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

342++++++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1m∑l 1 ,l 2 =1m∑l 1 =1[3Y jy l 1y l 2 − δj l 112X xy l 2]y l 12 y l 22 +[−δ j l 13X y l 2y l 3 − δ j l 212X y l 1y l 3]y l 11 y l 22 y l 32 +[4Y jy l 1y l 2 − δj l 14X xy l 2 − δ j l 212X xy l 1]y l 11 y l 23 +[−δ j l 14X y l 2y l 3 − δ j l 36X y l 1y l 2]y l 11 y l 21 y l 33 +[−δ j l 14X y l 2 − δ j l 26X y l 1]y l 12 y l 23 +[Y jy l 1 − δj l 14X x]y l 14 +m∑l 1 ,l 2 =1m∑l 1 =1[−δ j l 1X y l 2 − δ j l 24X y l 1]y l 11 y l 24 .[]4Y jxy − l 1 δj l 16X x 2 y l 13 +4.9. Inductive elaboration of the general formula. Now we compare theformula (2.9) for Y 4 with the above formula (4.8) for Y j 4 . The goal isto find the rules of transformation and of development by inspecting severalinstances, in order to devise how to transform and to develope the formula(2.25) to several dependent variables.First of all, we have to develope the general monomial (y λ1 ) µ1 · · ·(y λd ) µ d .In every monomial present in the expressions of Y j 1 , of Yj 2 , of Yj 3 and of∑Y j 4 above, we see that the number α of indices l β appearing in all the sumsml 1 ,...,l is exactly equal to µ α=1 1 + · · · + µ d . To denote these µ 1 + · · · + µ dindices l β , we shall use the notation:(4.10) l 1:1 , . . .,l 1:µ1 , . . .,l d:1 , . . .,l d:µd} {{ } } {{ }µ 1 µ} {{d}µ 1 +···+µ d,inspired by Convention 3.33. With such a choice of notation, we may avoidlong subscripts in the indices l β , like l µ1 +···+µ d. It follows that the developmentof the general monomial (y λ1 ) µ1 · · ·(y λd ) µ d to several dependent variablesyields m µ 1+···+µ d possible choices:∏∏(4.11)y l 1:ν 1λ 1· · · · · · y l d:ν dλ d,1ν 1 µ 1 1ν d µ dwhere the indices l 1:1 , . . .,l 1:µ1 , . . .,l d:1 , . . .,l d:µd take their values in the set{1, 2, . . ., m}. Consequently, the general expression of Yκ j must be of the