P.J. Adams and W. Hereman, Symbolic computation of conserved densities and fluxes for systems of partial differential equations with transcendental nonlinearities (2002).
An algorithm for the symbolic computation of polynomial conserved densities for systems of nonlinear evolution equations has previously been submitted by Ü. Göktas and Willy Hereman. Here, we submit alterations to the algorithm when transcendental nonlinearities (such as the sine function) are introduced. The algorithm is implemented in Mathematica under the name TransPDEDensityFlux.m. The software automatically carries out the lengthy symbolic computations for the construction of conserved densities and associated fluxes, and was tested on many transcendental systems of partial differential equations involving well-known equations from soliton theory. The existence of a sequence of conserved densities is a predictor for integrability (solvability) of the PDE system, via the Inverse Scattering Transform.
H. Eklund and W. Hereman, Symbolic computation of conserved densities and fluxes for nonlinear systems of differential-difference equations (2002).
Two algorithms are presented to find conserved densities and fluxes of nonlinear systems of differential-difference equations. Both algorithms utilize the scaling properties of lattice equations to reduce the problem to a calculus and linear algebra problem. The two algorithms are illustrated for the Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices. The first method leads to a three step algorithm which utilizes the dilation invariance of the conservation laws to construct the form of the density. For this method, the discrete Euler operator or discrete variational derivative is an advantageous tool. The algorithm is implemented in Mathematica. The package is called DDEDensityFlux.m. The key applications are to analyze the discretizations of the Korteweg-de Vries (KdV), and modified Korteweg-de Vries (mKdV) lattices. A combination of the KdV and mKdV lattices is also considered. The second method leads to a five step algorithm that is primarily useful in determining fluxes. Both of the algorithms presented could be used to investigate the integrability of semi-discrete lattices.
D. Baldwin, Ü. Göktas, W. Hereman, L. Hong, R. Martino, and J.C. Miller, A Mathematica program for the symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for systems of nonlinear partial differential equations (2002).
The Mathematica implementation of the tanh and sech-methods for computing exact travelling wave solutions of nonlinear partial differential equations (PDEs) is presented. These methods also apply to ordinary differential equations (ODEs).
New algorithms are given to compute polynomial solutions of ODEs and PDEs in terms of the Jacobi elliptic functions. An adaptation of the tanh-method to nonlinear differential-difference equations (DDEs) is also presented.
The new Mathematica packages, PDESpecialSolutions.m and DDESpecialSolutions.m, automatically compute closed-form solutions which are expressible as polynomials in the tanh, sech, sn and cn functions.
For systems of ODEs, PDEs, or DDEs with constant parameters, the software finds the conditions on the parameters so that the given differential equations admit solutions involving tanh, sech, both, sn or cn.
Unal Goktas, Willy Hereman and Grant Erdmann, Computation of conserved densities for systems of nonlinear differential-difference equations, submitted to Phys. Lett A, (1997).
A new method for the computation of conserved densities of nonlinear differential-difference equations is applied to Toda lattices and discretizations of the Korteweg-de Vries and nonlinear Schrodinger equations. The algorithm, which can be implemented in computer algebra languages such as Mathematica, can be used as an indicator of integrability.
Unal Goktas and Willy Hereman, Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations, J. Symbolic Computation, (1997).
A new algorithm for the symbolic computation of polynomial-type conserved densities for systems of nonlinear evolution equations is presented. The algorithm is implemented in Mathematica . The program condens.m automatically carries out the lengthy symbolic computations for the construction of conserved densities and associated fluxes. The code is tested on several well-known equations from soliton theory. For systems with parameters, condens.m can be used to determine the conditions on these parameters so that a sequence of conserved densities will exist. The existence of a sufficiently large number of conservation laws is a predictor for integrability of the system.
Frank Verheest and Willy Hereman, Conservation laws and solitary wave solutions for generalized Schamel equations, Physica Scripta 50 , 611-614 (1995).
The solitary wave solution is given for nonlinear equations, generalizing the standard and modified Korteweg--de Vries and Schamel equations, as recently investigated by Xiao. A search for conservation laws of a slightly more general class of nonlinear evolution equations reveals that the generalized Schamel equations can have no more than three polynomial invariants. The method is based on obtaining suitable building blocks for conserved densities under scalings which leave the evolution equations invariant.
Ralph Willox, Willy Hereman and Frank Verheest, Complete integrability of a modified vector derivative nonlinear Schrodinger equation, Physica Scripta 52 , 21-26 (1995).
Oblique propagation of magnetohydrodynamic waves in warm plasmas is described by a modified vector derivative nonlinear Schrodinger equation, if charge separation in Poisson's equation and the displacement current in Ampere's law are properly taken into account. This modified equation cannot be reduced to the standard derivative nonlinear Schrodinger equation and hence its possible integrability and related properties need to be established afresh. Indeed, the new equation is shown to be integrable by the existence of a bi-Hamiltonian structure, which yields the recursion operator needed to generate an infinite sequence of conserved densities. Some of these have been found explicitly by symbolic computations based on the symmetry properties of the new equation.
Willy Hereman, Symbolic Software for Lie Symmetry Analysis. In: CRC Handbook of Lie Group Analysis of Differential Equations. Vol. III, Chapter 13, Ed.: N.H. Ibragimov, CRC Press, Boca Raton, Florida (1995).
A survey of techniques and symbolic programs for the determination of Lie symmetry groups of systems of differential equations is presented. The purpose, methods and algorithms of symmetry analysis are outlined. An exhaustive review of the literature, including old and modern books and papers presenting key concepts is given. Special attention is paid to methods for reducing the determining equations into standard form, and their subsequent integration. Several examples illustrate the use of the Lie symmetry software. Throughout the paper, new trends in the development of symbolic packages for Lie symmetry analysis are indicated.
