(D2)               CSM$USERS:[WHEREMAN.HIROTA]H_KDV.OUT;4

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

Batching the file hir_sing.max
Batchload done.

(C4) n:3$

(C5) B(f,g):=Dxt[1,1](f,g) + Dx[4](f,g)$

(C6) name:KdV$

(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  KDV  EQUATION                 */

/*              WITH HIROTA'S METHOD                     */

/*        Version 1, released on May 29, 1995             */

/*                 Copyright 1995                         */

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

The equation in f corresponding to the given bilinear operator is 

   4          3        2                  2
  d F     dF d F      d F 2   dF dF      d F
F --- - 4 -- --- + 3 (---)  - -- -- + F -----  = 0
    4     dX   3        2     dT dX     dT dX
  dX         dX       dX

                                                   4
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  KDV  equation, 

We use the dispersion relation 

              3
 OMEGA[I] =  K
              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   =  10
 1

K   =  2
 2

K   =  15
 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   =  9
 1

K   =  3
 2

K   =  5
 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   =  18
 1

K   =  5
 2

K   =  14
 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   =  3
 1

K   =  6
 2

K   =  11
 3

K   =  16
 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   =  12
 1

K   =  17
 2

K   =  4
 3

K   =  15
 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.

                                    4
The polynomial is P(K,-OMEGA,L) =  K  - K OMEGA

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

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

                                     2          2          2
                            (K  - K )  (K  - K )  (K  - K )
                              2    1     3    1     3    2
The coefficient b[1,2,3] =  --------------------------------
                                     2          2          2
                            (K  + K )  (K  + K )  (K  + K )
                              2    1     3    1     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= 67750 msecs

(C8) /* end of command file */

Accumulated Computation Time= 71750 msecs
Time= 72070 msecs

/* Rest of the output deleted */
/* Output of graphics is missing here */

(D8)                                  DONE

(C9) QUIT();