(D15) CSM$USERS:[WHEREMAN.PROGRAMS.NPAINLEVE.SINGLE]P_CYLKDV.OUT;2 (C16) batch("np_exec.max"); (C17) /* ************************************************************************* */ /* Batch file NP_EXEC.MAX */ /* ************************************************************************* */ exec_painleve (eq, alpha, do_resonances, max_resonance, do_simplification)$ ---------------------------------------------------------------- PAINLEVE ANALYSIS OF EQUATION, F A(T) + F + 6 F F + F = 0 X X X X T ---------------------------------------------------------------- ALPHA SUBSTITUTE U G FOR f IN ORIGINAL EQUATION. 0 MINIMUM POWERS OF g ARE [2 ALPHA - 1, ALPHA - 3] 2 ALPHA - 1 2 * COEFFICIENT OF G IS 6 U ALPHA G 0 X NOTE : THIS TERM VANISHES FOR ALPHA = 0 , VERIFY IF ALPHA = 0 LEADS TO DOMINANT BEHAVIOR, IF IT DOES THEN RUN THE PROGRAM AGAIN WITH THIS USER SUPPLIED VALUE OF ALPHA. HENCE, PUT BETA = 0 . ALPHA - 3 3 * COEFFICIENT OF G IS U (ALPHA - 2) (ALPHA - 1) ALPHA (G ) 0 X NOTE : THIS TERM VANISHES FOR [ALPHA = 0, ALPHA = 1, ALPHA = 2] , VERIFY IF ANY OF THESE VALUES FOR ALPHA LEADS TO DOMINANT BEHAVIOR, IF IT DOES THEN RUN THE PROGRAM AGAIN WITH THIS VALUE AS USER SUPPLIED ALPHA, CALLED BETA. ---------------------------------------------------------------- FOR EXPONENTS ( 2 ALPHA - 1 ) AND ( ALPHA - 3 ) OF g, WITH alpha = - 2 , POWER OF g is - 5 ----> SOLVE FOR U 0 2 1 TERM - 12 U G (2 (G ) + U ) -- IS DOMINANT 0 X X 0 5 G IN EQUATION. ---------------------------------------------------------------- 2 1 ) WITH U = - 2 (G ) ----> FIND RESONANCES 0 X ALPHA R + ALPHA SUBSTITUTE U G + U G FOR f IN EQUATION 0 R 3 R - 5 TERM ( (G ) (R - 6) (R - 4) (R + 1) ) U G IS DOMINANT X R IN EQUATION. THE 2 NON-NEGATIVE INTEGRAL ROOTS ARE [R = 4, R = 6] WITH MAXIMUM RESONANCE = 6 ----> CHECK RESONANCES. 6 ==== \ K - 2 SUBSTITUTE POWER SERIES > G U FOR f IN EQUATION. / K ==== K = 0 2 WITH U = - 2 (G ) 0 X 1 2 2 * COEFFICIENT OF -- IS 6 (G ) ((- 2 (G ) ) - 6 G G + 5 U G ) 4 X X X X X 1 X G X U = 2 G 1 X X 1 2 2 * COEFFICIENT OF -- IS 4 G (4 G G - 3 (G ) + 6 U (G ) 3 X X X X X X X 2 X G + G G ) T X 2 4 G G - 3 (G ) + G G X X X X X X T X U = - ------------------------------- 2 2 6 (G ) X 1 3 2 * COEFFICIENT OF -- IS - 2 ((G ) A(T) + (G ) G 2 X X X X X X G 3 4 2 - 4 G G G + 3 (G ) - G G G - 6 U (G ) + G (G ) )/G X X X X X X X X T X X X 3 X T X X X 3 2 3 U = ((G ) A(T) + (G ) G - 4 G G G + 3 (G ) 3 X X X X X X X X X X X X X X 2 4 - G G G + G (G ) )/(6 (G ) ) T X X X T X X X 1 * COEFFICIENT OF - IS 0 G U IS ARBITRARY ! 