with(DEtools):

# === Hirota Bilinear Method ===
# This script provides tools to work with Hirota bilinear forms.
# It includes functions for handling bilinear derivatives, converting
# expressions to bilinear form, and printing results.

# === Bilinear Derivative / Hirota Operator ===
BD := proc(FF, DD)
    local f, g, x, m, opt;
    # Parse input arguments
    if nargs = 1 then
        return ‘*‘(FF[]);
    fi;
    f, g := FF[];
    x, m := DD[];
    opt := args[3..-1];
    
    # Handle case where m = 0 (base case)
    if m = 0 then
        return procname(FF, opt);
    fi;
    
    # Recursive definition of the Hirota operator
    procname([diff(f, x), g], [x, m-1], opt)
    - procname([f, diff(g, x)], [x, m-1], opt);
end:

# === Display Bilinear Derivative ===
‘print/BD‘ := proc(FF, DD)
    local f, g, x, m, i;
    f, g := FF[];
    f := cat(f, ‘ ‘, g);
    g := product(D[args[i][1]]^args[i][2], i=2..nargs);
    if g <> 1 then
        f := ‘‘(g) * ‘‘(f);
    fi;
    f;
end:

# === Collect Terms ===
getFnumer := proc(df, f, pow::posint := 1)
    local i, g, fdenom;
    # Ensure the expression is a sum
    if type(df, ‘+‘) then
        g := [op(df)];
        fdenom := map(denom, g);
        
        # Locate the term with denominator f^pow
        for i to nops(fdenom) while fdenom[i] <> f^pow do od;
        if i > nops(fdenom) then
            lprint(fdenom);
            error "No term(s) or numer=0 when denom=%1", op(0, f)^pow;
        fi;
        
        g := numer(g[i]);
        if not type(expand(g), ‘+‘) then
            lprint(g);
            error "Expected more than 1 term about Hirota D-operator";
        fi;
        
        return g;
    fi;
    
    lprint(df);
    error "Expected 1st argument to be type ‘+‘.";
end:

# === Extract Variable Powers ===
getvarpow := proc(df::function)
    local i, f, var, dif, pow;
    if op(0, df) <> diff then
        lprint(df);
        error "Expected diff function";
    fi;
    
    f := convert(df, D);
    var := [op(f)];
    dif := [op(op([0, 0], f))];
    pow := [0$nops(var)];
    f := op(op(0, f))(var[]);
    
    for i to nops(var) do
        dif := selectremove(member, dif, {i});
        pow[i] := nops(dif[1]);
        dif := dif[2];
    od;
    
    pow := zip((x, y) -> [x, y], var, pow);
    pow := remove(has, pow, {0});
    [[f, f], pow[]];
end:

# === Convert to Hirota Bilinear Form ===
HBF := proc(df)
    local i, c, f;
    
    if type(df, ‘+‘) then
        f := [op(df)];
        return map(procname, f);
    fi;
    
    if type(df, ‘*‘) then
        f := [op(df)];
        f := selectremove(hasfun, f, diff);
        c := f[2];
        f := f[1];
        if nops(f) <> 1 then
            lprint(df);
            error "Need only one diff function factor.";
        fi;
        f := f[];
        c := ‘*‘(c[]);
        f := getvarpow(f);
        f := [c, f];
        return f;
    fi;
    
    if op(0, df) = diff then
        f := getvarpow(df);
        f := [1, f];
        return f;
    fi;
    
    lprint(df);
    error "Unexpected type.";
end:

# === Print Hirota Bilinear Form ===
printHBF := proc(PL::list)
    local j, DD, f, C, tmp, gcdC;
    C := map2(op, 1, PL);
    gcdC := 1;
    
    if nops(C) > 1 then
        tmp := [seq(cat(_Z, i), i=1..nops(C))];
        gcdC := tmp *~ C;
        gcdC := ‘+‘(gcdC[]);
        gcdC := factor(gcdC);
        tmp := selectremove(has, gcdC, tmp);
        gcdC := tmp[2];
        if gcdC = 0 then
            gcdC := 1;
        fi;
        gcdC := gcdC * content(tmp[1]);
    fi;
    
    if gcdC <> 1 then
        C := C /~ gcdC;
    fi;
    
