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

Convention 5.2. The letters R will denote various functions of (x, y, y 1 ),changing with the context. Similarly, r = r(x, y), excluding the pure jetvariable y 1 . Hence, symbolically:(5.3) R = r + y 1 r + (y 1 ) 2 r + (y 1 ) 3 r + · · · .So the skeleton is(5.4) y 2 = F(x, y, y 1 ) = y 1 R = y 1 r + (y 1 ) 2 r + (y 1 ) 3 r + · · · .Applying L (2) , see (2.3)(II) for its expression, we get:(5.5) 0 = −Y 2 + X F x + Y F y + Y 1 F y1 .Observe that F x = (y 1 R) x = r y 1 +r (y 1 ) 2 +· · · and similarly for F y , but that(y 1 R) y1 = r +r y 1 +r (y 1 ) 2 + · · · . Inserting above Y 1 , Y 2 given by (2.6)(II),replacing y 2 by y 1 R and computing mod (y 1 ) 4 , we get:(5.6)0 ≡ − Y xx + [ − 2 Y xy + X xx]y1 + [ − Y yy + 2 X xy](y1 ) 2 + [ X yy](y1 ) 3 ++ [ − Y y + 2 X x](y1 r + (y 1 ) 2 r + (y 1 ) 3 r ) + [ 3 X y] ((y1 ) 2 r + (y 1 ) 3 r ) ++ [ X ]( y 1 r + (y 1 ) 2 r + (y 1 ) 3 r ) + [ Y ]( y 1 r + (y 1 ) 2 r + (y 1 ) 3 r ) ++ [ Y x] (r + y1 r + (y 1 ) 2 r + (y 1 ) 3 r ) ++ [ Y y − X x](y1 r + (y 1 ) 2 r + (y 1 ) 3 r ) + [ − X y]((y1 ) 2 r + (y 1 ) 3 r ) .We gather the powers cst., y 1 , (y 1 ) 2 and (y 1 ) 3 , equating their coefficients to0:(5.7)0 = −Y xx + P ( Y x),0 = −2 Y xy + X xx + P ( Y y , X x , X , Y , Y x),0 = −Y yy + 2 X xy + P ( Y y , X x , X y , X , Y , Y x),0 = X yy + P ( Y y , X x , X y , X , Y , Y x)Convention 5.8. The letter P will denote various linear combinations ofsome precise partial derivatives of X , Y which have analytic coefficientsin (x, y).By cross-differentiations and substitutions in the above system, allthird, fourth, fifth, etc. order derivatives of X , Y may be expressed asP ( X , Y , X x , X y , Y x , Y y , Y xy , Y yy).Proposition 5.9. An infinitesimal Lie symmetry X ∂∂x + Y ∂∂y of (E 1) isuniquely determined by the eight initial Taylor coefficients:(5.10) X (0), Y (0), X x (0), X y (0), Y x (0), Y y (0), Y xy (0), Y yy (0).251