Nonlocal conservation laws of PDEs possessing differential coverings

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.