Integrable semi-discretizations of the Davey-Stewartson system and a $(2+1)$-dimensional Yajima-Oikawa system. I

Abstract

The integrable Davey-Stewartson system is a linear combination of the two elementary flows that commute: $\mathrm{i}{q}_{t_1}+q_{xx}+2q{\partial }_y^{-1}{\partial }_x(|q{|}^2)=0$ and $\mathrm{i}{q}_{t_2}+q_{yy}+2q{\partial }_x^{-1}{\partial }_y(|q{|}^2)=0$. In the literature, each elementary Davey-Stewartson flow is often called the Fokas system because it was studied by Fokas in the early 1990s. In fact, the integrability of the Davey-Stewartson system dates back to the work of Ablowitz and Haberman in 1975; the elementary Davey-Stewartson flows, as well as another integrable $(2+1)$-dimensional nonlinear Schrödinger equation $\mathrm{i}{q}_t+q_{xy}+2q{\partial }_y^{-1}{\partial }_x(|q{|}^2)=0$ proposed by Calogero and Degasperis in 1976, appeared explicitly in Zakharov's article published in 1980. By applying a linear change of the independent variables, an elementary Davey-Stewartson flow can be identified with a $(2+1)$-dimensional generalization of the integrable long wave-short wave interaction model, called the Yajima-Oikawa system: $\mathrm{i}{q}_t+q_{xx}+uq=0$, $u_t+c{u}_y=2(|q{|}^2{)}_x$. In this paper, we propose a new integrable semi-discretization (discretization of one of the two spatial variables, say $x$) of the Davey-Stewartson system by constructing its Lax-pair representation; the two elementary flows in the semi-discrete case indeed commute. By applying a linear change of the continuous independent variables to an elementary flow, we also obtain an integrable semi-discretization of the $(2+1)$-dimensional Yajima-Oikawa system.

17 pages; (v2) improved introduction and added one reference

References in corpus (1)

• On a new integrable generalization of the Toda lattice and a discrete Yajima-Oikawa system