Willy Hereman, Review of Symbolic Software for Lie Symmetry Analysis, to appear in: Mathematical and Computer Modelling, vol. 20, Special Issue on Algorithms for Nonlinear Systems, (1995).
Small Computer algebra packages and tools that aid in the computation of Lie symmetries of differential equations are reviewed. The methods and algorithms of Lie symmetry analysis are briefly outlined. Examples illustrate the use of the symbolic software.
Willy Hereman and Wuning Zhuang, Symbolic Software for Soliton Theory, Acta Appl. Math. 39 , 361-378 (1995). Also: Proc. of KdV '95 Conf., 361-378. (M. Hazewinkel, H.W. Capel, E.M. de Jager, eds.) Dordrecht, The Netherlands: Kluwer Academic Publishers.
Four symbolic programs, in Macsyma or Mathematica language, are presented. The first program tests for the existence of solitons for nonlinear PDEs. It explicitly constructs solitons using Hirota's bilinear method. In the second program, the Painleve integrability test for ODEs and PDEs is implemented. The third program provides an algorithm to compute conserved densities for nonlinear evolution equations. The fourth software package aids in the computation of Lie symmetries of systems of differential and difference-differential equations. Several examples illustrate the capabilities of the software.
Willy Hereman and Wuning Zhuang, Symbolic computation of exact solutions of nonlinear evolution and wave equations with Macsyma, Journal of Symbolic Computation (1995).
Hirota's bilinear method for finding exact soliton solutions of nonlinear evolution and wave equations is discussed and illustrated. The explicit code of the MACSYMA program HIROTA_SINGLE.MAX is included. This program automatically carries out the lengthy algebraic computations for the symbolic calculation of one-, two- and three-soliton solutions of bilinear equations of KdV type. The program also allows to test if three and four-soliton solutions exist for such equations. The MACSYMA program is tested by constructing exact solutions of various nonlinear partial differential equations from soliton theory, such as the Korteweg-de Vries equation and its higher order generalizations; the Sawada-Kotera, the Kadomtsev-Petviashvili, the Boussinesq and shallow water wave equations.
W. Hereman, Symbolic software for the study of nonlinear partial differential equations, Computer Methods for Partial Differential Equations, Proceedings of the IMACS PDE7 International Conference, New Brunswick, New Jersey, June 22-24, 1992. Ed.: G. Richter (1992).
Three entirely symbolic MACSYMA programs are presented. The first one carries out the Painleve integrability test, the second computes solitons and the third program aids in finding the Lie symmetry group for systems of differential equations.
Willy Hereman and Wuning Zhuang, Symbolic computation of solitons with Macsyma, Computational and Applied Mathematics II: Differential Equations. Proceedings of the 13th IMACS World Congress, Dublin, July 22-26, 1991, Eds.: W.F. Ames and P.J. van der Houwen, Elsevier, Amsterdam, pp. 287-296 (1992).
Hirota's method for constructing soliton solutions of nonlinear evolution and wave equations is discussed and illustrated. The Macsyma program HIROTA_SINGLE automatically calculates N-soliton solutions for N=1, 2 or 3. The program also allows to test the necessary conditions for the existence of four-soliton solutions. Exact analytical solutions of various nonlinear PDEs from soliton theory are presented.
B. Champagne, W. Hereman and P. Winternitz, The computer calculation of Lie point symmetries of large systems of differential equations, Computer Physics Communications, vol. 66, pp. 319-340 (1991).
A MACSYMA program is presented that greatly helps in the calculation of Lie symmetry groups of large systems of differential equations. The program calculates the determining equations for systems of m differential equations of order k, with p independent and q dependent variables, where m, k, p and q are arbitrary positive integers. The program automatically produces a list of determining equations for the coefficients of the vector field. This list has been parsed so that it is free of duplicate equations and trivial differential redundancies. Numerical factors and non zero parameters occurring as factors are also removed. From the solution of these determining equations one can construct the Lie symmetry group. An example shows the use the program in batch mode. It also illustrates a feedback mechanism, that not only allows the treatment of a large number of complicated partial differential equations but also aids in solving the determining equations step by step.
Willy Hereman and Wuning Zhuang, A MACSYMA program for the Hirota method, Proceedings of the 13th IMACS World Congress on Computation and Applied Mathematics, Dublin, July 22-26, 1991, Eds: R. Vichnevetsky and J.J.H. Miller, Criterion Press, Dublin, vol. 2, pp. 842-843 (1991).
Hirota's method for finding soliton solutions of nonlinear evolution and wave equations is briefly discussed and illustrated. A MACSYMA program that automatically carries out the lengthy algebraic computations is included.
Willy Hereman and Sigurd Angenent, The Painleve test for nonlinear ordinary and partial differential equations, MACSYMA Newsletter, vol. 6, pp. 11-8 (1989).
A MACSYMA program is presented which determines whether a given single nonlinear ODE or PDE with (real) polynomial terms fulfills the necessary conditions for having the Painlev\'e property. Together with some mathematical background, we give a synopsis of the algorithm for the program, its scope and limitations. Various examples of typical output of the program are provided.
Willy Hereman and Eric Van den Bulck, MACSYMA program for the Painleve test of nonlinear ordinary and partial differential equations, Proceedings of the Workshop on Finite Dimensional Integrable Nonlinear Dynamical Systems, Eds.: P.G.L. Leach and W.H. Steeb, Johannesburg, South Africa, January 11-15, 1988, World Scientific, Singapore, pp. 117-129 (1988).
No abstract available.
Back to Personal Home Page of Willy Hereman
Updated: 2/8/2005