Cauchy Problem for Integrable Discrete Equations on Quad-Graphs
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Cauchy Problem for Integrable Discrete Equations on Quad-Graphs Dedicated to S. P. Novikov on his 65 birthday V. E. ADLER1, and A. P. VESELOV2
1 Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany. e-mail: [email protected] 2 Loughborough University, Loughborough, Leicestershire LE11 3TU, UK and Landau Institute for Theoretical Physics, Russia. e-mail: [email protected]
(Received: 24 September 2003) Abstract. Initial value problems for the integrable discrete equations on quad-graphs are investigated. We give a geometric criterion of when such a problem is well-posed. In the basic example of the discrete KdV equation an effective integration scheme based on the matrix factorization problem is proposed and the interaction of the solutions with the localized defects in the regular square lattice are discussed in details. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented. Mathematics Subject Classifications (2000): 37K35, 37K40, 37K60. Key words: Cauchy problem, quad-graph, discrete integrable equations.
1. Introduction The discrete potential KdV (dKdV) equation (vm+1,n+1 − vm,n )(vm+1,n − vm,n+1 ) = αm − βn ,
m, n ∈ Z
(1)
can be considered as a simplest representative of the integrable nonlinear discrete equations in two dimensions. This field is widely studied and many other examples can be found in the literature, e.g., [3, 11, 18, 25, 24, 6, 20]. However, we restrict ourselves to this simplest model, since our aim here is to analyze the generalizations of another kind. More precisely, we will discuss the role of the support of the discrete equations, that is, roughly speaking, the set where the independent variables live. Of course, the most natural way to discretize 2-dimensional PDE is to consider equations on the regular square grid Z2 as above, but recent results [1, 4, 21, 19, 2, 22, 23, 9] demonstrate that possibly more general synonym of ‘2D’ in the discrete case is ‘planar graphs’. On leave from Landau Institute for Theoretical Physics, Chernogolovka, Russia.
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V. E. ADLER AND A. P. VESELOV
It should be noted that nonstandard planar graphs were considered in Mathematical Physics already quite a while ago (e.g., Korepin’s works [15, 16] where the solvable spin models on the quasi-crystallic tilings were investigated). A systematic theory of the linear difference operators on the graphs in relation to soliton theory has been initiated by S. P. Novikov (see [22, 23, 9, 10, 17]). We start in Section 2 with the necessary information about integrable equations on quad-graphs, which are planar graphs with quadrilateral faces. In this case the integrability can be understood as the so-called 3D consistency property, introduced and studied in [4, 21, 19, 2]. The discrete potential KdV equation on quad-graph reads locally exactly as on the square lattice: (v12 − v)(v1 − v2 ) = α1 − α2 ,
(2)
but now it is assumed that the fields v are assigned to the vertices of the quad-graph and the parame
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