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

375Sixthly:(3.22) ⎧δ k 1j 1Θ n+1y = −L k 1j 1 ,y + 2 Mk 1+ x j 1⎪⎨ + 2 ∑ ∑H k 1j 1 ,k 2M k 2− δ k 1j 1k 2⎪⎩k 2H k 2k 2 ,k 2M k 2− 1 2∑+ δ k 1j 1M k 2Θ k 2+ 1 2 δk 1j 1Θ n+1 Θ n+1 .k 2∑L k 2j 1L k 1k 2+k 23.23. Solving Θ j 1, x j 2 Θj 1 y , Θ n+1 and Θ n+1x j 1 y . From the six families of equations(3.17), (3.18), (3.19), (3.20), (3.21) and (3.22), we can solve Θ j 1, x j 2Θ j 1 y , Θ n+1 and Θ n+1x j 1 y . Not mentioning the (hard) intermediate computations,we obtain firstly:(3.24)⎧⎪⎨Θ j 1x j 2 = −2 G j 1 ,j 2 ,y + H j 1j 1 ,j 1 ,x j 2 + ∑ lG j2 ,l L l j 1+ 1 2 Hj 1j 1 ,j 1H j 2j 2 ,j 2− ∑ lH l j 1 ,j 2H l l,l−⎪⎩− G j1 ,j 2Θ n+1 − 1 2 H j 1 ,j 1Θ j 1− 1 2 Hj 2j 2 ,j 2Θ j 1+ ∑ lH l j 1 ,j 2Θ l + 1 2 Θj 1Θ j 2.Secondly:(3.25) ⎧Θ j 1y = − 1 3 Hj 1j 1 ,j 1 ,y + 2 3 Lj 1+ 4j 1 ,x j 13 G j 1 ,j 1M j 1+ 2 3⎪⎨⎪⎩+ 2 3∑l+ 1 2 Θj 1Θ n+1 .H j 1j 1 ,l Ll j 1− 2 3∑l∑G j1 ,l M l − 1 ∑Hl,l l L l j21+ll∑L l j 1Θ l +H l j 1 ,j 1L j 1l− 1 2 Hj 1j 1 ,j 1Θ n+1 + 1 2lThirdly:(3.26) ⎧Θ n+1 = − 2 x j 13 Hj 1j 1 ,j 1 ,y + 1 3 Lj 1+ 2j 1 ,x j 13 G j 1 ,j 1M j 1+ 4 3⎪⎨⎪⎩+ 1 3∑l+ 1 2 Θj 1Θ n+1 .H j 1j 1 ,l Ll j 1− 1 3∑l∑G j1 ,l M l − 1 ∑Hl,l l 2Ll j 1+ll∑L l j 1Θ l +H l j 1 ,j 1L j 1l− 1 2 Hj 1j 1 ,j 1Θ n+1 + 1 2l