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Working paper

Loewner equations and reductions of dispersionless hierarchies

math. arxive. Cornell University, 2020. No. 2010.02277.
Akhmedova V., Takebe T., Zabrodin A.
The equations of Loewner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric onformal maps and the other one is the theor of integrable systems.In this paper we compare the both approaches. After recalling the derivation of Loewner equations based on complex analysis we review one- and multi-variable reductions of dispersionless integrable hierarchies (dKP, dBKP, dToda and dDKP). The one-variable reductions are described by solutions of different versions of Loewner equation: chordal (rational) for dKP, quadrant for dBKP, radial (trigonometric) for dToda and elliptic for DKP.We also discuss multi-variable reductions which are given by a system of Loewner equations supplemented by a system of patial differential equations of hydrodynamic type. The solvability of the hydrodynamic type system can be proved by means of the generalized hodograph method.