Time= 40 msecs

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

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

Batching the file hir_sing.max
Batchload done.

Time= 4170 msecs

(C4) n:3$

Time= 0 msecs

(C5) B(f,g):= Dxt[1,1](f,g) + Dx[6](f,g)$

Time= 0 msecs

(C6) name:Sawada_Kotera$

Time= 0 msecs

(C7) hirota(b,name,n,3,2,true,true)$

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

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

/*            BY WILLY HEREMAN AND WUNING ZHUANG         */

/*              FOR THE CALCULATION OF SOLITONS          */

/*             OF THE  SAWADA_KOTERA  EQUATION                 */

/*              WITH HIROTA'S METHOD                     */

/*        Version 1.0, released on May 29, 1995            */

/*                  Copyright 1995                         */

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

The equation in f corresponding to the given bilinear operator is 


   6          5        2   4         3                  2
  d F     dF d F      d F d F       d F 2   dF dF      d F
F --- - 6 -- --- + 15 --- --- - 10 (---)  - -- -- + F -----  = 0
    6     dX   5        2   4         3     dT dX     dT dX
  dX         dX       dX  dX        dX

                                                   6
For this equation the polynomial P(K,-OMEGA,L) =  K  - K OMEGA

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

For the  SAWADA_KOTERA  equation, 

We use the dispersion relation 

              5
 OMEGA[I] =  K
              I

WARNING!

The degree of K[I] in the dispersion law is  5

The random number test(s) may create very large integers.

This renders the test unreliable on some computers,

(due to storage problems with large integers).

Set the random-number test(s) equal to zero, and run

the symbolic test(s). 

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   =  8
 1

K   =  3
 2

K   =  12
 3

The equation passed the random number test(s) for the existence

of a 3 soliton solution.

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   =  3
 2

K   =  8
 3

The equation passed the random number test(s) for the existence

of a 3 soliton solution.

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   =  3
 2

K   =  12
 3

The equation passed the random number test(s) for the existence

of a 3 soliton solution.

Starting the symbolic test for the existence of a 3 soliton solution.

The equation passed the symbolic test for the existence

of a 3 soliton solution.

Starting the random test(s) for the existence of a 4 soliton solution.

Wavenumbers k[i] selected for the random number test(s): 

K   =  4
 1

K   =  6
 2

K   =  12
 3

K   =  14
 4

The equation passed the random number test(s) for the existence

of a 4 soliton solution.

Starting the random test(s) for the existence of a 4 soliton solution.

Wavenumbers k[i] selected for the random number test(s): 

K   =  14
 1

K   =  12
 2

K   =  6
 3

K   =  3
 4

The equation passed the random number test(s) for the existence

of a 4 soliton solution.

Starting the symbolic test for the existence of a 4 soliton solution.

The equation passed the symbolic test for the existence

of a 4 soliton solution.

Starting the construction of the three-soliton solution.

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

                                    6
The polynomial is P(K,-OMEGA,L) =  K  - K OMEGA


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


The coefficient b[1,2,3] is calculated via the polynomial form.

                                     2   2            2           2          2
The coefficient b[1,2,3] =  (K  - K )  (K  - K  K  + K ) (K  - K )  (K  - K )
                              2    1     2    1  2    1    3    1     3    2

   2            2    2            2            2   2            2           2
 (K  - K  K  + K ) (K  - K  K  + K )/((K  + K )  (K  + K  K  + K ) (K  + K )
   3    1  3    1    3    2  3    2     2    1     2    1  2    1    3    1

          2   2            2    2            2
 (K  + K )  (K  + K  K  + K ) (K  + K  K  + K ))
   3    2     3    1  3    1    3    2  3    2


                             THETA  + THETA  + THETA            THETA  + THETA
                                  3        2        1                3        2
The function f =  B        %E                         + A     %E
                   1, 2, 3                               2, 3

           THETA  + THETA      THETA            THETA  + THETA      THETA
                3        1          3                2        1          2
 + A     %E                + %E       + A     %E                + %E
    1, 3                                 1, 2

     THETA
          1
 + %E       + 1


At the end of the computations the form of the function f

and the coefficients a[i,j] and b[1,2,3] are available.

The explicit factored form of a[1,2] and b[1,2,3] can be obtained

by entering factor(a[1,2]); and factor(b[1,2,3]); 

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= 163250 msecs

(C8) kill(all)$

Time= 30 msecs

(C1) closefile();