Time= 30 msecs

(D2)              CSM$USERS:[WHEREMAN.HIROTA]H_ITOB8.OUT;1

(C3) batchload("hir_sing.max")$

Batching the file hir_sing.max
Batchload done.

Time= 4220 msecs

(C4) n:3$

Time= 0 msecs

(C5) B(f,g):= - Dxt[2,2](f,g) - Dxt[0,2](f,g) + Dx[2](f,g)$

Time= 0 msecs

(C6) name:Ito_b8$

Time= 10 msecs

(C7) hirota(b,name,n,1,1,true,true)$

/*********************************************************/

/*    WELCOME TO THE MACSYMA PROGRAM HIR_SING.MAX        */

/*            BY WILLY HEREMAN AND WUNING ZHUANG         */

/*              FOR THE CALCULATION OF SOLITONS          */

/*             OF THE  ITO_B8  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   2       2                3              4         2           3
d F d F     d F    dF 2      d F   dF       d F       d F     dF  d F
--- --- - F --- + (--)  - 2 ------ -- + F ------- + F --- - 2 -- ------
  2   2       2    dX         2    dX       2   2       2     dT      2
dT  dX      dX              dT  dX        dT  dX      dT         dT dX

                                                              2
                                                             d F  2    dF 2
                                                       + 2 (-----)  - (--)   = 0
                                                            dT dX      dT


                                                      2           2    2
For this equation the polynomial P(K,-OMEGA,L) =  (- K  - 1) OMEGA  + K

The equation has at least a one- and two-soliton solution.

For the  ITO_B8  equation, 

there are  2  different dispersion relations.

                 K                      K
                  I                      I
[OMEGA  = - ------------, OMEGA  = ------------]
      I           2            I         2
            SQRT(K  + 1)           SQRT(K  + 1)
                  I                      I

We use the dispersion relation 

                    K
                     I
 OMEGA[I] =  - ------------
                     2
               SQRT(K  + 1)
                     I

In the expansion of f we use THETA = K X - OMEGA T + 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   =  21
 1

K   =  8
 2

K   =  17
 3

The equation did not pass the random number test(s) for 

the existence of a 3 soliton solution.

The condition  162505211904 (481022565 SQRT(290) SQRT(442)

 - 826179090 SQRT(65) SQRT(442) + 2099992138 SQRT(65) SQRT(290) - 339776325320)

/435358893125  = 0 must be satisfied.

Starting the construction of the two-soliton solution.

The coefficient a[i,j] is calculated via the polynomial form.

                                       2           2    2
The polynomial is P(K,-OMEGA,L) =  (- K  - 1) OMEGA  + K

                                   2       2       2
The coefficient a[i,j] =  (2 SQRT(K  + 1) K  SQRT(K  + 1)
                                   I       J       J

              2               2           2       2            2
 - 4 K  SQRT(K  + 1) K  SQRT(K  + 1) + 2 K  SQRT(K  + 1) SQRT(K  + 1)
      I       I       J       J           I       I            J

           2            2            3      2  2    3
 + 2 SQRT(K  + 1) SQRT(K  + 1) - K  K  + 2 K  K  - K  K  - 2 K  K  - 2)
           I            J         I  J      I  J    I  J      I  J

          2       2       2                   2               2
/(2 SQRT(K  + 1) K  SQRT(K  + 1) + 4 K  SQRT(K  + 1) K  SQRT(K  + 1)
          I       J       J           I       I       J       J

      2       2            2                2            2            3
 + 2 K  SQRT(K  + 1) SQRT(K  + 1) + 2 SQRT(K  + 1) SQRT(K  + 1) + K  K
      I       I            J                I            J         I  J

      2  2    3
 + 2 K  K  + K  K  + 2 K  K  - 2)
      I  J    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= 23950 msecs

(C8) kill(all)$

Time= 40 msecs

(C1) closefile();