(D13) CSM$USERS:[WHEREMAN.PROGRAMS.NPAINLEVE.SINGLE]P_FHN.OUT;4 (C14) batch("np_exec.max"); (C15) /* ************************************************************************* */ /* Batch file NP_EXEC.MAX */ /* ************************************************************************* */ exec_painleve (eq, alpha, do_resonances, max_resonance, do_simplification)$ SUBSTITUTE X ----> G + X0 ---------------------------------------------------------------- 3 PAINLEVE ANALYSIS OF EQUATION, SQRT(2) F + C F - 2 SQRT(2) F G G G 2 + (2 A + 2) F - SQRT(2) A F = 0 ---------------------------------------------------------------- ALPHA SUBSTITUTE U G FOR f IN ORIGINAL EQUATION. 0 MINIMUM POWERS OF g ARE [3 ALPHA, 2 ALPHA, ALPHA - 2] 3 ALPHA 3 * COEFFICIENT OF G IS - 2 SQRT(2) U 0 2 ALPHA 2 * COEFFICIENT OF G IS 2 U (A + 1) 0 ALPHA - 2 * COEFFICIENT OF G IS SQRT(2) U (ALPHA - 1) ALPHA 0 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. ---------------------------------------------------------------- FOR EXPONENTS ( 3 ALPHA ) AND ( 2 ALPHA ) OF g, POWER OF g IS NOT MINIMAL -- SKIP THIS VALUE OF ALPHA. ---------------------------------------------------------------- FOR EXPONENTS ( 3 ALPHA ) AND ( ALPHA - 2 ) OF g, WITH alpha = - 1 , POWER OF g is - 3 ----> SOLVE FOR U 0 1 TERM - 2 SQRT(2) (U - 1) U (U + 1) -- IS DOMINANT 0 0 0 3 G IN EQUATION. ---------------------------------------------------------------- 1 ) WITH U = 1 ---> FIND RESONANCES 0 ALPHA R + ALPHA SUBSTITUTE U G + U G FOR f IN EQUATION 0 R R - 3 TERM ( SQRT(2) (R - 4) (R + 1) ) U G IS DOMINANT R IN EQUATION. THE ONLY NON-NEGATIVE INTEGRAL ROOT IS [R = 4] WITH MAXIMUM RESONANCE = 4 ----> CHECK RESONANCES. 4 ==== \ K - 1 SUBSTITUTE POWER SERIES > U G FOR f IN EQUATION. / K ==== K = 0 WITH U = 1 0 1 * COEFFICIENT OF -- IS - (C - 2 A + 6 SQRT(2) U - 2) 2 1 G C - 2 A - 2 U = - ----------- 1 6 SQRT(2) 2 2 SQRT(2) (C - 4 A + 4 A + 72 U - 4) 1 2 * COEFFICIENT OF - IS - ------------------------------------- G 12 2 2 C - 4 A + 4 A - 4 U = - ------------------- 2 72 * COEFFICIENT OF 1 IS 3 2 3 2 C - 3 A C + 3 A C - 3 C - 2 A + 3 A + 3 A + 108 SQRT(2) U - 2 3 - ------------------------------------------------------------------ 27 (C - 2 A + 1) (C + A - 2) (C + A + 1) U = - ------------------------------------- 3 108 SQRT(2) C (C - 2 A + 1) (C + A - 2) (C + A + 1) * COEFFICIENT OF G IS - --------------------------------------- 27 SQRT(2) U IS ARBITRARY ? 4 C (C - 2 A + 1) (C + A - 2) (C + A + 1) COMPATIBILITY CONDITION: - --------------------------------------- 27 SQRT(2) = 0, *** CONDITION IS NOT SATISFIED. *** *** CHECK FOR FREE PARAMETERS OR PRESENCE OF U . *** 0 ---------------------------------------------------------------- 2 ) WITH U = - 1 ---> FIND RESONANCES 0 ALPHA R + ALPHA SUBSTITUTE U G + U G FOR f IN EQUATION 0 R R - 3 TERM ( SQRT(2) (R - 4) (R + 1) ) U G IS DOMINANT R IN EQUATION. THE ONLY NON-NEGATIVE INTEGRAL ROOT IS [R = 4] WITH MAXIMUM RESONANCE = 4 ----> CHECK RESONANCES. 4 ==== \ K - 1 SUBSTITUTE POWER SERIES > U G FOR f IN EQUATION. / K ==== K = 0 WITH U = - 1 0 1 * COEFFICIENT OF -- IS C + 2 A - 6 SQRT(2) U + 2 2 1 G C + 2 A + 2 U = ----------- 1 6 SQRT(2) 2 2 SQRT(2) (C - 4 A + 4 A - 72 U - 4) 1 2 * COEFFICIENT OF - IS ------------------------------------- G 12 2 2 C - 4 A + 4 A - 4 U = ------------------- 2 72 * COEFFICIENT OF 1 IS 3 2 3 2 C - 3 A C + 3 A C - 3 C + 2 A - 3 A - 3 A - 108 SQRT(2) U + 2 3 ------------------------------------------------------------------ 27 (C - A - 1) (C - A + 2) (C + 2 A - 1) U = ------------------------------------- 3 108 SQRT(2) C (C - A - 1) (C - A + 2) (C + 2 A - 1) * COEFFICIENT OF G IS --------------------------------------- 27 SQRT(2) U IS ARBITRARY ? 4 C (C - A - 1) (C - A + 2) (C + 2 A - 1) COMPATIBILITY CONDITION: --------------------------------------- = 0, 27 SQRT(2) *** CONDITION IS NOT SATISFIED. *** *** CHECK FOR FREE PARAMETERS OR PRESENCE OF U . *** 0 ---------------------------------------------------------------- FOR EXPONENTS ( 2 ALPHA ) AND ( ALPHA - 2 ) OF g, POWER OF g IS NOT MINIMAL -- SKIP THIS VALUE OF ALPHA. ---------------------------------------------------------------- (C16) 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 <= [4] and 1 <= l <= [2] ) 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 <= [2] ) 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 <= [2] ) j stands for j_th alpha,k stands for k_th solution set for u[0] ---------------------------------------------------------------- TO SEE THIS MENU AGAIN JUST TYPE < output() > ---------------------------------------------------------------- (C17) /* ************************** END of NP_EXEC.MAX ************************** */ (D17) DONE (C18) closefile();