On a nonlinear integrable difference equation on the square

We present a nonlinear partial difference equation defined on a square which is obtained by combining the Miura transformations between the Volterra and the modified Volterra differential-difference equations. This equation is not symmetric with respect to the exchange of the two discrete variables. Its integrability is proved by constructing its Lax pair. The uncovery of new nonlinear integrable completely discrete equations is always a very challenging problem as, by proper continuous limits, many other results on differential-difference and partial differential equations can be obtained. In the case of differential equations by now a lot is known starting from the pioneering works by Gardner, Green, Kruskal and Miura. A summary of these results is already of public domain and presented for example in the Encyclopedia of Mathematical Physics [5] or in the Encyclopedia of Nonlinear Science [6]. Among those results let us mention the classification scheme of nonlinear integrable partial differential equations introduced by Shabat using the formal symmetry approach, see [11] for a review. The classification of differential-difference equations has also been carried out using the formal symmetry approach by Yamilov [18] and it is a well defined procedure which can be easily computerized for many families of equations [10,19]. In the completely discrete case the situation is different. Many researchers have tried to carry out the approach of formal symmetries introduced by Shabat, without