Bilinear, trilinear forms, and exact solution of certain fourth order integrable difference equations

Abstract

A systematic investigation of finding bilinear or trilinear representations of fourth order autonomous ordinary difference equation, x(n+4)=F(x(n),x(n+1),x(n+2),x(n+3)) or xn+4=F(xn,xn+1,xn+2,xn+3), is made. As an illustration, we consider fourth order symplectic integrable difference equations reported by [Capel and Sahadevan, Physica A 289, 86 (2001)] and derived their bilinear or trilinear forms. Also, it is shown that the obtained bilinear representations admit exact solution of rational form.