Time= 40 msecs

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

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

Batching the file hir_sing.max
Batchload done.

Time= 4140 msecs

(C4) n:3$

Time= 0 msecs

(C5) B(f,g):= Dxt[3,1](f,g) - Dx[2](f,g) - Dxt[1,1](f,g)$

Time= 0 msecs

(C6) name:Shallow_Water$

Time= 0 msecs

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

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

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

/*            BY WILLY HEREMAN AND WUNING ZHUANG         */

/*              FOR THE CALCULATION OF SOLITONS          */

/*             OF THE  SHALLOW_WATER  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 


    3        2    2       2                3                     4          2
dF d F      d F  d F     d F    dF 2      d F   dF   dF dF      d F        d F
-- --- - 3 ----- --- + F --- - (--)  + 3 ------ -- - -- -- - F ------ + F ----- 
dT   3     dT dX   2       2    dX            2 dX   dT dX          3     dT dX
   dX            dX      dX              dT dX                 dT dX

                                                                             = 0


                                                        3           2
For this equation the polynomial P(K,-OMEGA,L) =  (K - K ) OMEGA - K

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

For the  SHALLOW_WATER  equation, 

We use the dispersion relation 

                 K
                  I
 OMEGA[I] =  - ------
                2
               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   =  10
 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   =  15
 1

K   =  8
 2

K   =  5
 3

K   =  20
 4

The equation passed the random number test(s) 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.

                                         3           2
The polynomial is P(K,-OMEGA,L) =  (K - K ) OMEGA - K

                                   2   2            2
                          (K  - K )  (K  - K  K  + K  - 3)
                            J    I     J    I  J    I
The coefficient a[i,j] =  --------------------------------
                                   2   2            2
                          (K  + K )  (K  + K  K  + K  - 3)
                            J    I     J    I  J    I

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

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

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

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

   2            2
 (K  + K  K  + K  - 3))
   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= 90430 msecs

(C8) kill(all)$

Time= 40 msecs

(C1) closefile();