A family of integrable evolution equations of third order

TL;DR

This paper introduces a new family of third-order integrable evolution equations involving transcendental functions, linked to the Krichever-Novikov equation through geometric methods related to genus two curves.

Contribution

It constructs a novel family of integrable third-order equations using geometric techniques involving genus two curves and their Jacobians, expanding the class of known integrable systems.

Findings

Established a new family of integrable equations with transcendental nonlinearities.

Connected these equations to the Krichever-Novikov equation via differential substitutions.

Utilized algebraic geometry of genus two curves in the construction process.

Abstract

We construct a family of integrable equations of the form $v_t=f(v,v_x,v_{xx},v_{xxx})$ such that $f$ is a transcendental function in $v,v_x,v_{xx}$. This family is related to the Krichever-Novikov equation by a differential substitution. Our construction of integrable equations and the corresponding differential substitutions involves geometry of a family of genus two curves and their Jacobians.