Multilinear operators: the natural extension of Hirota's bilinear formalism

Abstract We introduce multilinear operators, which generalize Hirota's bilinear D operator, based on the principle of gauge invariance of the τ functions. We show that these operators can be constructed systematically using the bilinear D as building blocks. We concentrate in particular on the trilinear case and study the possible integrability of equations with one dependent variable. The fifth order equation of the Lax hierarchy as well as Satsuma's lowest-order gauge invariant equation are shown to have simple trilinear expressions. The formalism can be extended to an arbitrary degree of multilinearity.