Fundamental Properties of the Binary Operators in Soliton Theory and Their Generalization

Abstract

We have introduced a binary operator to transform a class of nonlinear evolution equations into the bilinear form

$$F({D_x},{D_t},{\mkern 1mu} ...)f\cdot g = 0$$

The binary operator D ⁿ_x operating on a pair of arbitrary functions f and g is defined by

$${D_x}^nf\left( x \right)\cdot g\left( x \right) = {\left( {\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial x'}}} \right)^n}f\left( x \right)g\left( {x'} \right)\left| {x'} \right. = x.$$

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