Defining the Hirota operator

Question

The Hirota $D$-operator (derivative) is mathematically defined as follows:

$$D_x^n f\cdot g=\left.\frac{\partial^n}{\partial s ^n} f(x+s)g(x-s)\right|_{s=0} $$

An example of this operator acting on two functions $a(x)$ and $b(x)$ is the following:

$$D_x a(x)\cdot b(x) = a'(x)b(x)-a(x)b'(x)$$

I'm trying to make a Hirota $D$-operator function in Mathematica. What I've tried is the following

HirotaD[a[x_], b[x_], n_] :=
  Module[{},
   sol = D[a[x + y]*b[x - y], {y, n}] /. y -> 0 //

     TraditionalForm;
   Print[("\!\(\*SubscriptBox[\(D\), \(x\)]\)")^n, "=", sol]
   ];

This appears to work at first when I simply use arbitrary functions a[x] and b[z] as input functions:

HirotaD[a[x], b[x], 1]
(* ==> Subscript[D, x]=b(x)a'(x)-a(x) b'(x)  *)

However, it fails to output anything when I use any predefined functions.

f[x_]:=Sin[x];
g[x_]:=Cos[x];

HirotaD[f[x],g[x],1]
(* ==> HirotaD[Sin[x], Cos[x], 1] *)

How do I make it work on predefined functions?

Answer 1

The following definition takes as arguments two pure functions, a and b, their argument x and the parameter n.

HirotaD[a_, b_, x_, n_] := 
  Module[{}, 
   sol = D[a[x + y]*b[x - y], {y, n}] /. y -> 0 // TraditionalForm;
   Print[("\!\(\*SubscriptBox[\(D\), \(x\)]\)")^n, "=", sol]];

It works on general functions, not yet defined.

HirotaD[a, b, x, 1]
(* Subscript[D, x]=b(x) a'(x)-a(x) b'(x) *)

On built in functions, like Sin and Cos

HirotaD[Sin, Cos, x, 1]
(* Subscript[D, x]=sin^2(x)+cos^2(x) *)

And on products of functions, but for more complicated uses, one has to define a pure function, as follows:

HirotaD[a[#] b[#] &, u, x, 1]
(* Subscript[D, x]=b(x) u(x) a'(x)+a(x) u(x) b'(x)-a(x) b(x) u'(x) *)

Answer 2

I thought I'd offer a definition that works more like Mathematica's D function:

HirotaD[a_, b_, x_, n_] := Module[{fa, fb, d, s}, (
   fa = a /. (x -> (x + s));
   fb = b /. (x -> (x - s));
   d = D[fa fb, {s, n}];
   d /. s -> 0 // Simplify
   )]

HirotaD[a_, b_, x_] := HirotaD[a, b, x, 1]

Which you can call like

HirotaD[a[x], b[x], x] (* General Example *)
HirotaD[Sin[x], Cos[x], x] (* Should give 1 *)
HirotaD[Sin[y], Cos[y], x] (* df[y]/dx = 0, should give 0 *)
HirotaD[a[x], b[x], x, 2] (* Second order *)

which gives

$b(x) a'(x) - a(x) b'(x)$

$0$

$1$

$b(x) a''(x)-2 a'(x) b'(x)+a(x) b''(x)$