Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

Assuming F = F(x, y x ) to be independent of y, or equivalently assumingM (E1 ) to be:(1.5) y = b + Π(x, a),the author has checked that equivalence to Y = B + XA holds if and onlyif two differential rational expressions annihilate:0 = Π x 2 a( ) 44− 6 Π x 2 a 3 Π xa( ) 25+ 15 Π ( ) 2x 2 a 2 Πxa 2( ) 6− 4 Π x 2 a 2 Π xa( ) 35Πxa Πxa Πxa Πxa357(1.6)− Π x 2 a Π xa 4( ) 5+ 10 Π xa 3 Π x 2 a Π xa 2( ) 6− 15 Π ( 3x 2 a Πxa 2)( ) 7andΠxa Πxa Πxa0 = Π x 4 a 2(Πxa) 2− 6 Π x 3 a 2 Π x 2 a(Πxa) 3− 4 Π x 3 a Π x 2 a 2(Πxa) 3− Π x 4 a Π xa 2(Πxa) 3++ 15 Π ( ) 2x 2 a 2 Πx 2 a( ) 4+ 10 Π ( ) 3x 3 a Π x 2 a Π xa 2 Πx( ) 4− 152 a Πxa 2( ) 5.Πxa Πxa ΠxaAs an application, this characterizes local sphericity of a rigid hypersurfacew = ¯w + i Θ(z, ¯z) of C 2 . The answer for a general y = Π(x, a, b), togetherwith a proof, will appear elsewhere.A modern restitution of Lie’s original proof of Theorem 1.1 may be foundin [Me2004]. In this reference, we generalize Theorem 1.1 to several dependentvariables y = (y 1 , y 2 , . . .,y m ). In the present Part III, we will insteadpass to several independent variables x = (x 1 , x 2 , . . ., x n ).Theorem 1.7. Let K = R or C, let n ∈ N, suppose n 2 and consider asystem of completely integrable partial differential equations in n independentvariables x = (x 1 , . . .,x n ) ∈ K n and in one dependent variable y ∈ Kof the form:(1.8) y x j 1x j 2 (x) = F j 1,j 2(x, y(x), yx 1(x), . . .,y x n(x) ) , 1 j 1 , j 2 n,where F j 1,j 2= F j 2,j 1. Under a local change of coordinates (x, y) ↦→(X, Y ) = (X(x, y), Y (x, y)), this system (1.8) is equivalent to the simplest“flat” system(1.9) Y X j 1X j 2 = 0, 1 j 1 , j 2 n,if and only if there exist arbitrary functions G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1and M k 1ofthe variables (x, y), for 1 j 1 , j 2 , k 1 n, satisfying the two symmetryconditions G j1 ,j 2= G j2 ,j 1and H k 1j 1 ,j 2= H k 1j 2 ,j 1, such that the equation (1.8)