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

(X 1 , X 2 , Y , Xy 1,X 2x, Y 2 x 1, Y y , X 2x 1 x, Y 1 x 1 x 1, Y yy), namely:⎧X 1x = 1 P, X 2 2 31 x = P, Y 1 x 2 = P,(8.86)⎪⎨⎪⎩X 1x = 4 P, X 2 52 y = P,X 1x 1 yX 1x 2 y6= P, X 2x 2 x 2 7 = P, Y x 1 x 2 8 = P,9= P, X 2 1011x 2 y = P, Y x 1 y = P,Xyy1 1213= P, Y x 2 y = P,X 2x 1 x 1 x 2 14= P, Y x 1 x 1 x 1 15= P,X 2x 1 x 2 x 2 16= P, Y x 1 x 1 x 2 17= P,X 2 1819x 1 x 2 y = P, Y x 1 x 1 y = P,Y x 1 yyY x 2 yy20= P,21= P,Y yyy22= P.Then the expressions P are stable under differentiation with respectto x 1 , to x 2 , to y and moreover, all other, higher orderpartial derivatives of X 1 , of X 2 , of Y may be expressed asP ( X 1 , X 2 , Y , Xy 1,X 2x, Y 2 x 1, Y y , X 2x 1 x, Y 2 x 1 x 1, Y yy).Corollary 8.87. Every infinitesimal Lie symmetry of the PDE system (E 5 ) isuniquely determined by the ten initial Taylor coefficients(8.88)X 1 (0), X 2 (0), Y (0), Xy 1 (0), X 2x 2(0), Y x 1(0), Y y(0), X 2x 1 x 2(0), Y x 1 x 1(0), Y yy(0).Proof of the proposition. At first, (8.83) 1 yields (8.86) 15 ; (8.81) 1 yields(8.86) 3 ; differentiating (8.81) 1 with respect to x 1 yields (8.86) 8 ; differentiating(8.81) 1 with respect to y yields (8.86) 13 ; differentiating (8.81) 1 withrespect to x 1 x 1 yields (8.86) 17 ; and differentiating (8.81) 1 with respect to yyyields (8.86) 21 . Also, rewriting (8.81) 2 as(8.89) X 1x 2 = −1 2 Y x 1,we get (8.86) 4 ; and rewriting (8.81) 3 as(8.90) X 1x 1 = 1 2 X 2x 2 + 1 2 Y y + r Y x 1 + s ∗ Y x 1 x 1,we get (8.86) 1 .Next, differentiating (8.81) 2 with respect to x 1 and (8.81) 3 with respect tox 2 , we get, taking account of 0 = Y x 2 y = Y x 1 x 2 = Y x 1 x 1 x 2, replacing X x 1 x 2283