Time= 30 msecs (D2) CSM$USERS:[WHEREMAN.HIROTA]H_ITOB6.OUT;1 (C3) batchload("hir_sing.max")$ Batching the file hir_sing.max Batchload done. Time= 4790 msecs (C4) n:3$ Time= 0 msecs (C5) B(f,g):= Dxt[0,2](f,g) + Dxt[3,1](f,g) +Dy[2](f,g)$ Time= 10 msecs (C6) name:Ito_b6$ Time= 0 msecs (C7) hirota(b,name,n,2,2,true,true)$ /*********************************************************/ /* WELCOME TO THE MACSYMA PROGRAM HIR_SING.MAX */ /* BY WILLY HEREMAN AND WUNING ZHUANG */ /* FOR THE CALCULATION OF SOLITONS */ /* OF THE ITO_B6 EQUATION */ /* WITH HIROTA'S METHOD */ /* Version 1.0, released on May 29, 1995 */ /*********************************************************/ The equation in f corresponding to the given bilinear operator is 2 3 2 2 3 2 4 d F dF 2 dF d F d F d F d F dF d F d F dF 2 F --- - (--) - -- --- + 3 ----- --- - 3 ------ -- + F --- + F ------ - (--) 2 dY dT 3 dT dX 2 2 dX 2 3 dT dY dX dX dT dX dT dT dX = 0 2 3 2 For this equation the polynomial P(K,-OMEGA,L) = OMEGA - K OMEGA + L The equation has at least a one- and two-soliton solution. For the ITO_B6 equation, there are 2 different dispersion relations. 6 2 3 6 2 3 SQRT(K - 4 L ) - K SQRT(K - 4 L ) + K I I I I I I [OMEGA = - --------------------, OMEGA = --------------------] I 2 I 2 We use the dispersion relation 6 2 3 SQRT(K - 4 L ) - K I I I OMEGA[I] = - -------------------- 2 In the expansion of f we use THETA = K X - OMEGA T + L Y + CST. Starting the random test(s) for the existence of a 3 soliton solution. Wavenumbers k[i] selected for the random number test(s): K = 6 1 K = 4 2 K = 9 3 Wavenumbers l[i] selected for the random number test(s): L = 4 1 L = 9 2 L = 12 3 The equation did not pass the random number test(s) for the existence of a 3 soliton solution. The condition 6718464 (1321236 SQRT(182) SQRT(943) SQRT(58985) + 50297649 SQRT(943) SQRT(58985) - 21006728 SQRT(182) SQRT(58985) - 2891324712 SQRT(58985) - 233115900 SQRT(182) SQRT(943) - 13563413451 SQRT(943) + 7651085448 SQRT(182) + 672844873936) = 0 must be satisfied. Starting the construction of the two-soliton solution. The coefficient a[i,j] is calculated via the polynomial form. 2 3 2 The polynomial is P(K,-OMEGA,L) = OMEGA - K OMEGA + L 2 6 2 2 6 2 The coefficient a[i,j] = (3 K K SQRT(K - 4 L ) - 3 K K SQRT(K - 4 L ) I J J J I J J J 6 2 6 2 5 2 4 3 3 + SQRT(K - 4 L ) SQRT(K - 4 L ) + 4 L L - 3 K K + 3 K K - K K I I J J I J I J I J I J 6 2 2 4 2 2 6 2 5 - 3 K SQRT(K - 4 L ) K + 3 K K + 3 K SQRT(K - 4 L ) K - 3 K K ) I I I J I J I I I J I J 2 6 2 2 6 2 /(3 K K SQRT(K - 4 L ) + 3 K K SQRT(K - 4 L ) I J J J I J J J 6 2 6 2 5 2 4 3 3 + SQRT(K - 4 L ) SQRT(K - 4 L ) + 4 L L - 3 K K - 3 K K - K K I I J J I J I J I J I J 6 2 2 4 2 2 6 2 5 + 3 K SQRT(K - 4 L ) K - 3 K K + 3 K SQRT(K - 4 L ) K - 3 K K ) I I I J I J I I I J I J THETA + THETA THETA THETA 2 1 2 1 The function f = A %E + %E + %E + 1 1, 2 At the end of the computations the form of the function f and the coefficient a[1,2] are explicitly available. The explicit factored form of a[1,2] can be obtained by typing factor(a[1,2]); The explicit forms of theta[i] and omega[i] are also available. The form of f can be obtained by typing f; . The explicit form of f can be obtained by typing expression(f); . Time= 22860 msecs (C8) kill(all)$ Time= 40 msecs (C1) closefile();