Integrable Hierarchies and Dispersionless Limit

TL;DR

This paper explores the dispersionless limits of integrable hierarchies like KP and Toda, introducing dressing operations as canonical transformations and providing a twistor-based solution construction.

Contribution

It presents a novel approach to dispersionless hierarchies, linking dressing operations to canonical transformations and offering a new twistor construction for solutions.

Findings

Dressing operations in dispersionless hierarchies are canonical transformations.

Quantization of dressing operations relates dispersionless and ordinary hierarchies.

New twistor construction for solutions of KP and Toda hierarchies.

Abstract

Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied. Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is dressing operators of the ordinary KP and Toda hierarchy. An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quatization of the similar construction of solutions of dispersionless hierarchies. These results as well as those obtained in previous papers are presented with proofs and necessary technical details.