Hierarchies of Manakov-Santini Type by Means of Rota-Baxter and Other Identities

TL;DR

This paper extends the Lax-Sato approach to integrable systems of Manakov-Santini type by introducing operators satisfying Rota-Baxter identities, leading to classification and new examples of such hierarchies.

Contribution

It formalizes the Lax-Sato approach for a broader class of integrable systems using Rota-Baxter operators, and constructs new Manakov-Santini type hierarchies.

Findings

Classified hierarchies of Manakov-Santini type systems.

Constructed new examples related to dispersionless KP and r-th systems.

Illustrated the theory with Laurent series algebra.

Abstract

The Lax-Sato approach to the hierarchies of Manakov-Santini type is formalized in order to extend it to a more general class of integrable systems. For this purpose some linear operators are introduced, which must satisfy some integrability conditions, one of them is the Rota-Baxter identity. The theory is illustrated by means of the algebra of Laurent series, the related hierarchies are classified and examples, also new, of Manakov-Santini type systems are constructed, including those that are related to the dispersionless modified Kadomtsev-Petviashvili equation and so called dispersionless r-th systems.