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

Takayuki Tsuchida·Thomas Wolf
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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.

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Takayuki Tsuchida, Thomas Wolf. 2005-07-14. Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I. https://doi.org/10.1088/0305-4470%2F38%2F35%2F006

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