The method of contact integrable extensions is used to find new differential coverings for the generalized (2 + 1)-dimensional dispersionless Dym equation and corresponding Bäcklund transformations.
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We apply Cartan’s method of equivalence to find a Bäcklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation.
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To a domain with conical points Ω, we associate a natural C*-algebra that is motivated by the study of boundary value problems on Ω, especially using the method of layer potentials. In two dimensions, we allow Ω to be a domain with ramified cracks. We construct an explicit groupoid associated to ∂Ω and use the theory of pseudodifferential operators on groupoids and its representations to obtain our layer potentials C*-algebra. We study its structure, compute the associated K-groups, and prove Fredholm conditions for the natural pseudodifferential operators affiliated to this C*-algebra.
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