Symbolic computation of conservation laws of nonlinear partial differential equations in multi‐dimensions

Abstract

A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi-dimensions is presented. The method avoids advanced differential-geometric tools. Instead, it is solely based on calculus, variational calculus, and linear algebra.Densities are constructed as linear combinations of scaling homogeneous terms with undetermined coefficients. The variational derivative (Euler operator) is used to compute the undetermined coefficients. The homotopy operator is used to compute the fluxes.The method is illustrated with nonlinear PDEs describing wave phenomena in fluid dynamics, plasma physics, and quantum physics. For PDEs with parameters, the method determines the conditions on the parameters so that a sequence of conserved densities might exist. The existence of a large number of conservation laws is a predictor for complete integrability. The method is algorithmic, applicable to a variety of PDEs, and can be implemented in computer algebra systems such as Mathematica, Maple, and REDUCE.

Similarity: generalizations, applications and open problems

“…[10,42,[48][49][50][51][52][53][54][55]); -efficiently solve the (overdetermined) linear systems of determining equations for symmetries or conservation law multipliers and solve the nonlinear systems of determining equations for the nonclassical and related methods through the development of symbolic manipulation software (cf. [42,[56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71]); -develop numerical schemes that effectively use symmetries and/or conservation laws for ODE's (cf. [72]), for difference equations (cf.…”

Some Approaches to the Calculation of Conservation Laws for a Telegraph System and Their Comparisons

“…Several systematic approaches have been established for the calculation of conservation laws, such as the symmetry/adjoint symmetry pair method (SA method) [5][6][7], the direct construction method (multiplier approach, variational derivatives approach) [3][4][5][7][8][9], symmetry action on a known conservation law method [5,10], Ibragimov's conservation theorem [11] (equivalent to the SA method [12]) and Cheviakov's recursion formul [13]. Wolf, Hereman, Temuerchaolu and Cheviakov [14][15][16][17] have presented some effective computer programmes to calculate conservation laws for PDEs.…”

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

“…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.…”