    DD := map2(op, 2, PL);
    f := op(0, DD[1][1][1]);
    DD := map(z -> product(D[z[i][1]]^z[i][2], i=2..nops(z)), DD);
    DD := zip(‘*‘, C, DD);
    DD := ‘+‘(DD[]);
    gcdC * ‘‘(DD) * cat(f, ‘ ‘, f);
end:

# === Print Hirota Bilinear Transform ===
printHBT := proc(uf, u, f, i, j, PL, alpha := 1)
    local DD, g, C, tmp, pl;
    pl := printHBF(PL);
    if j > 0 then
        print(u = 2 * alpha * ’diff’(ln(f), x$j));
    else
        print(u = 2 * alpha * ln(f));
    fi;
    
    if i > 0 then
        print(’diff’(pl/f^2, x$i) = 0);
    else
        print(pl/f^2 = 0);
    fi;
    
    NULL;
end:

# === Guess Differential Order ===
guessdifforder := proc(PL::list, x::name)
    local L, minorder, maxorder, tmp;
    L := map2(op, 2, PL);
    L := map(z -> z[2..-1], L);
    tmp := map(z -> map2(op, 2, z), L);
    tmp := map(z -> ‘+‘(z[]), tmp);
    tmp := selectremove(type, tmp, even);
    
    minorder := 0;
    if nops(tmp[1]) < nops(tmp[2]) then
        minorder := 1;
    fi;
    
    tmp := map(z -> select(has, z, {x}), L);
    tmp := map(z -> map2(op, 2, z), tmp);
    if has(tmp, {[]}) then
        maxorder := 0;
    else
        tmp := map(op, tmp);
        maxorder := min(tmp[]);
    fi;
    
    if type(maxorder - minorder, odd) then
        maxorder := maxorder - 1;
    fi;
    
    [minorder, maxorder];
end:

# === Guess Alpha Value ===
guessalpha := proc(Res, uf, u, f, i, j, PL, alpha)
    local tmp, res, pl, flag, k;
    flag := 1;
    tmp := [op(Res)];
    tmp := map(numer, tmp);
    tmp := gcd(tmp[1], tmp[-1]);
    
    if type(tmp, ‘*‘) then
        tmp := remove(has, tmp, f);
    fi;
    
    if tmp <> 0 and has(tmp, {alpha}) then
        tmp := solve(tmp/alpha^difforder(uf), {alpha});
        
        if tmp <> NULL and has(tmp, {alpha}) then
            lprint(tmp);
            for k to nops([tmp]) while flag = 1 do
                res := collect(expand(subs(tmp[k], Res)), f, factor);
                if res = 0 then
                    pl := subs(tmp[k], PL);
                    printHBT(uf, u, f, i, j, pl, rhs(tmp[k]));
                    flag := 0;
                fi;
            od;
        fi;
    fi;
    
    PL;
end:

Error, missing operator or `;`

## Hirota Bilinear Method
 ## Bilinear Derivative / Hirota Operator

BD := proc(FF, DD)
    local f, g, x, m, opt;

    # If only one argument is provided, return the product of elements in FF
    if nargs = 1 then
        return ‘*‘(FF[]);
    fi;

    # Extract elements from FF into f and g, and from DD into x and m
    f, g := FF[];
    x, m := DD[];

    # Capture any additional arguments passed to the procedure
    opt := args[3..-1];

    # If m (derivative order) is zero, recursively call the procedure with FF and optional arguments
    if m = 0 then
        return procname(FF, opt);
    fi;

    # Compute the difference of two recursive calls:
    # 1. Differentiate f with respect to x, and keep g unchanged, reducing m by 1
    # 2. Keep f unchanged and differentiate g with respect to x, reducing m by 1
    return procname([diff(f, x), g], [x, m-1], opt) - procname([f, diff(g, x)], [x, m-1], opt);
end:

