(d24) /home/ugoktas/macsyma/painleve/unal/system/sp_hirsa.out (c25) batch("sp_exec.max"); (c26) /* sp_exec.max */ exec_painleve(eqlist,do_resonances,max_resonance,do_simplification)$ YOU ARE USING THE SIMPLIFICATION SUGGESTED BY KRUSKAL. You selected G(T,X,...) = X - H(T,...). ------------------------------------------------------------------------ PAINLEVE ANALYSIS OF THE EQUATION(S), f 1 x x x - 6 f f + ------- + 3 f f - f = 0 2 2 2 1 1 1 x x t - f - 3 f f - f = 0 2 1 2 2 x x x x t ------------------------------------------------------------------------ alpha 1 SUBSTITUTE u g FOR f 1, [0] 1 alpha 2 SUBSTITUTE u g FOR f 2, [0] 2 IN THE ORIGINAL EQUATION(S). THE MINIMUM POWERS OF g IN EQUATION 1 ARE [2 alpha - 1, alpha - 3, 2 alpha - 1, alpha - 1] 2 1 1 1 THE MINIMUM POWERS OF g IN EQUATION 2 ARE [alpha - 3, alpha + alpha - 1, alpha - 1] 2 2 1 2 By balancing any two or more terms, we obtain the solution set: [[alpha = - 2], [alpha = - 2, alpha = - 2]] 1 1 2 Now, we separate the cases above into different groups. The set: [[alpha = - 2]] 1 contains the case(s) where you have freedom. Note that if only one value of alpha is specified then the other alphas (the ones that are not listed) are arbitrary. The program will not automatically perform the remaining steps of the Painleve test for the case(s) in the above list. If you want to continue with these cases, you can determine appropriate values for the alphas, and give them to the program as a user-supplied alphalist in the datafile. To do so, give the selected values in the betalist. Continuing the next steps with : [[alpha = - 2, alpha = - 2]] 1 2 THE POWER OF g is - 5 IN EQUATION 1 . THE POWER OF g is - 5 IN EQUATION 2 . ----> SOLVE FOR u 1, [0] ----> SOLVE FOR u 2, [0] ------------------------------------------------------------------------ 2 2 1 TERM 6 (2 u - u - 2 u ) -- IS DOMINANT 2, [0] 1, [0] 1, [0] 5 g IN EQUATION 1 . 1 TERM 6 (u + 4) u -- IS DOMINANT 1, [0] 2, [0] 5 g IN EQUATION 2 . ------------------------------------------------------------------------ WITH [u = - 4, u = - 2] ------> FIND RESONANCES 1, [0] 2, [0] alpha r + alpha 1 1 SUBSTITUTE u g + u g FOR f 1, [0] 1, [r] 1 IN THE EQUATION(S). alpha r + alpha 2 2 SUBSTITUTE u g + u g FOR f 2, [0] 2, [r] 2 IN THE EQUATION(S). THIS IS THE MATRIX FOR RESONANCES: [ 2 ] [ (r - 4) (r - 5 r - 18) ] [ ----------------------- 12 (r - 4) ] [ 2 ] [ ] [ 2 ] [ - 2 3 - (r - 7) (r - 2) r ] THIS IS THE EQUATION FOR RESONANCES: (r - 8) (r - 6) (r - 4) (r - 3) (r + 1) (r + 2) - ----------------------------------------------- = 0 2 THESE ARE THE RESONANCES: [r = - 2, r = - 1, r = 3, r = 4, r = 6, r = 8] WITH MAXIMUM RESONANCE = 8 ----> CHECK RESONANCES. 8 ==== \ k - 2 SUBSTITUTE POWER SERIES > u g FOR f IN THE EQUATION(S). / 1, [k] 1 ==== k = 0 8 ==== \ k - 2 SUBSTITUTE POWER SERIES > u g FOR f IN THE EQUATION(S). / 2, [k] 2 ==== k = 0 ------------------------------------------------------------------------ * u = - 4 1, [0] * u = - 2 2, [0] ------------------------------------------------------------------------ * u = 0 1, [1] ------------------------------------------------------------------------ * u = 0 2, [1] ------------------------------------------------------------------------ h t * u = -- 1, [2] 3 ------------------------------------------------------------------------ 2 h t * u = ---- 2, [2] 3 ------------------------------------------------------------------------ * THE COEFFICIENT MATRIX FOR THE VECTOR [u , u ] 1, [3] 2, [3] HAS A DETERMINANT EQUAL TO ZERO. THE COMPATIBILITY CONDITION IS SATISFIED ! u = u 1, [3] 2, [3] THERE IS/ARE 1 FREE FUNCTION(S) ! ------------------------------------------------------------------------ ------------------------------------------------------------------------ * THE COEFFICIENT MATRIX FOR THE VECTOR [u , u ] 1, [4] 2, [4] HAS A DETERMINANT EQUAL TO ZERO. THE COMPATIBILITY CONDITION IS SATISFIED ! u = 2 u 1, [4] 2, [4] THERE IS/ARE 1 FREE FUNCTION(S) ! ------------------------------------------------------------------------ ------------------------------------------------------------------------ h + 30 u h t t 2, [3] t * u = - -------------------- 1, [5] 63 ------------------------------------------------------------------------ h - 12 u h t t 2, [3] t * u = -------------------- 2, [5] 63 ------------------------------------------------------------------------ * THE COEFFICIENT MATRIX FOR THE VECTOR [u , u ] 1, [6] 2, [6] HAS A DETERMINANT EQUAL TO ZERO. THE COMPATIBILITY CONDITION IS SATISFIED ! 