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Polynomial dynamical systems and the Korteweg—de Vries equation

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Abstract

We explicitly construct polynomial vector fields L k , k = 0, 1, 2, 3, 4, 6, on the complex linear space C6 with coordinates X = (x 2, x 3, x 4) and Z = (z 4, z 5, z 6). The fields L k are linearly independent outside their discriminant variety Δ ⊂ C6 and are tangent to this variety. We describe a polynomial Lie algebra of the fields L k and the structure of the polynomial ring C[X,Z] as a graded module with two generators x 2 and z 4 over this algebra. The fields L 1 and L 3 commute. Any polynomial P(X,Z) ∈ C[X,Z] determines a hyperelliptic function P(X,Z)(u 1, u 3) of genus 2, where u 1 and u 3 are the coordinates of trajectories of the fields L 1 and L 3. The function 2x 2(u 1, u 3) is a two-zone solution of the Korteweg–de Vries hierarchy, and ∂ z 4(u 1, u 3)/∂u 1 = ∂x 2(u 1, u 3)/∂u 3.

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  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    V. M. Buchstaber

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Correspondence to V. M. Buchstaber.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 294, pp. 191–215.

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Buchstaber, V.M. Polynomial dynamical systems and the Korteweg—de Vries equation. Proc. Steklov Inst. Math. 294, 176–200 (2016). https://doi.org/10.1134/S0081543816060110

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