Aspects of nonlinear evolution PDEs

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During the three-month research visit of Professor Himonas to Cambridge we will investigate two problems:(a) The Cauchy problem for a certain integrable generalisation of the nonlinear Schroedinger (NLS)equation. This equation is actually related with the NLS in the same way that the Camassa-Holmequation is related to the Korteweg-de Vries equation. We will use both the inverse scattering transform method and PDE techniques for the analysis of the Cauchy problem. (b) Initial-boundary value problems on the half-line for a large class of nonlinear evolution PDEs whose linear part is dispersive. We will consider the nonlinearity as forcing and we will use a general formula obtained recently for the solution of linear dispersive equations at the half-line to formulate an integral equation. This latter equation will be analysed using PDE techniques.