Scattering for the quartic generalised Korteweg-de Vries equation

We show that the quartic generalised KdV equation

ut+uxxx+(u⁴x₎=0     

A new high-order energy-preserving scheme for the modified Korteweg-de Vries equation

In this paper, a new high-order energy-preserving scheme is proposed for the modified Korteweg-de Vries equation. The proposed scheme is constructed by using the Hamiltonian boundary value methods in time, and Fourier pseudospectral method in space. Exploiting this method, we get second-order and fourth-order energy-preserving integrators. The proposed schemes not only have high accuracy, but also exactly conserve the total mass and energy in the discrete level. Finally, single solitary wave and the interaction of two solitary waves are presented to illustrate the effectiveness of the proposed schemes.     

Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations

Numerical schemes for initial value problems of stochastic differential equations (SDEs) are considered so as to derive the order conditions of ROW-type schemes in the weak sense. Rooted tree analysis, the well-known useful technique for the counterpart of the ordinary differential equation case, is extended to be applicable to the SDE case. In our analysis, the roots are bi-colored corresponding to the ordinary and stochastic differential terms, whereas the vertices have four kinds of label corresponding to the terms derived from the ROW-schemes. The analysis brings a transparent way for the weak order conditions of the scheme. An example is given for illustration.     

A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping

An accurate and efficient numerical approach, based on a finite difference method with Crank-Nicolson time stepping, is proposed for the Landau-Lifshitz equation without damping. The phenomenological Landau-Lifshitz equation describes the dynamics of ferromagnetism. The Crank-Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau-Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau-Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.     

A new high accuracy locally one-dimensional scheme for the wave equation

In this paper, a new locally one-dimensional (LOD) scheme with error of O(Δt⁴+h⁴) for the two-dimensional wave equation is presented. The new scheme is four layer in time and three layer in space. One main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step. The stability and dispersion analysis of the new scheme are given. The computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed, which makes the scheme suitable for performing practical simulation in wave propagation modeling. Furthermore, a comparison of our new scheme and the traditional finite difference scheme is given, which shows the superiority of our new method.     

Eigenvalue problem for a coupled channel Schrödinger equation with application to the description of deformed nuclear systems

We discuss the accurate computation of the eigensolutions of systems of coupled channel Schrödinger equations as they appear in studies of real physical phenomena like fission, alpha decay and proton emission. A specific technique is used to compute the solution near the singularity in the origin, while on the rest of the interval the solution is propagated using a piecewise perturbation method. Such a piecewise perturbation method allows us to take large steps even for high energy-values. We consider systems with a deformed potential leading to an eigenvalue problem where the energies are given and the required eigenvalue is related to the adjustment of the potential, viz, the eigenvalue is the depth of the nuclear potential. A shooting technique is presented to determine this eigenvalue accurately.     

A note on error expressions for reflected and averaged implicit Runge-Kutta methods

In addition to their usefulness in the numerical solution of initial value ODE's, the implicit Runge-Kutta (IRK) methods are also important for the solution of two-point boundary value problems. Recently, several classes of modified IRK methods which improve significantly on the efficiency of the standard IRK methods in this application have been presented. One such class is the Averaged IRK methods; a member of the class is obtained by applying an averaging operation to a non-symmetric IRK method and its reflection. In this paper we investigate the forms of the error expressions for reflected and averaged IRK methods. Our first result relates the expression for the local error of the reflected method to that of the original method. The main result of this paper relates the error expression of an averaged method to that of the method upon which it is based. We apply these results to show that for each member of the class of the averaged methods, there exists an embedded lower order method which can be used for error estimation, in a formula-pair fashion.This work was supported by the Natural Science and Engineering Research Council of Canada.     

A numerical method for the Benjamin-Ono equation

This paper is concerned with the numerical solution of the Cauchy problem for the Benjamin-Ono equationu _t +uu _x −Hu _xx =0, whereH denotes the Hilbert transform. Our numerical method first approximates this Cauchy problem by an initial-value problem for a corresponding 2L-periodic problem in the spatial variable, withL large. This periodic problem is then solved using the Crank-Nicolson approximation in time and finite difference approximations in space, treating the nonlinear term in a standard conservative fashion, and the Hilbert transform by a quadrature formula which may be computed efficiently using the Fast Fourier Transform.     