4 COMPATIBILITY CONDITION IS SATISFIED ! 4 3 2 * COEFFICIENT OF 1 IS (2 (G ) G A(T) - 3 (G ) (G ) A(T) X X X X X X X 4 4 3 - G (G ) A(T) - (G ) G + 9 (G ) G G T X X X X X X X X X X X X X X X X 3 2 2 3 + 17 (G ) G G - 48 (G ) (G ) G + 2 G (G ) G X X X X X X X X X X X X X X X T X X X X X 2 2 3 2 - 70 (G ) G (G ) + 174 G (G ) G - 17 G (G ) G G X X X X X X X X X X X X T X X X X X X 3 5 3 2 2 + 8 G (G ) G - 81 (G ) + 21 G G (G ) - 21 G (G ) (G ) T X X X X X X X T X X X T X X X X 6 3 2 2 8 - 36 U (G ) G + 9 G (G ) G - (G ) (G ) G - 36 U (G ) 4 X X X T X X X X X T X X X 5 X 7 4 4 3 5 - 36 U (G ) - G (G ) - 2 G (G ) + 2 G G (G ) )/(6 (G ) ) 4 X T T X T X X X X T T X X X X 4 3 2 4 U = (2 (G ) G A(T) - 3 (G ) (G ) A(T) - G (G ) A(T) 5 X X X X X X X T X 4 3 3 - (G ) G + 9 (G ) G G + 17 (G ) G G X X X X X X X X X X X X X X X X X X X X X X X 2 2 3 2 2 - 48 (G ) (G ) G + 2 G (G ) G - 70 (G ) G (G ) X X X X X X X T X X X X X X X X X X X 3 2 3 + 174 G (G ) G - 17 G (G ) G G + 8 G (G ) G X X X X X X T X X X X X X T X X X X X 5 3 2 2 6 - 81 (G ) + 21 G G (G ) - 21 G (G ) (G ) - 36 U (G ) G X X T X X X T X X X X 4 X X X 3 2 2 7 4 + 9 G (G ) G - (G ) (G ) G - 36 U (G ) - G (G ) T X X X X X T X X X 4 X T T X X 4 3 8 - 2 G (G ) + 2 G G (G ) )/(36 (G ) ) T X X X X T T X X X 2 A(T) + 2 A (T) T * COEFFICIENT OF G IS --------------- 6 G X U IS ARBITRARY ? 6 2 A(T) + 2 A (T) T COMPATIBILITY CONDITION: --------------- = 0, 6 G X *** CONDITION IS NOT SATISFIED. *** *** CHECK FOR FREE PARAMETERS OR PRESENCE OF U . *** 0 ---------------------------------------------------------------- (C18) output()$ ---------------------------------------------------------------- AT THE END OF THE COMPUTATIONS THE FOLLOWING ARE AVAILABLE: * U VALUE(S) (type uval[j,k,l] where 1 <= j <= 1 and 0 <= k <= [6] and 1 <= l <= [1] ) j stands for j_th alpha,k stands for u[k],l stands for l_th solution set for u[0] * ALPHA VALUE(S) (type alpha[j] where 1 <= j <= 1 ) j stands for j_th alpha * COMPATIBILITY CONDITION(S) (type compcond[j,k] where 1 <= j <= 1 and 1 <= k <= [1] ) j stands for j_th alpha,k stands for k_th solution set for u[0] * RESONANCE(S) (type res[j,k] where 1 <= j <= 1 and 1 <= k <= [1] ) j stands for j_th alpha,k stands for k_th solution set for u[0] ---------------------------------------------------------------- TO SEE THIS MENU AGAIN JUST TYPE < output() > ---------------------------------------------------------------- (C19) /* ************************** END of NP_EXEC.MAX ************************** */ (D19) DONE (C20) closefile();