One Geometric Model for Non-local Transformations

A geometric formulation of an important class of non-local transformations is presented. The contemporary geometric theory of Bäcklund transformations (BT) [1–4] is based on certain integrability conditions for a system of differential equations which are determine it. These equations are written in terms of connections in the submanifold of a jet-bundle foliation. One of the first nontrivial examples of BT in three dimensions was described in paper [1]. In this paper, a geometric formulation of an important class of non-local transformations is presented. The generalization is obtained due to the use of non-local transformations of dependent variables and integrability conditions of a more general type. 1. In the fibered submanifold (Ej ,M ′ , ρ ′ ) with the k-jet bundle J1(M ′ N ′ ), we set an equation of order t = 2 ≤ l: Lq2 ( y, v, v 1 , v 2 ) = 0, v = { v } , M ′ = R (1, . . . , n− 1) , ( q = 1, . . . ,m ′ , B = 1, . . . ,m ′) . (1) We use here such notations: ∂μu = ∂u ∂xμ , {xμ} = (x0, x1, . . . , xn−1). Let now equation (1) be written in the form of n-th order exterior differential forms (n-forms) α = 1 n! α μ0...μn−1(y, v)dy μ0 ∧ dy1 ∧ . . . ∧ dyn−1 , (c = 1, . . . , r). (2) When this system of forms αc is equal to zero, then (1) is fulfiled. So, the system of αc = 0 forms generates an ideal I = {αc}. The condition for system (2) to be closed is dαc ⊂ I. Let now consider 1-forms of connections which generate the standard basis of contact forms ω = du −HA μ ( x, v, v 1 ;u ) dx, (μ = 0, . . . , n− 1, A = 1, . . . ,m) . (3) Let x = y, (μ = 0, . . . , n− 1).