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Willy Hereman

Publications and source records attributed to Willy Hereman.

28 records · Page 2Linked to original sources
arXiv2003

Symbolic Computation of Conserved Densities, Generalized Symmetries, and Recursion Operators for Nonlinear Differential-Difference Equations

Algorithms for the symbolic computation of conserved densities, fluxes, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations are presented. In the algorithms we use discrete versions of the Frechet and variational derivatives, as well as discrete Euler and homotopy operators. The algorithms are illustrated for prototypical nonlinear lattices, including the Kac-van Moerbeke (Volterra) and Toda lattices. Results are shown for the modified Volterra and Ablowitz-Ladik lattices.

nlin.SI↗
arXiv1998

The Painlevé Integrability Test

The Painlevé test is a widely applied and quite successful technique to investigate the integrability of nonlinear ODEs and PDEs by analyzing the singularity structure of the solutions. The test is named after the French mathematician Paul Painlevé ....

solv-int↗
arXiv1998

Algorithmic Integrability Tests for Nonlinear Differential and Lattice Equations

Three symbolic algorithms for testing the integrability of polynomial systems of partial differential and differential-difference equations are presented. The first algorithm is the well-known Painlevé test, which is applicable to polynomial systems of ordinary and partial differential equations. The second and third algorithms allow one to explicitly compute polynomial conserved densities and higher-order symmetries of nonlinear evolution and lattice equations. The first algorithm is implemented in the symbolic syntax of both Macsyma and Mathematica. The second and third algorithms are available in Mathematica. The codes can be used for computer-aided integrability testing of nonlinear differential and lattice equations as they occur in various branches of the sciences and engineering. Applied to systems with parameters, the codes can determine the conditions on the parameters so that the systems pass the Painlevé test, or admit a sequence of conserved densities or higher-order symmetries.

solv-int↗
arXiv1998

Computation of Higher-order Symmetries for Nonlinear Evolution and Lattice Equations

A straightforward algorithm for the symbolic computation of higher-order symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the higher-order symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first-order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi-discrete lattice equations. With our Integrability Package, higher-order symmetries are obtained for several well-known systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of higher-order symmetries exist. The existence of a sequence of such symmetries is a predictor for integrability.

solv-int↗
arXiv1998

Invariants and Symmetries for Partial Differential Equations and Lattices

Methods for the computation of invariants and symmetries of nonlinear evolution, wave, and lattice equations are presented. The algorithms are based on dimensional analysis, and can be implemented in any symbolic language, such as Mathematica. Invariants and symmetries are shown for several well-known equations. Our Mathematica package allows one to automatically compute invariants and symmetries. Applied to systems with parameters, the package determines the conditions on these parameters so that a sequence of invariants or symmetries exists. The software can thus be used to test the integrability of model equations for wave phenomena.

solv-int↗
arXiv1998

Computation of conservation laws for nonlinear lattices

An algorithm to compute polynomial conserved densities of polynomial nonlinear lattices is presented. The algorithm is implemented in Mathematica and can be used as an automated integrability test. With the code diffdens.m, conserved densities are obtained for several well-known lattice equations. For systems with parameters, the code allows one to determine the conditions on these parameters so that a sequence of conservation laws exist.

solv-int↗
arXiv1997

Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations

A new algorithm for the symbolic computation of polynomial 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. The code is tested on several well-known partial differential 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 might exist. The existence of a large number of conservation laws is a predictor for integrability of the system.

solv-int↗
arXiv1994

Complete integrability of a modified vector derivative nonlinear Schroedinger equation

Oblique propagation of magnetohydrodynamic waves in warm plasmas is described by a modified vector derivative nonlinear Schroedinger 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 Schroedinger 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. Since the new equation includes as a special case the derivative nonlinear Schroedinger equation, the recursion operator for the latter one is now readily available.

solv-int↗