#restart;

# Inladen van pakketten
with(PDEtools):
with(DEtools):
NULL;

# Declaraties
declare(u(x, t));
declare(f(x, t));
alpha := 1;

# Definities
KdV := diff(u(x, t), t) + 6 * u(x, t) * diff(u(x, t), x) + diff(u(x, t), x, x, x);
Hirota := u(x, t) = 2 * alpha * diff(ln(f(x, t)), x);

# Substitutie van Hirota-transformatie
KdV_Hirota := simplify(subs(Hirota, KdV));
KdV_Hirota := expand(KdV_Hirota / (f(x, t)^2));

# Berekening van bilineaire termen met BD
BD := proc(FF, DD)
    local f, g, x, m, opt;

    # Controle op invoer
    if nargs = 1 then
        return mul(FF[]);
    fi;

    # Ontleden van invoer
    f, g := op(FF);
    x, m := op(DD);
    opt := args[3 .. -1];

    # Basisgeval
    if m = 0 then
        return procname(FF, opt);
    fi;

    # Recursieve berekening
    procname([diff(f, x), g], [x, m-1], opt)
    - procname([f, diff(g, x)], [x, m-1], opt);
end:

# Debugging
debug_msg := proc(msg, val)
    print("[DEBUG]:", msg, val);
end:

# Voorbeeldberekening
try
    # FF en DD instellen
    FF := [f(x, t), f(x, t)];
    DD := [x, 2];

    # Bilineaire afgeleide berekenen
    BD_result := BD(FF, DD);
    debug_msg("Bilineaire afgeleide van de tweede orde", BD_result);

    # Hergebruiken van BD in KdV-Hirota
    Hirota_Bilinear := simplify(KdV_Hirota + BD_result);
    debug_msg("Herschreven KdV-vergelijking met bilineaire termen", Hirota_Bilinear);

    # Resultaat printen
    print("Bilineaire afgeleide:", BD_result);
    print("Herschreven vergelijking:", Hirota_Bilinear);

catch:
    debug_msg("Fout opgetreden", lasterror());
end try;

#restart;
with(PDEtools):
with(DEtools):
NULL;

# Declare variables
declare(u(x, t));  # Original function u(x, t)
declare(f(x, t));  # Auxiliary function f(x, t)
alpha := 1;        # Set alpha value for Hirota transformation

# Define the SK equation
SK := diff(u(x, t), t)
     + 45 * u(x, t)^2 * diff(u(x, t), x)
     + 15 * diff(u(x, t), x, x) * diff(u(x, t), x)
     + 15 * u(x, t) * diff(u(x, t), x, x, x)
     + diff(u(x, t), x, x, x, x, x) = 0;

# Hirota transformation
Hirota := u(x, t) = 2 * diff(diff(ln(f(x, t)), x), x);

# Substitute Hirota transformation into SK equation
SK_Hirota := subs(Hirota, SK);

# Simplify and normalize by f(x, t)^2
SK_Hirota := simplify(SK_Hirota / f(x, t)^2);

# Debugging output
debug_msg := proc(msg, val)
    print("[DEBUG]:", msg, val);
end:

debug_msg("Transformed SK Equation", SK_Hirota);

# Example: Compute bilinear derivative for the second-order term
FF := [f(x, t), f(x, t)];  # Input functions for BD
DD := [x, 2];             # Second-order derivative
try
    BD_result := BD(FF, DD);
    debug_msg("Bilinear derivative for second order", BD_result);
catch:
    debug_msg("Error in BD computation", lasterror());
end try;

# Combine SK_Hirota with bilinear derivative results
try
    SK_Bilinear := simplify(SK_Hirota + BD_result);
    debug_msg("Rewritten SK Equation with bilinear terms", SK_Bilinear);

    # Print final result
    print("Bilinear derivative result:", BD_result);
    print("Rewritten SK Equation:", SK_Bilinear);
catch:
    debug_msg("Error combining bilinear terms with SK equation", lasterror());
end try;