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

g j i 1 ,...,i λand similarly for h i1 ,...,i λ. For arbitrary indices l 1 , . . .,l λ = 1, . . .,m,∂the partial derivativeλ fwill be abbreviated by f ∂y l 1 ···∂y l λ l 1 ,...,l λ.Appying the chain rule, we may compute:(5.17)m∑ ]h i1 = f l1[g l 1i1,h i1 ,i 2=h i1 ,i 2 ,i 3=h i1 ,i 2 ,i 3 ,i 4=l 1 =1m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1+m∑l 1 =1f l1 ,l 2[g l 1i1g l 2i2]+m∑l 1 ,l 2 ,l 3 ,l 4 =1m∑l 1 ,l 2 ,l 3 =1+++m∑l 1 =1f l1 ,l 2 ,l 3[g l 1i1g l 2i2g l 3i3]+f l1[g l 1i1 ,i 2 ,i 3],m∑l 1 ,l 2 =1m∑l 1 ,l 2 =1m∑l 1 =1Introducing the derivationsF 2i :=F 3i :=n∑f l1[g l 1i1 ,i 2],m∑l 1 ,l 2 =1f l1 ,l 2 ,l 3 ,l 4[g l 1i1g l 2i2g l 3i3g l 4i4]+353f l1 ,l 2[g l 1i1g l 2i2 ,i 3+ g l 1i2g l 2i1 ,i 3+ g l 1i3g l 2i1 ,i 2]+f l1 ,l 2 ,l 3[g l 1i2g l 2i3g l 3i1 ,i 4+ g l 1i3g l 2i1g l 3i2 ,i 4+ g l 1i1g l 2i2g l 3i3 ,i 4++g l 1i1g l 2i4g l 3i2 ,i 3+ g l 1i2g l 2i4g l 3i3 ,i 1+ g l 1i3g l 2i4g l 3i1 ,i 2]+f l1 ,l 2[g l 1i1 ,i 2g l 2i3 ,i 4+ g l 1i1 ,i 3g l 2i2 ,i 4+ g l 1i1 ,i 4g l 2i2 ,i 3]+f l1 ,l 2[g l 1i1g l 2i2 ,i 3 ,i 4+ g l 1i2g l 2i1 ,i 3 ,i 4+ g l 1i3g l 2i1 ,i 2 ,i 4+ g l 1i4g l 2i1 ,i 2 ,i 3]+f l1[g l 1i1 ,i 2 ,i 3 ,i 4].m∑k 1 =1 l 1 =1n∑k 1 =1 l 1 =1+m∑l 1 =1m∑g l 1k1 ,ig l 1k1 ,i⎛g l 1 ⎝ i∂∂g l 1∂∂g l 1m∑l 2 =1k1+k1+f l1 ,l 2m∑l 1 =1n∑⎛g l 1 ⎝ im∑k 1 ,k 2 =1 l 1 =1∂∂f l2+m∑l 2 =1m∑l 2 ,l 3 =1f l1 ,l 2g l 1k1 ,k 2 ,if l1 ,l 2 ,l 3⎞∂⎠ ,∂f l2∂∂g l +1k1 ,k 2⎞∂⎠,∂f l2 ,l 3(5.18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .