Integrable evolution equations with constant separant

TL;DR

This paper classifies integrable one-field evolution equations of orders 2, 3, and 5 with constant separant, using necessary conditions derived from formal recursion operators, and presents new recurrent formulas and stronger results for fifth order equations.

Contribution

It provides the first published proofs and recurrent formulas for classifying integrable equations with constant separant, especially strengthening results for fifth order cases.

Findings

Classification of integrable equations of orders 2, 3, and 5

Recurrent formulas for necessary integrability conditions

Stronger results for fifth order equations

Abstract

The survey provides classification results for integrable one-field evolution equations of orders 2, 3 and 5 with the constant separant. The classification is based on necessary integrability conditions following from the existence of the formal recursion operator for integrable equations. Recurrent formulas for the whole infinite sequence of necessary conditions are presented for the first time. The most of the classification statements can be found in papers by S.I. Svinilupov and V.V. Sokolov but the proofs have never been published before. The result concerning the fifth order equations is stronger than obtained before.