Cauchy problem for integrable discrete equations on quad-graphs

Abstract

Initial value problems for the integrable discrete equations on quad-graphs are investigated. A geometric criterion of the well-posedness of such a problem is found. The effects of the interaction of the solutions with the localized defects in the regular square lattice are discussed for the discrete potential KdV and linear wave equations. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.

Corrected version with the assumption of nonsingularity of solutions explicitly stated

References in corpus (4)

• Classification of integrable equations on quad-graphs. The consistency approach
• Lax pair for the Adler (lattice Krichever-Novikov) System
• Yang-Baxter maps and integrable dynamics
• Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy

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• Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps
• Singularities of Type-Q ABS Equations
• Algebraic entropy computations for lattice equations: why initial value problems do matter
• On the lattice-geometry and birational group of the six-point multi-ratio equation
• Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings
• Periodic continued fractions and hyperelliptic curves
• On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times 3$ Determinants