"/*********************************************************/"
"/*    WELCOME TO THE MATHEMATICA PROGRAM HIROTA.M        */"
"/*            BY WILLY HEREMAN AND WUNING ZHUANG         */"
"/*              FOR THE CALCULATION OF SOLITONS          */"
"/*           OF THE ""Ito-b8"" 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]^2 - 
   f[x, y, z, t]*Derivative[0, 0, 0, 2][f][x, y, z, t] - 
   Derivative[1, 0, 0, 0][f][x, y, z, t]^2 - 
   2*Derivative[1, 0, 0, 1][f][x, y, z, t]^2 + 
   2*Derivative[1, 0, 0, 0][f][x, y, z, t]*
    Derivative[1, 0, 0, 2][f][x, y, z, t] + 
   f[x, y, z, t]*Derivative[2, 0, 0, 0][f][x, y, z, t] - 
   Derivative[0, 0, 0, 2][f][x, y, z, t]*
    Derivative[2, 0, 0, 0][f][x, y, z, t] + 
   2*Derivative[0, 0, 0, 1][f][x, y, z, t]*
    Derivative[2, 0, 0, 1][f][x, y, z, t] - 
   f[x, y, z, t]*Derivative[2, 0, 0, 2][f][x, y, z, t]" = 0"
"For this equation the polynomial P(K,-OMEGA,L)= "K^2 - OMEGA^2 - K^2*OMEGA^2
"The equation has at least a one- and two-soliton solution."
"For the ""Ito-b8"" equation, "
"we use the dispersion relation: "
" OMEGA[I] = "-(K[I]/Sqrt[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"] = "7
"for this test K["2"] = "11
"for this test K["3"] = "5
"The equation did not pass the random number test(s) for "
"the existence of a "3" soliton solution."
"The condition "(-10124075808*(1107738469 - 299773500*Sqrt[13] - 
       23820500*Sqrt[61] + 6425679*Sqrt[793]))/12466931425" = 0 must be \
satisfied."
"Starting the construction of the two-soliton solution."
"The coefficient a[I,J] is calculated via the polynomial form."
"The polynomial is P[K,-OMEGA,L] = "K^2 - OMEGA^2 - K^2*OMEGA^2
"The coefficient a[I,J] = "(-2 - 2*K[I]*K[J] - K[I]^3*K[J] + 
     2*K[I]^2*K[J]^2 - K[I]*K[J]^3 + 2*Sqrt[1 + K[I]^2]*Sqrt[1 + K[J]^2] + 
     2*K[I]^2*Sqrt[1 + K[I]^2]*Sqrt[1 + K[J]^2] - 
     4*K[I]*Sqrt[1 + K[I]^2]*K[J]*Sqrt[1 + K[J]^2] + 
     2*Sqrt[1 + K[I]^2]*K[J]^2*Sqrt[1 + K[J]^2])/
   (-2 + 2*K[I]*K[J] + K[I]^3*K[J] + 2*K[I]^2*K[J]^2 + K[I]*K[J]^3 + 
     2*Sqrt[1 + K[I]^2]*Sqrt[1 + K[J]^2] + 
     2*K[I]^2*Sqrt[1 + K[I]^2]*Sqrt[1 + K[J]^2] + 
     4*K[I]*Sqrt[1 + K[I]^2]*K[J]*Sqrt[1 + K[J]^2] + 
     2*Sqrt[1 + K[I]^2]*K[J]^2*Sqrt[1 + K[J]^2])
"The function f = "1 + E^THETA[1] + E^THETA[2] + 
   E^(THETA[1] + THETA[2])*a[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 forms of OMEGA[i] and THETA[i] are also available."
"The Explicit form of f can be obtained by typing EXPRF."