BUBBLETONS ARE NOT EMBEDDED

Abstract

We discuss constant mean curvature bubbletons in Euclidean 3-space via dressing with simple factors, and prove that single bubbletons are not embedded. A key feature of an integrable system is the presence of an algebraic transformation method which generates new solutions from old ones. In particular even by starting with a trivial solution one obtains a hierarchy of interesting global solutions. For the KdV equation one thus obtains the solitons via a Backlund transform. Solitons are solitary traveling waves with localized energy that are stable when interacting with each other. Many of the modern techniques in integrable systems theory stem from classical surface theory, developed by Backlund, Bianchi and Darboux amongst others for the structure equations of special surface classes. Away from umbilic points the structure equation of constant mean curvature (cmc) surfaces is the sinh-Gordon equation, whose trivial solution gives rise to the round cylinder. The term 'bubbleton' is due to Sterling and Wente (17), and the bubbletons are the solitons of the sinh-Gordon equation. The single bubbletons are obtained by transforming the standard cylinder by a Bianchi-Backlund transform. The resulting transformed cmc cylinder globally looks like the standard cylinder except for a localized part in which bubble-like pieces are added to the underlying surface, see figure 1.1. A video of how bubbles interact when they move through each other can be seen at (13).