Evolution equations for polynomials and rational functions which are conformal on the unit disk

摘要

On the unit disk D in the complex plane C two evolution equations for conformal mappings Ω(z,t),z∈D,t⩾0 are studied: The quasi-linear Löwner–Kufarev (L–K) equation and the quasi-linear Hopper equation. The first one has ‘Hamiltonian’, say, fΩ, the second one FΩ. The L–K equation has the property that for any initial condition Ω0(z) which is conformal on D, the solution Ω(z,t) remains conformal as long as it exists (Section 1). The H equation has the property that for any initial condition Ω0(z) which is polynomial/rational on D, the solution Ω(z,t) remains polynomial/rational as long as it exists (Section 2). We find conditions on the pair of Hamiltonians {fΩ,FΩ}, such that both the L–K and the H equations describe one and the same evolution phenomenon. This implies that both the properties of being conformal and of being polynomial/rational persist (Section 3). We show that ‘compatible pairs’ {fΩ,FΩ} are not rare. They can both be found in physics and be ‘artificially’ constructed. In this paper the emphasis is on algebraic properties, although analysis cannot be avoided altogether.