Classification of 3-dimensional integrable scalar discrete equations

Abstract

We classify all integrable 3-dimensional scalar discrete quasilinear equations Q=0 on an elementary cubic cell of the 3-dimensional lattice. An equation Q=0 is called integrable if it may be consistently imposed on all 3-dimensional elementary faces of the 4-dimensional lattice. Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only nontrivial (non-linearizable) integrable equation from this class is the well-known dBKP-system. (Version 2: A small correction in Table 1 (p.7) for n=2 has been made.) (Version 3: A few small corrections: one more reference added, the main statement stated more explicitly.)

20 p. LaTeX + 1 EPS figure

References in corpus (5)

• New features of FORM
• Classification of integrable equations on quad-graphs. The consistency approach
• Yang-Baxter maps and symmetries of integrable equations on quad-graphs
• Discrete differential geometry. Consistency as integrability
• On organizing principles of Discrete Differential Geometry. Geometry of spheres

Cited by in corpus (5)

• Symbolic Computation of Lax Pairs of Partial Difference Equations Using Consistency Around the Cube
• Multi-Component Extension of CAC Systems
• Hyperdeterminants as integrable discrete systems
• Towards the classification of integrable differential-difference equations in 2 + 1 dimensions
• On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times 3$ Determinants