Reduction of the Laplace sequence of differential-difference equations

Abstract

The method for constructing a Lax pair and a recursion operator for nonlinear integrable equations of sine-Gordon type, developed in the works of I.T. Habibullin et al. (J. Phys. A, Math. Theor. \textbf{49}, 035202 (2016)), is adapted to the case of differential-difference equations of soliton type. As an illustration of the method, we consider the equation
$$
u_{n+1,x}=-u_{n,x}-(u_{n+1}-u_n)^2,
$$construct for this equation a Lax pair and a recursion operator that describes the hierarchy of higher symmetries.