Abstract
A basic question in the theory of nonlinear partial differential equations is: when does a solution form a singularity and what is its nature? The aim of this lecture series is to offer an introduction into the analytic study of these questions for unidirectional shallow water waves models. Two particular models are investigated: the famous Korteweg–de Vries equation and the more recent Camassa–Holm equation. A rather classical approach to the Korteweg–de Vries equation is presented showing that this flow is globally well-posed for large classes of initial conditions. As a consequence wave breaking cannot be observed within the Korteweg–de Vries regime. In pronounced contrast to that a different picture may be seen in the case of the Camassa–Holm flow: while some solutions exist for ever other develop a wave breaking in finite time. Finally a geometric picture of the Camassa–Holm equation is presented as a geodesic flow on a Frechet–Lie group consisting of smooth diffeomorphisms of the real line.
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