Computer generation of necessary integrability conditions for polynomial-nonlinear evolution systems

  • Authors:

  • V. P. Gerdt;A. Y. Zharkov

  • Affiliations:

  • Laboratory of Computing Techniques and Automation, Joint Institute for Nuclear Research, Head Post Office, P.O. Box 79, Moscow, USSR;Karl-Marx- Universitat, Sektion Informatik, Karl-Marx-Platz, Leipzig, DDR-7010

  • Venue:
  • ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
  • Year:
  • 1990

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Abstract

We use the symmetry approach to establish an efficient program in REDUCE for verifying necessary integrability conditions for polynomial-nonlinear evolution equations and systems in one-spatial and one-temporal dimensions. These conditions follow from the existence of higher infinitesimal symmetries and conservation law densities. We briefly consider the mathematical background of the symmetry approach to the problem of integrability. In the description of our algorithms and their implementation in REDUCE we present in particular the basic algorithm for reversing the operator of the total derivative with respect to the spatial variable. One of the most interesting application of the present program is the problem of classification when the complete list of integrable equations from a given multiparametric family is needed. In this case the program generates necessary integrability conditions in form of a system of nonlinear algebraic equations in the parameters present in the initial equations. In spite of their often complicated structure, there are systems for which the solution can be found in exact form by applying the technique of Groebner basis. We present three examples of evolution equations for which this system can in fact be solved.