Details:
Title  Solving algebraic systems which arise as necessary integrability conditions for polynomialnonlinear evolution equations  Author(s)  Vladimir P. Gerdt, Nikolai V. Khutornoy, Alexey Yu. Zharkov  Type  Article in Conference Proceedings  Abstract  The investigation of the problem of integrability of polynomialnonlinear 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 KdVlike equations, the seventh order MKdVlike equations, and the third order coupled KdVlike systems. All our computations have been carried out by using the computer algebra system REDUCE (version 3.2) on an IBM PC ATlike 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 builtin 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 shows that 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].  ISBN  0201548925 
URL 
http://doi.acm.org/10.1145/96877.96968 
Language  English  Pages  299  Publisher  ACM Press  Address  New York, NY, USA  Year  1990  Translation 
No  Refereed 
No  Conferencename  International Conference on Symbolic and Algebraic Computation 
