Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I

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.Comment: 60 pages, 6 tables; added one remark in section 4.2.17 (p.33) plus several minor changes, to appear in J.Phys.

Integrable discretization of recursion operators and unified bilinear forms to soliton hierarchies

In this paper, we give a procedure of how to discretize the recursion operators by considering unified bilinear forms of integrable hierarchies. As two illustrative examples, the unified bilinear forms of the AKNS hierarchy and the KdV hierarchy are presented from their recursion operators. Via the compatibility between soliton equations and their auto-B\"acklund transformations, the bilinear integrable hierarchies are discretized and the discrete recursion operators are obtained. The discrete recursion operators converge to the original continuous forms after a standard limit.Comment: 11Page