed with ε ≈ 0.4. For St ≤ 0.005 the flow is quasi-steady throughout the observed range of Re (360 < Re < 1260). For St > 0.005 a propagating train of waves appears, during every cycle, in the core flow downstream of the indentation, and closed eddies form in the separated flow regions on the walls beneath their crests and above their troughs. Later in the cycle, a second, corotating eddy develops upstream of the first in the same separated-flow region (‘eddy doubling’), and, later still, three-dimensional disturbances appear, before being swept away downstream to leave undisturbed parallel flow at the end of the cycle. The longitudinal positions of the wave crests and troughs and of the vortex cores are measured as functions of time for many values of the parameters; they vary with St but not with Re. Our inviscid, long-wavelength, small-amplitude theory predicts the formation of a wavetrain during each cycle, in which the displacement of a core-flow streamline satisfies the linearized Kortewegde Vries equation downstream of the indentation. The waves owe their existence to the non-zero vorticity gradient in the oncoming flow. Eddy formation and doubling are not described by the theory. The predicted positions of the wave crests and troughs agree well with experiment for the larger values of St used (up to 0.077), but less well for small values. Analysis of the viscous boundary layers indicates that the inviscid theory is self-consistent for sufficiently small time, the time of validity increasing as St increases (for fixed ε).

Conservation laws for second-order invariant variational problems

In the literature, a symmetry requirement in the mixed, second partials appearing in the Lagrangian is needed before conservation laws for second-order variational problems can be obtained. In this paper this assumption is removed and an example of a modified, linear Korteweg-de Vries equation is analysed.

On a simple wave approximation of a set of linear dispersive wave equations

The validity of an approximation α0 of one of the solutions α of a set of two linear coupled dispersive wave equations has been discussed. α0 is the solution of a linear Korteweg-de Vries equation and satisfies the same initial condition as α. It is shown that for square integrable solutions having a spectral range not exceeding [−Δ, Δ] the approximation is useful if Δ5μ2t«1 in the sense that ‖α−α0‖(t)« ‖α‖(t)(L 2 -norm).μ is a measure for the dispersion. The approximation fails in that sense ast→ ∞. Some remarks to a similar nonlinear problem are made.