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

64been able to achieve systematic corrections of our computations by alwayschecking them alongside with the existence of the change of coordinates(x, y) ↦→ (X, Y ).Coming back to the definition (3.35) and to the approximation (3.58), wehave:(4.9)G l 1= −□ l 1xx∼ = −Yl 1xx ,H l 1l1= −2□ l 1+ xy l 1 □0 ∼ xx = −2 Y l 1+ X xy l 1 xx,L l 1l1 ,l 1= −□ l 1y l 1y l 1 + 2 □0 xy l 1∼ = −Yl 1y l 1y l 1 + 2 X xy l 1.Differentiating the first two lines with respect to y l 1and the third line withrespect to x, and replacing the sign ∼ = by the sign ≡ (in a non-rigorous way,this corresponds essentially to neglecting the derivatives of order 0, 1, 2 and3 of X, Y j and to neglecting the difference between the Jacobian matrix ofthe transformation and the identity matrix), we get:G l 1y l 1y l 1 ≡ −Y l 1xxy l 1y l 1 ,(4.10)H l 1l 1 ,xy l 1 ≡ −2Y l 1xy l 1xy l 1 + X xxxy l 1,L l 1l1 ,l 1 ,xx ≡ −Y l 1y l 1y l 1xx + 2X xy l 1xx .Hence the linear combination −2 · (4.10) 1 + 4 3 · (4.10) 2 − 2 3 (4.10) 3 yieldsthe desired result:(4.11)0 ≡?≡ −2G l 1y l 1y l 1 + 4 3 Hl 1l 1 ,xy l 1 − 2 3 Ll 1l1 ,l 1 ,xx≡ 2Y l 1− 8 xxy l 1y l1 a 3 Y l 1+ 4 xy l 1xy l 1a 3 X xxxy l 1 + 2b 3 Y l 1y l 1y l1xx − 4a 3 X xy l 1xxb≡ 0, indeed!Thanks to this straightforward computation, we guess that the approximatepartial differential relations (4.8) are a consequence of the approximate relations(3.106), (3.108), (3.110) and (3.96), namely:(4.12)(3.106) mod : 0 ≡ −2 G j + 2 y l 1 δj l 1G l 2+ y l 2 Hj l 1 ,x − δj l 1H l 2l2 ,x ,(3.108) mod : 0 ≡ − 1 2 Hj l 1 ,y l 2 + 1 6 δj l 1H l 2l 2 ,y l 2 + 1 3 δj l 2H l 1l 1 ,y l 1 ++ L j l 1 ,l 2 ,x − 1 3 δj l 1L l 2l2 ,l 2 ,x − 2 3 δj l 2L l 1l1 ,l 1 ,x ,(3.110) mod : 0 ≡ L j l 1 ,l 2 ,y l 3 − Lj l 1 ,l 3 ,y l 2 + δj l 3M l1 ,l 2 ,x − δ j l 2M l1 ,l 3 ,x,(3.96) mod : 0 ≡ M l1 ,l 2 ,y l 3 − M l1 ,l 3 ,y l 2.