Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I
Abstract
We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov-Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of 2nd order systems with a 3rd order or a 4th order symmetry and 3rd order systems with a 5th order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made.
Keywords
Cite
@article{arxiv.nlin/0412003,
title = {Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I},
author = {Takayuki Tsuchida and Thomas Wolf},
journal= {arXiv preprint arXiv:nlin/0412003},
year = {2007}
}
Comments
60 pages, 6 tables; added one remark in section 4.2.17 (p.33) plus several minor changes, to appear in J.Phys.A