Solving algebraic systems which arise as necessary integrability conditions for polynomial-nonlinear evolution equations

摘要

The investigation of the problem of integrability of polynomial-nonlinear evolution equations, in particular, verifying the existence of the higher symmetries and conservation laws can often be reduced to the problem of finding the exact solution of a complicated system of nonlinear algebraic equations. It is remarkable that these algebraic equations can be not only obtained completely automatically by computer [1] but also often not only completely solved by computer, in spite of their complicated structure and often infinitely many solutions.

We demonstrate this fact using the Groebner basis method [2] and obtain all (infinitely many) solutions of the systems of algebraic equations which are equivalent to integrability of three different multiparametric families of NLEEs [1]: the seventh order scalar KdV-like equations, the seventh order MKdV-like equations, and the third order coupled KdV-like systems.

All our computations have been carried out by using the computer algebra system REDUCE (version 3.2) on an IBM PC AT-like computer. Because of the fact that the computer algebra system REDUCE 3.2 (in particular on IBM PC, and unlike REDUCE 3.3), has no built-in package for computation of Groebner basis, we have written our own program in Rlisp in order to solve systems of algebraic equations using Buchberger's algorithm [2]. To make the program effecient we have used the distributive form for the internal representation of polynomials together with multivariate factorization. In order to obtain the (infinitely many) solutions, we construct a lexicographic Groebner basis, then we compute, according to [3], the dimension and independent sets of variables for the ideal which is generated by the input system.

Thereafter, we consider each set of variables as free parameters and compute a Groebner basis leaving the order of the others unchanged. As a result we obtain a set of Groebner bases with a simple structure, and the solution can be found in an easy way.

Our analysis showsthat the Groebner basis method allows us to obtain the complete set of exact solutions for systems of nonlinear algebraic equations which are the necessary integrability conditions for NLEEs and therefore to select all integrable evolution equations. It is clear that the solvability of the above systems and of even more complicated ones is closely connected with the property of integrability. In addition to their importance in the theory and application of NLEEs, such systems are very useful for testing different computer algebra algorithms. One of our system is in a list of examples for Groebner basis computations [4].