Nonlocal conservation laws of PDEs possessing differential coverings

Abstract: In his 1892 paper "Sulla trasformazione di Bäcklund per le superfici pseudosferiche" (Rend. Mat. Acc. Lincei, s. 5, v. 1 (1892) 2, pp. 3-12; Opere, vol. 5, pp. 163-173) Luigi Bianchi noticed, among other things, that quite simple transformations of the formulas that describe the Bäcklund 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 Bäcklund transformations, Acta Appl. Math. 15 (1989) 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.

mathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

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Geometry of differential equations seminar