The Gould-Hopper Polynomials in the Novikov-Veselov equation

TL;DR

This paper explores the use of Gould-Hopper polynomials to analyze the Novikov-Veselov equation, deriving root dynamics, Lax pairs, and rational solutions, and connecting to models like the Gold-fish and Liouville equations.

Contribution

It introduces a novel application of Gould-Hopper polynomials to the NV equation, including root dynamics, Lax pair derivation, and explicit rational solutions.

Findings

Root dynamics of the $\sigma$-flow analyzed using GH polynomials

Lax pair for the NV equation derived

Explicit smooth rational solutions constructed

Abstract

We use the Gould-Hopper (GH) polynomials to investigate the Novikov-Veselov (NV) equation. The root dynamics of the $\sigma$-flow in the NV equation is studied using the GH polynomials and then the Lax pair is found. In particulr, when $N=3,4,5$, one can get the Gold-fish model. The smooth rational solutions of the NV equation are also constructed via the extended Moutard transformation and the GH polynomials. The asymptotic behavior is discussed and then the smooth rational solution of the Liouville equation is obtained.