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

234m for q = 1, . . ., p. In the present Part I, we study the Lie symmetries of ageneral system of analytic partial differential equations of the form:((E ) y j x α(x) = F αj x, y(x), ( y j(q) (x) ) ),x β(q) 1qpwhere j with 1 j m and α ∈ N n satisfy( ) ( ) ( )(1.2) j, α ≠ (j, 0) and j, α ≠ j(q), β(q) .In particular, all (κ + 1)-th partial derivatives of the unknown y = y(x)depend on a certain precise set of derivatives of order κ: the system iscomplete. In addition, all the other partial derivatives of order κ do alsodepend on the same precise set of derivatives.Here, we assume that u = 0 is a local solution of the system (E ) and thatthe functions Fα j are K-algebraic (in the sense of Nash) or K-analytic, in aneighborhood of the origin in K n+m+p . Even if our concern will be localthroughout, we will not introduce any special notation to speak of open subsetsand simply refer to various K µ . We will study five concrete instances,the first three ones being classical.Example 1.3. With n = m = κ = 1, a second order ordinary differentialequation(E 1 ) y xx = F(x, y, y x ),and more generally y x κ+1 = F(x, y, y x , . . .,y x κ), where x, y ∈ K,see [Lie1883, EL1890, Tr1896, Ca1924, Se1931, Ca1932a, Ol1986,Ar1988, BK1989, GTW1989, HK1989, Ib1992, Ol1995, N2003].Example 1.4. With n 2, m = 1 and κ = 1, a complete system of secondorder equations(E 2 ) y x i 1x i 2 = F i1 ,i 2(x i , y, y x k), 1 i1 , i 2 n,see [Ha1937, Ch1975, Su2001] and Part III below.Example 1.5. Dually, with n = 1, m 2 and κ = 1, an ordinary system ofsecond order(E 3 ) yxx j = F j( x, y j 1), y j 1x , j = 1, . . ., m,see [Fe1995, Me2004] and the references therein.Example 1.6. With n = 1, m = 2 and κ = 1, a system of the form{y2x = F ( x, y 1 , y 2 , yx)1(E 4 )y 1 xx = G( x, y 1 , y 2 , y 1 x).