On integrability of semi-discrete Tzitzeica equation

Abstract

In the paper we consider a semi-discrete version of Tzitzeica equation $$\frac{du_{n+1}}{dx}=\frac{du_{n}}{dx}+(e^{-2u_n} +e^{-2u_{n+1}})+\sqrt{e^{2u_n}+e^{2u_{n+1}}},$$ which was found in a recent paper [R.N. Garifullin and I.T. Habibullin 2021 J. Phys. A: Math. Theor. 54 205201]. It was shown that this equation possessed generalized symmetries along the discrete and continuous directions. These generalized symmetries are equations of Sawada-Kotera equation type and of discere Sawada-Kotera equation type. In this work we construct the Lax pair for this equation and for its generalized symmetries. The found Lax pair is written out in terms of $3\times 3$ matrices and this indicates the integrability of the found equations. To solve this problem, we employ the known relation between one of the generalized symmetries with a well-studied Kaup-Kupershmidt equation. The found Lax pairs can be employed in further studies of this equation, namely, for finding its conservations laws, the recursion operators and wide classes of solutions. Moreover, we write out two Lax representations in the form of scalar operators. The first representation is written in terms of the powers of the differentiation operators with respect to the continuous variable $x$, while the other is written via the powers of the operator of the shift along the discrete variable $n$.