(D15) CSM$USERS:[WHEREMAN.PROGRAMS.NPAINLEVE.SINGLE]P_SWKDV.OUT;8 (C16) batch("np_exec.max"); (C17) /* ************************************************************************* */ /* Batch file NP_EXEC.MAX */ /* ************************************************************************* */ exec_painleve (eq, alpha, do_resonances, max_resonance, do_simplification)$ You are using the simplification suggested by KRUSKAL You selected G(T,X,...) = X - H(T,...) ---------------------------------------------------------------- PAINLEVE ANALYSIS OF EQUATION, B F F - F + A F F + F - F T X X X X T X X T X X X T X = 0 ---------------------------------------------------------------- ALPHA SUBSTITUTE U G FOR f IN ORIGINAL EQUATION. 0 MINIMUM POWERS OF g ARE [2 ALPHA - 3, ALPHA - 4] 2 ALPHA - 3 2 2 * COEFFICIENT OF G IS - U (ALPHA - 1) ALPHA (B + A) H 0 T NOTE : THIS TERM VANISHES FOR [ALPHA = 0, ALPHA = 1] , 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. ALPHA - 4 * COEFFICIENT OF G IS - U (ALPHA - 3) (ALPHA - 2) (ALPHA - 1) 0 ALPHA H T NOTE : THIS TERM VANISHES FOR [ALPHA = 0, ALPHA = 1, ALPHA = 2, ALPHA = 3] , 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 - 3 ) AND ( ALPHA - 4 ) OF g, WITH alpha = - 1 , POWER OF g is - 5 ----> SOLVE FOR U 0 1 TERM 2 U (U B + U A - 12) H -- IS DOMINANT 0 0 0 T 5 G IN EQUATION. ---------------------------------------------------------------- 12 1 ) WITH U = ----- ----> FIND RESONANCES 0 B + A ALPHA R + ALPHA SUBSTITUTE U G + U G FOR f IN EQUATION 0 R R - 5 TERM ( - H (R - 6) (R - 4) (R - 1) (R + 1) ) U G IS DOMINANT T R IN EQUATION. THE 3 NON-NEGATIVE INTEGRAL ROOTS ARE [R = 1, R = 4, R = 6] WITH MAXIMUM RESONANCE = 6 ----> CHECK RESONANCES. 6 ==== \ K - 1 SUBSTITUTE POWER SERIES > G U FOR f IN EQUATION. / K ==== K = 0 12 WITH U = ----- 0 B + A 1 * COEFFICIENT OF -- IS 0 4 G U IS ARBITRARY ! 1 COMPATIBILITY CONDITION IS SATISFIED ! 24 (U B H + U A H - H - U B + 1) 2 T 2 T T 1 1 T * COEFFICIENT OF -- IS - --------------------------------------- 3 B + A G H + U B - 1 T 1 T U = -------------- 2 (B + A) H T 1 2 * COEFFICIENT OF -- IS - 12 (2 U B H - U A B H - 2 B H 2 1 T T 1 T T T T G T T 2 3 3 2 3 2 + A H + 2 U B (H ) + 4 U A B (H ) + 2 U A (H ) - 2 U B H T T 3 T 3 T 3 T 1 T T T 2 2 + U A B H )/((B + A) (H ) ) 1 T T T T (2 B - A) (U B H - H - U B H ) 1 T T T T 1 T T T T U = - ------------------------------------------ 3 2 3 2 (B + A) (H ) T 1 * COEFFICIENT OF - IS - 12 (B - A) (2 B - A) G 2 2 (U B H H - H H - 3 U B (H ) + 3 (H ) + 3 U B H H 1 T T T T T T T T 1 T T T T 1 T T T T T T T 2 3 4 - U B (H ) )/((B + A) (H ) ) 1 T T T T T U IS ARBITRARY ? 4 COMPATIBILITY CONDITION: - 12 (B - A) (2 B - A) 2 2 (U B H H - H H - 3 U B (H ) + 3 (H ) + 3 U B H H 1 T T T T T T T T 1 T T T T 1 T T T T T T T 2 3 4 - U B (H ) )/((B + A) (H ) ) = 0, 1 T T T T T *** CONDITION IS NOT SATISFIED. *** *** CHECK FOR FREE PARAMETERS OR PRESENCE OF U . *** 0 2 2 2 * COEFFICIENT OF 1 IS (U B H H - B H H - (U ) A B H 1 T T T T T T 1 T T T T 2 4 4 2 4 + 2 U A B H - A H + 24 U B (H ) + 48 U A B (H ) + 24 U A (H ) 1 T T T T 5 T 5 T 5 T T 2 3 2 3 2 2 2 + 30 U B (H ) - 30 U A (H ) - U B (H ) + U U A B H 4 T 4 T 1 T 1 1 T T T T T T T T 2 3 - U A B H )/((B + A) (H ) ) 1 T T T T 2 2 2 U = - (U B H H - B H H - (U ) A B H + 2 U A B H 5 1 T T T T T T 1 T T 1 T T T T T 2 3 2 3 2 2 - A H + 30 U B (H ) - 30 U A (H ) - U B (H ) T T 4 T 4 T 1 T T T T T 2 2 4 + U U A B H - U A B H )/(24 (B + A) (H ) ) 1 1 T 1 T T T T T T T * COEFFICIENT OF G IS - (B - A) (2 B - A) 2 2 2 2 2 3 ((U ) B (H ) - 2 U B (H ) + (H ) - 30 U B (H ) H 1 T T 1 T T T T 4 T T T T T T 3 2 - 30 U A (H ) H - 2 U U B H H + 2 U B H H 4 T T T 1 1 T T T 1 T T T T T T T T T 4 4 2 2 2 3 5 + 30 U B (H ) + 30 U A (H ) + (U ) B (H ) )/((B + A) (H ) ) 4 T 4 T 1 T T T T T T T T U IS ARBITRARY ? 6 COMPATIBILITY CONDITION: - (B - A) (2 B - A) 2 2 2 2 2 3 ((U ) B (H ) - 2 U B (H ) + (H ) - 30 U B (H ) H 1 T T 1 T T T T 4 T T T T T T 3 2 - 30 U A (H ) H - 2 U U B H H + 2 U B H H 4 T T T 1 1 T T T 1 T T T T T T T T T 4 4 2 2 2 3 5 + 30 U B (H ) + 30 U A (H ) + (U ) B (H ) )/((B + A) (H ) ) 4 T 4 T 1 T T T T T T T T = 0, *** 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();