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On zeros of differences of meromorphic functions

Yong Liu;  HongXun Yi — 2011

Annales Polonici Mathematici

Let f be a transcendental meromorphic function and g ( z ) = f ( z + c ₁ ) + ⋯ + f ( z + c k ) - k f ( z ) and g k ( z ) = f ( z + c ₁ ) ⋯ f ( z + c k ) - f k ( z ) . A number of results are obtained concerning the exponents of convergence of the zeros of g(z), g k ( z ) , g(z)/f(z), and g k ( z ) / f k ( z ) .

Multiple end solutions to the Allen-Cahn equation in ℝ 2

Michał Kowalczyk;  Yong Liu;  Frank Pacard

Séminaire Laurent Schwartz — EDP et applications

An entire solution of the Allen-Cahn equation Δ u = f ( u ) , where f is an odd function and has exactly three zeros at ± 1 and 0 , e.g. f ( u ) = u ( u 2 - 1 ) , is called a 2 k end solution if its nodal set is asymptotic to 2 k half lines, and if along each of these half lines the function u looks (up to a multiplication by - 1 ) like the one dimensional, odd, heteroclinic solution H , of H ' ' = f ( H ) . In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space of such solutions....

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