Numerical analysis of a multiscale finite element scheme for the resolution of the stationary Schrödinger equation

A numerical method for the resolution of the one-dimensional Schrödinger equation with open boundary conditions was presented in N. Ben Abdallah and O. Pinaud (Multiscale simulation of transport in an open quantum system: resonances and WKB interpolation. J. Comp. Phys. 213(1), 288–310 (2006)). The main attribute of this method is a significant reduction of the computational cost for a desired accuracy. It is based particularly on the derivation of WKB basis functions, better suited for the approximation of highly oscillating wave functions than the standard polynomial interpolation functions. The present paper is concerned with the numerical analysis of this method. Consistency and stability results are presented. An error estimate in terms of the mesh size and independent on the wavelength λ is established. This property illustrates the importance of this method, as multiwavelength grids can be chosen to get accurate results, reducing by this manner the simulation time.     

A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials

In this paper, the quadratic Riccati differential equation is solved by He's variational iteration method with considering Adomian's polynomials. Comparisons were made between Adomian's decomposition method (ADM), homotopy perturbation method (HPM) and the exact solution. In this application, we do not have secular terms, and if λ$\lambda$, Lagrange multiplier, is equal -1$-1$ then the Adomian's decomposition method is obtained. The results reveal that the proposed method is very effective and simple and can be applied for other nonlinear problems.     

On the Korteweg-de Vries equation

Existence, uniqueness, and continuous dependence on the initial data are proved for the local (in time) solution of the (generalized) Korteweg-de Vries equation on the real line, with the initial function in the Sobolev space of order s>3/2 and the solution u(t) staying in the same space, s= being included For the proper KdV equation, existence of global solutions follows if s2. The proof is based on the theory of abstract quasilinear evolution equations developed elsewhere.Dedicated to Hans Lewy and Charles B. Morrey Jr.Partially supported by NSF Grant MCS76-04655.     

A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems

In this article, we develop an explicit symmetric linear phase-fitted four-step method with a free coefficient as parameter. The parameter is used for the optimization of the method in order to solve efficiently the Schrödinger equation and related oscillatory problems. We evaluate the local truncation error and the interval of periodicity as functions of the parameter. We reveal a direct relationship between the periodicity interval and the local truncation error. We also measure the efficiency of the new method for a wide range of possible values of the parameter and compare it to other well known methods from the literature. The analysis and the numerical results help us to determine the optimal values of the parameter, which render the new method highly efficient.     

Neumann problem for the Korteweg-de Vries equation

We consider Neumann initial-boundary value problem for the Korteweg-de Vries equation on a half-line

(0.1)     

A second-order difference scheme for a parameterized singular perturbation problem

In this paper we consider a singularly perturbed quasilinear boundary value problem depending on a parameter. The problem is discretized using a hybrid difference scheme on Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.     

Remarks on the Korteweg-de Vries equation

We show for the Korteweg-de Vries equation an existence uniqueness theorem in Sobolev spaces of arbitrary fractional orders≧2, provided the initial data is given in the same space.     

Stability of a family of multi-time-level difference schemes for the advection equation

Summary For the linear advection equation we consider explicit multi-time-level schemes of highest order which are one step in space direction only. If a stencil involvesk time steps we show that it is stable in theL ₂-sense for Courant numbers in the interval (0, 1/k). Since the order is 2k–1 one can use these schemes for high order discretization of the boundary conditions in hyperbolic initial value problems.Part of this work has been performed in the project Mehrschritt-Differenzenschemata of the Schwerpunktprogramm Finite Approximationen in der Strömungsmechanik which has been supported by the DFG     

On the generalized Korteweg-de Vries equation

We prove that the local L² norm of the solution of the generalized Korteweg-de Vries equation $$u_t + (F(u) + \sum\limits_{s = 0}^m {( - 1)^s D_x^{2s} u)_x = 0,m \geqslant 2,} $$ with nice initial datum, where F satisfies certain general conditions, for example, P(u) = u^p, where p is an odd integer ≧3, decays t o zero as time goes to infinity.     

Asymptotic behavior of the Korteweg-de Vries equation posed in a quarter plane

The purpose of this work is to study the exponential stabilization of the Korteweg-de Vries equation in the right half-line under the effect of a localized damping term. We follow the methods in [G.P. Menzala, C.F. Vasconcellos, E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math. LX (1) (2002) 111-129] which combine multiplier techniques and compactness arguments and reduce the problem to prove the unique continuation property of weak solutions. Here, the unique continuation is obtained in two steps: we first prove that solutions vanishing on the support of the damping function are necessarily smooth and then we apply the unique continuation results proved in [J.C. Saut, B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations 66 (1987) 118-139]. In particular, we show that the exponential rate of decay is uniform in bounded sets of initial data.     

The discrete Korteweg-de Vries equation

We review the different aspects of integrable discretizations in space and time of the Korteweg-de Vries equation, including Miura transformations to related integrable difference equations, connections to integrable mappings, similarity reductions and discrete versions of Painlevé equations as well as connections to Volterra systems.