Abstract
In his 1892 paper, L. 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, 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.
Highlights
The covering τ I is ψx = −(w − u)ψ, ψt = 2 u xx ψ − (λ + u x )(w − u) ψ, which leads to the nonlocal conservation law ω = −(w − u) dx + 2 u xx ψ − (λ + u x )(w − u) dt of the potential KdV equation
Consider the equation uyy = utx + uy u xx − u x u xy that arises in the theory of integrable hydrodynamical chains, see [21]
We described a procedure that allows one to associate, in an algorithmic way, with any nontrivial finite-dimensional covering over a differentially connected equation a nonlocal conservation law
Summary
Reformulated in modern language, this means that the 1-form ω = cos(u + w) dx + cos(u − w) dy is a nonlocal conservation law for Equation (1). It became clear much later, some 100 years after the publication of [1], that nonlocal conservation laws are important invariants of PDEs and are used in numerous applications, e.g.,: numerical methods [2,3], sociological models [4,5], integrable systems [6], electrodynamics [7,8], mechanics [9,10,11], etc.
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