Karlsruhe 2011 – wissenschaftliches Programm
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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik
MP 2: Classical Field Theory
MP 2.1: Vortrag
Dienstag, 29. März 2011, 09:00–09:20, 30.45: 201
Combinatorics of KP line solitons: a tropical approach — Folkert Müller-Hoissen1 and •Aristophanes Dimakis2 — 1Max-Planck-Institute for Dynamics and Self-Organization, Bunsenstrasse 10, D-37073 Göttingen — 2Department of Financial and Management Engineering, University of the Aegean, 41 Kountourioti Str., GR-82100 Chios
The Kadomtsev-Petviashvili (KP) equation in particular models certain network patterns formed by waves on shallow water in terms of line soliton solutions. The simplest class of such solutions corresponds, in a tropical approximation, to chains of rooted binary trees, and it turns out that they realize maximal chains in Tamari lattices (which are poset structures on associahedra). The analysis also makes contact with ``higher-order" versions of Tamari lattices. A general line soliton network solution can be described, in good approximation, as a superimposition of solutions from the tree class, with rather simple modifications. All this yields a characterization of possible evolutions of wave network patterns on shallow water, provided that the KP approximation applies. It is based on our publication J. Phys. A: Math. Theor. 44 (2011) 025203.