"/*********************************************************/"
"/*    WELCOME TO THE MATHEMATICA PROGRAM HIROTA.M        */"
"/*            BY WILLY HEREMAN AND WUNING ZHUANG         */"
"/*              FOR THE CALCULATION OF SOLITONS          */"
"/*           OF THE ""Shallow_Water"" EQUATION                   */"
"/*              WITH HIROTA'S METHOD                     */"
"/*     Version 1.0 firts released on May 29, 1995        */"
"/*      Last updated on January 25, 2007                 */"
"/*               Copyright 1995-2007                     */"
"/*********************************************************/"
"The equation in f corresponding to the given bilinear operator is "
Derivative[0, 0, 0, 1][f][x, y, z, t]*Derivative[1, 0, 0, 0][f][x, y, z, t] + 
   Derivative[1, 0, 0, 0][f][x, y, z, t]^2 - 
   f[x, y, z, t]*Derivative[1, 0, 0, 1][f][x, y, z, t] - 
   f[x, y, z, t]*Derivative[2, 0, 0, 0][f][x, y, z, t] + 
   3*Derivative[1, 0, 0, 1][f][x, y, z, t]*
    Derivative[2, 0, 0, 0][f][x, y, z, t] - 
   3*Derivative[1, 0, 0, 0][f][x, y, z, t]*
    Derivative[2, 0, 0, 1][f][x, y, z, t] - 
   Derivative[0, 0, 0, 1][f][x, y, z, t]*
    Derivative[3, 0, 0, 0][f][x, y, z, t] + 
   f[x, y, z, t]*Derivative[3, 0, 0, 1][f][x, y, z, t]" = 0"
"For this equation the polynomial P(K,-OMEGA,L)= "-K^2 + K*OMEGA - K^3*OMEGA
"The equation has at least a one- and two-soliton solution."
"For the ""Shallow_Water"" equation, "
"we use the dispersion relation: "
" OMEGA[I] = "-(K[I]/(-1 + K[I]^2))
"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): "
"for this test K["1"] = "20
"for this test K["2"] = "18
"for this test K["3"] = "9
"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): "
"for this test K["1"] = "16
"for this test K["2"] = "2
"for this test K["3"] = "14
"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): "
"for this test K["1"] = "8
"for this test K["2"] = "16
"for this test K["3"] = "18
"for this test K["4"] = "2
"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): "
"for this test K["1"] = "9
"for this test K["2"] = "6
"for this test K["3"] = "19
"for this test K["4"] = "14
"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."
"The polynomial is P[K,-OMEGA,L] = "-K^2 + K*OMEGA - K^3*OMEGA
"The coefficient a[I,J] = "((K[I] - K[J])^2*
     (-3 + K[I]^2 - K[I]*K[J] + K[J]^2))/
   ((K[I] + K[J])^2*(-3 + K[I]^2 + K[I]*K[J] + K[J]^2))
"The coefficient b[1,2,3] is calculated via the polynomial form."
"The coefficient b[1,2,3] = "((K[1] - K[2])^2*
     (-3 + K[1]^2 - K[1]*K[2] + K[2]^2)*(K[1] - K[3])^2*(K[2] - K[3])^2*
     (-3 + K[1]^2 - K[1]*K[3] + K[3]^2)*(-3 + K[2]^2 - K[2]*K[3] + K[3]^2))/
   ((K[1] + K[2])^2*(-3 + K[1]^2 + K[1]*K[2] + K[2]^2)*(K[1] + K[3])^2*
     (K[2] + K[3])^2*(-3 + K[1]^2 + K[1]*K[3] + K[3]^2)*
     (-3 + K[2]^2 + K[2]*K[3] + K[3]^2))
"The function f = "1 + E^THETA[1] + E^THETA[2] + E^THETA[3] + 
   E^(THETA[1] + THETA[2])*a[1, 2] + E^(THETA[1] + THETA[3])*a[1, 3] + 
   E^(THETA[2] + THETA[3])*a[2, 3] + 
   E^(THETA[1] + THETA[2] + THETA[3])*b[1, 2, 3]
"At the end of the computations the form of the function f"
"and the coefficients a[i,j] and b[1,2,3] are explicitly available."
"The explicit forms of OMEGA[i] and THETA[i] are also available."
"The Explicit form of f can be obtained by typing EXPRF."