Algorithmic integrability tests for nonlinear differential and lattice equations

Abstract

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.

Symbolic Software for the Painlevé Test of Nonlinear Ordinary and Partial Differential Equations

“…The implementations described in [34,35,37] are limited to ODEs, while the implementations discussed in [16,[45][46][47] allow the testing of PDEs directly using the WTC algorithm. The implementation for PDEs written in Mathematica by Hereman et al [16] is limited to two independent variables (x and t) and is unable to find all the dominant behaviors in systems with undetermined exponents α i (as is the case with the Hirota-Satsuma system). Our package PainleveTest.m [4] written in Mathematica syntax, allows the testing of polynomial PDEs (and ODEs) with no limitation on the number of differential equations or the number of independent variables (except where limited by memory).…”

A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

“…Algorithms for computing conserved densities and generalized symmetries are described in [26,27,28,30,33,34,35]. Our code, PDERecursionOperator.m [9], uses these algorithms to compute the densities and generalized symmetries needed to construct the non-local part of the operator.…”

“…Inspection of the ranks of generalized symmetries usually provides a hint on how to select the gap. If R is a recursion operator for (5), then the Lie derivative [35,54,59] of R is zero, which leads to the following defining equation:…”

“…The package PDERecursionOperator.m [9] is part of our symbolic software collection for the integrability testing of nonlinear PDEs, including algorithms and Mathematica codes for the Painlevé test [6,7,8] and the computation of conservation laws [4,5,15,16,29,32,33,34,36], generalized symmetries [26,30] and recursion operators [10,26]. As a matter of fact, our package PDERecursionOperator.m builds on the code InvariantsSymmetries.m [27] for the computation of conserved densities and generalized symmetries for nonlinear PDEs.…”

“…Drawing on the analogies with the PDE case, we also developed methods, algorithms, and software to compute conservation laws [31,33,36,38] and generalized symmetries [30] of nonlinear differential-difference equations (DDEs). Although the algorithm is well-established [37], a Mathematica package that automatically computes recursion operators of nonlinear DDEs is still under development.…”