Nonlocal conservation laws of PDEs possessing differential coverings

TL;DR

This paper explores the general existence of nonlocal conservation laws in PDEs with differential coverings, building on Bianchi's classical observations and providing methods and examples for their construction.

Contribution

It introduces a general framework for constructing nonlocal conservation laws in PDEs with differential coverings, extending classical results.

Findings

Nonlocal conservation laws can be systematically constructed for PDEs with differential coverings.

The methods are illustrated with several explicit examples.

The approach generalizes Bianchi's classical observations on the sine-Gordon equation.

Abstract

In his 1892 paper [L. Bianchi, Sulla trasformazione di B\"{a}cklund per le superfici pseudosferiche, Rend. Mat. Acc. Lincei, s. 5, v. 1 (1892) 2, 3--12], L. Bianchi noticed, among other things, that quite simple transformations of the formulas that describe the B\"{a}cklund transformation of the sine-Gordon equation lead to what is called a nonlocal conservation law in modern language. Using the techniques of differential coverings [I.S. Krasil'shchik, A.M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and B\"{a}cklund transformations, Acta Appl. Math. v. 15 (1989) 1-2, 161--209], we show that this observation is of a quite general nature. We describe the procedures to construct such conservation laws and present a number of illustrative examples.