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Wydział Matematyki, Informatyki i Mechaniki Uniwersytetu Warszawskiego

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Sem. Równań Fiz. Mat.

 

Total integrals of solutions for inhomogeneous Painlevé II equation


Prelegent: Piotr Kokocki

2020-01-16 12:30

We establish a formula determining the value of the Cauchy integrals for the real and purely imaginary Ablowitz-Segur solutions for the inhomogeneous second Painlevé equation. Our approach relies on the Deift-Zhou steepest descent analysis of the corresponding Riemann-Hilbert problem and the construction of an appropriate parametrix in a neighborhood of the origin. The obtained results are used to provide a rigorous proof of a numerically predicted phenomena that an arbitrary logarithmic spiral is a finite time singularity developed by a geometric flow that approximates the vortex patch dynamics of the 2D Euler equation.