Non-existence of S-integrable three-point partial difference equations in the lattice plane (Q6063902)

In this short note the authors use \textit{R. Yamilov}'s theorem [J. Phys. A, Math. Gen. 39, No. 45, R541--R623 (2006; Zbl 1105.35136), Theorem 13] to show that lattice equations defined on three points cannot be S-integrable. An equation (continuous or discrete) is said to be S-integrable if it is integrable by means of the inverse scattering transform method, while it is said to be C-integrable if it is linearisable through a change of variables, see [\textit{F. Calogero}, in: What is integrability, Springer Ser. Nonlinear Dyn. 1--62 (1991; Zbl 0808.35001)]. Yamilov's theorem states that for any evolutionary differential-difference equation of the form \[ \frac{\mathrm{d}u_{n}}{\mathrm{d}t} = f(u_{n+N},u_{n+N-1},\dots,u_{n+M}), \qquad \frac{\partial f}{\partial u_{n+N}} \frac{\partial f}{\partial u_{n+M}}\neq 0, \tag{1} \] where \(f\) is a locally analytic function of its arguments and \(N\geq M\), a necessary condition for S-integrability is \(M=-N\). See also [the authors, Ufim. Mat. Zh. 13, No. 2, 158--165 (2021; Zbl 1488.39050)] and the monograph [\textit{D. Levi} et al., Continuous symmetries and integrability of discrete equations. Providence, RI: American Mathematical Society (AMS) (2022; Zbl 1525.39001)] for further details about this theorem. Let us now be more specific. The paper focuses on partial difference equations defined on a three-point stencil, i.e., functional relations of the following form: \[ \mathcal{E}_{n,m}(u_{n,m},u_{n+1,m},u_{n,m+1})=0, \tag{2} \] for an unknown field \(u_{n,m}\colon\mathbb{Z}^{2}\to\mathbb{C}\). The function \(\mathcal{E}_{n,m}(x,y,z,t)\) is multilinear in its arguments. In the literature some results on the C-integrability of partial difference equations of the form (2) have been obtained in [\textit{C. Scimiterna} and \textit{D. Levi}, J. Phys. A, Math. Theor. 45, No. 45, Article ID 025205, 13 p. (2012; Zbl 1266.39013)]. The question that this paper answers is ``can a partial difference equation of the form (2) be S-integrable?''. The answer is found to be negative. This result is proved by considering the partial continuous limit of Equation (2): \[ u_{n,m+1}=u_{n}(t+\varepsilon) = u_{n}(t) + \varepsilon\frac{\mathrm{d}u_{n}}{\mathrm{d}t} + \varepsilon\frac{\mathrm{d}^{2}u_{n}}{\mathrm{d}t^{2}} +O(\varepsilon^{3}). \tag{3} \] Indeed, plugging (3) into (2), after some algebraic simplification one obtains that the partial continuous limit is a differential-difference equation (1) with \(N=1\) and \(M=0\). This readily implies the statement. The paper is written concisely, but without sacrificing its clearness, and with a good amount of background material and references.