2 u - 24 u + 3 u 2, [3] 2, [6] 2, [3] t u = - --------------------------------- 1, [6] 12 THERE IS/ARE 1 FREE FUNCTION(S) ! ------------------------------------------------------------------------ ------------------------------------------------------------------------ u + 12 u u 2, [4] 2, [3] 2, [4] t * u = - ----------------------------- 1, [7] 12 ------------------------------------------------------------------------ 2 4 h h + u (8 (h ) - 84 u ) + 21 u t t t 2, [3] t 2, [4] 2, [4] t * u = -------------------------------------------------------- 2, [7] 504 ------------------------------------------------------------------------ * THE COEFFICIENT MATRIX FOR THE VECTOR [u , u ] 1, [8] 2, [8] HAS A DETERMINANT EQUAL TO ZERO. THE COMPATIBILITY CONDITION IS SATISFIED ! 2 u = - (h - 6 u h - 12 u h - 198 u h 1, [8] t t t 2, [3] t t 2, [3] t 2, [3] t t 2 + 3024 u + 756 u )/756 2, [8] 2, [4] THERE IS/ARE 1 FREE FUNCTION(S) ! ------------------------------------------------------------------------ WITH [u = - 4, u = 2] ------> FIND RESONANCES 1, [0] 2, [0] alpha r + alpha 1 1 SUBSTITUTE u g + u g FOR f 1, [0] 1, [r] 1 IN THE EQUATION(S). alpha r + alpha 2 2 SUBSTITUTE u g + u g FOR f 2, [0] 2, [r] 2 IN THE EQUATION(S). THIS IS THE MATRIX FOR RESONANCES: [ 2 ] [ (r - 4) (r - 5 r - 18) ] [ ----------------------- - 12 (r - 4) ] [ 2 ] [ ] [ 2 ] [ 2 3 - (r - 7) (r - 2) r ] THIS IS THE EQUATION FOR RESONANCES: (r - 8) (r - 6) (r - 4) (r - 3) (r + 1) (r + 2) - ----------------------------------------------- = 0 2 THESE ARE THE RESONANCES: [r = - 2, r = - 1, r = 3, r = 4, r = 6, r = 8] WITH MAXIMUM RESONANCE = 8 ----> CHECK RESONANCES. 8 ==== \ k - 2 SUBSTITUTE POWER SERIES > u g FOR f IN THE EQUATION(S). / 1, [k] 1 ==== k = 0 8 ==== \ k - 2 SUBSTITUTE POWER SERIES > u g FOR f IN THE EQUATION(S). / 2, [k] 2 ==== k = 0 ------------------------------------------------------------------------ * u = - 4 1, [0] * u = 2 2, [0] ------------------------------------------------------------------------ * u = 0 1, [1] ------------------------------------------------------------------------ * u = 0 2, [1] ------------------------------------------------------------------------ h t * u = -- 1, [2] 3 ------------------------------------------------------------------------ 2 h t * u = - ---- 2, [2] 3 ------------------------------------------------------------------------ * THE COEFFICIENT MATRIX FOR THE VECTOR [u , u ] 1, [3] 2, [3] HAS A DETERMINANT EQUAL TO ZERO. THE COMPATIBILITY CONDITION IS SATISFIED ! u = - u 1, [3] 2, [3] THERE IS/ARE 1 FREE FUNCTION(S) ! ------------------------------------------------------------------------ ------------------------------------------------------------------------ * THE COEFFICIENT MATRIX FOR THE VECTOR [u , u ] 1, [4] 2, [4] HAS A DETERMINANT EQUAL TO ZERO. THE COMPATIBILITY CONDITION IS SATISFIED ! u = - 2 u 1, [4] 2, [4] THERE IS/ARE 1 FREE FUNCTION(S) ! ------------------------------------------------------------------------ ------------------------------------------------------------------------ 30 u h - h 2, [3] t t t * u = -------------------- 1, [5] 63 ------------------------------------------------------------------------ h + 12 u h t t 2, [3] t * u = - -------------------- 2, [5] 63 ------------------------------------------------------------------------ * THE COEFFICIENT MATRIX FOR THE VECTOR [u , u ] 1, [6] 2, [6] HAS A DETERMINANT EQUAL TO ZERO. THE COMPATIBILITY CONDITION IS SATISFIED ! 2 u - 24 u - 3 u 2, [3] 2, [6] 2, [3] t u = --------------------------------- 1, [6] 12 THERE IS/ARE 1 FREE FUNCTION(S) ! ------------------------------------------------------------------------ ------------------------------------------------------------------------ 12 u u - u 2, [3] 2, [4] 2, [4] t * u = - ----------------------------- 1, [7] 12 ------------------------------------------------------------------------ 2 - 4 h h + u (8 (h ) + 84 u ) + 21 u t t t 2, [3] t 2, [4] 2, [4] t * u = ---------------------------------------------------------- 2, [7] 504 ------------------------------------------------------------------------ * THE COEFFICIENT MATRIX FOR THE VECTOR [u , u ] 1, [8] 2, [8] HAS A DETERMINANT EQUAL TO ZERO. THE COMPATIBILITY CONDITION IS SATISFIED ! 2 u = - (h + 6 u h + 12 u h - 198 u h 1, [8] t t t 2, [3] t t 2, [3] t 2, [3] t t 2 - 3024 u + 756 u )/756 2, [8] 2, [4] THERE IS/ARE 1 FREE FUNCTION(S) ! ------------------------------------------------------------------------ ------------------------------------------------------------------------ (c27) /* sp_exec.max */ (d27) done (c28) closefile();