(D13) CSM$USERS:[WHEREMAN.PROGRAMS.NPAINLEVE.SINGLE]P_VARCO.OUT;1 (C14) batch("np_exec.max"); (C15) /* ************************************************************************* */ /* Batch file NP_EXEC.MAX */ /* ************************************************************************* */ exec_painleve (eq, alpha, do_resonances, max_resonance, do_simplification)$ SUBSTITUTE X ----> G + X0 ---------------------------------------------------------------- 2 3 PAINLEVE ANALYSIS OF EQUATION, - 4 F A(G + X0) - 8 F (G + X0) + 2 F F G G 2 4 - (F ) - 3 F - 2 B = 0 G ---------------------------------------------------------------- ALPHA SUBSTITUTE U G FOR f IN ORIGINAL EQUATION. 0 MINIMUM POWERS OF g ARE [2 ALPHA - 2, 4 ALPHA, 3 ALPHA, 0] 2 ALPHA - 2 2 * COEFFICIENT OF G IS U (ALPHA - 2) ALPHA 0 NOTE : THIS TERM VANISHES FOR [ALPHA = 0, 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. 4 ALPHA 4 * COEFFICIENT OF G IS - 3 U 0 3 ALPHA 3 * COEFFICIENT OF G IS - 8 U X0 0 * COEFFICIENT OF 1 IS - 2 B ---------------------------------------------------------------- FOR EXPONENTS ( 2 ALPHA - 2 ) AND ( 4 ALPHA ) OF g, WITH alpha = - 1 , POWER OF g is - 4 ----> SOLVE FOR U 0 2 1 TERM - 3 (U - 1) U (U + 1) -- IS DOMINANT 0 0 0 4 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 - 4 TERM ( 2 (R - 3) (R + 1) ) U G IS DOMINANT R IN EQUATION. THE ONLY NON-NEGATIVE INTEGRAL ROOT IS [R = 3] WITH MAXIMUM RESONANCE = 3 ----> CHECK RESONANCES. 3 ==== \ K - 1 SUBSTITUTE POWER SERIES > U G FOR f IN EQUATION. / K ==== K = 0 WITH U = 1 0 1 * COEFFICIENT OF -- IS - 8 (X0 + U ) 3 1 G U = - X0 1 1 2 * COEFFICIENT OF -- IS - 2 (2 A(G + X0) - 3 X0 + 3 U + 4) 2 2 G 2 2 A(G + X0) - 3 X0 + 4 U = - ----------------------- 2 3 1 * COEFFICIENT OF - IS 8 X0 G U IS ARBITRARY ? 3 COMPATIBILITY CONDITION: 8 X0 = 0, *** CONDITION IS NOT SATISFIED. *** *** CHECK FOR FREE PARAMETERS OR PRESENCE OF U . *** 0 3 * COEFFICIENT OF X0 IN NUMERATOR IS 2 ---------------------------------------------------------------- 2 ) WITH U = - 1 ---> FIND RESONANCES 0 ALPHA R + ALPHA SUBSTITUTE U G + U G FOR f IN EQUATION 0 R R - 4 TERM ( - 2 (R - 3) (R + 1) ) U G IS DOMINANT R IN EQUATION. THE ONLY NON-NEGATIVE INTEGRAL ROOT IS [R = 3] WITH MAXIMUM RESONANCE = 3 ----> CHECK RESONANCES. 3 ==== \ K - 1 SUBSTITUTE POWER SERIES > U G FOR f IN EQUATION. / K ==== K = 0 WITH U = - 1 0 1 * COEFFICIENT OF -- IS 8 (X0 + U ) 3 1 G U = - X0 1 1 2 * COEFFICIENT OF -- IS - 2 (2 A(G + X0) - 3 X0 - 3 U - 4) 2 2 G 2 2 A(G + X0) - 3 X0 - 4 U = ----------------------- 2 3 1 * COEFFICIENT OF - IS 8 X0 G U IS ARBITRARY ? 3 COMPATIBILITY CONDITION: 8 X0 = 0, *** CONDITION IS NOT SATISFIED. *** *** CHECK FOR FREE PARAMETERS OR PRESENCE OF U . *** 0 3 * COEFFICIENT OF X0 IN NUMERATOR IS 2 ---------------------------------------------------------------- FOR EXPONENTS ( 2 ALPHA - 2 ) AND ( 3 ALPHA ) OF g, POWER OF g IS NOT MINIMAL -- SKIP THIS VALUE OF ALPHA. ---------------------------------------------------------------- FOR EXPONENTS ( 2 ALPHA - 2 ) AND ( 0 ) OF g, WITH alpha = 1 , POWER OF g is 0 ----> SOLVE FOR U 0 2 TERM - (2 B + U ) 1 IS DOMINANT 0 IN EQUATION. ---------------------------------------------------------------- 1 ) WITH U = - SQRT(2) SQRT(- B) ----> FIND RESONANCES 0 ALPHA R + ALPHA SUBSTITUTE U G + U G FOR f IN EQUATION 0 R R TERM ( - 2 SQRT(2) SQRT(- B) (R - 1) (R + 1) ) U G IS DOMINANT R IN EQUATION. THE ONLY NON-NEGATIVE INTEGRAL ROOT IS [R = 1] WITH MAXIMUM RESONANCE = 1 ----> CHECK RESONANCES. 1 ==== \ K + 1 SUBSTITUTE POWER SERIES > U G FOR f IN EQUATION. / K ==== K = 0 WITH U = - SQRT(2) SQRT(- B) 0 * COEFFICIENT OF G IS 0 U IS ARBITRARY ! 1 COMPATIBILITY CONDITION IS SATISFIED ! ---------------------------------------------------------------- 2 ) WITH U = SQRT(2) SQRT(- B) ----> FIND RESONANCES 0 ALPHA R + ALPHA SUBSTITUTE U G + U G FOR f IN EQUATION 0 R R TERM ( 2 SQRT(2) SQRT(- B) (R - 1) (R + 1) ) U G IS DOMINANT R IN EQUATION. THE ONLY NON-NEGATIVE INTEGRAL ROOT IS [R = 1] WITH MAXIMUM RESONANCE = 1 ----> CHECK RESONANCES. 1 ==== \ K + 1 SUBSTITUTE POWER SERIES > U G FOR f IN EQUATION. / K ==== K = 0 WITH U = SQRT(2) SQRT(- B) 0 * COEFFICIENT OF G IS 0 U IS ARBITRARY ! 1 COMPATIBILITY CONDITION IS SATISFIED ! ---------------------------------------------------------------- FOR EXPONENTS ( 4 ALPHA ) AND ( 3 ALPHA ) OF g, POWER OF g IS NOT MINIMAL -- SKIP THIS VALUE OF ALPHA. ---------------------------------------------------------------- FOR EXPONENTS ( 4 ALPHA ) AND ( 0 ) OF g, alpha = 0 ALREADY DONE! ---------------------------------------------------------------- FOR EXPONENTS ( 3 ALPHA ) AND ( 0 ) OF g, alpha = 0 ALREADY DONE! ---------------------------------------------------------------- (C16) output()$ ---------------------------------------------------------------- AT THE END OF THE COMPUTATIONS THE FOLLOWING ARE AVAILABLE: * U VALUE(S) (type uval[j,k,l] where 1 <= j <= 2 and 0 <= k <= [3, 1] and 1 <= l <= [2, 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 <= 2 ) j stands for j_th alpha * COMPATIBILITY CONDITION(S) (type compcond[j,k] where 1 <= j <= 2 and 1 <= k <= [2, 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 <= 2 and 1 <= k <= [2, 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();