Nonsemilinear one-dimensional PDEs: analysis of PT deformed models and numerical study of compactons

This thesis is based on the work done during my PhD studies and is roughly divided in two independent parts. The first part consists of Chapters 1 and 2 and is based on the two papers Cavaglia et al. [2011] and Cavaglia & Fring [2012], concerning the complex PT-symmetric deformations of the KdV equation and of the inviscid Burgers equation, respectively. The second part of the thesis, comprising Chapters 3 and 4, contains a review and original numerical studies on the properties of certain quasilinear dispersive PDEs in one dimension with compacton solutions. The subjects treated in the two parts of this work are quite different, however a common theme, emphasised in the title of the thesis, is the occurrence of nonsemilinear PDEs. Such equations are characterised by the fact that the highest derivative enters the equation in a nonlinear fashion, and arise in the modeling of strongly nonlinear natural phenomena such as the breaking of waves, the formation of shocks and crests or the creation of liquid drops. Typically, nonsemilinear equations are associated to the development of singularities and non-analytic solutions. Many of the complex deformations considered in the first two chapters are nonsemilinear as a result of the PT deformation. This is also a crucial feature of the compacton-supporting equations considered in the second part of this work. This thesis is organized as follows. Chapter 1 contains an introduction to the field of PT-symmetric quantum and classical mechanics, motivating the study of PT-symmetric deformations of classical systems. Then, we review the contents of Cavaglia et al. [2011] where we explore travelling waves in two family of complex models obtained as PT-symmetric deformations of the KdV equation. We also illustrate with many examples the connection between the periodicity of orbits and their invariance under PTsymmetry. Chapter 2 is based on the paper Cavaglia & Fring [2012] on the PTsymmetric deformation of the inviscid Burgers equation introduced in Bender & Feinberg [2008]. The main original contribution of this chapter is to characterise precisely how the deformation affects the gradient catastrophe. We also point out some incorrect conclusions of the paper Bender & Feinberg [2008]. Chapter 3 contains a review on the properties of nonsemilinear dispersive PDEs in one space dimension, concentrating on the compacton solutions discovered in Rosenau & Hyman [1993]. After an introduction, we present some original numerical studies on the K(2, 2) and K(4, 4) equations. The emphasis is on illustrating the different type of phenomena exhibited by the solutions to these models. These numerical experiments confirm previous results on the properties of compacton-compacton collisions. Besides, we make some original observations, showing the development of a singularity in an initially smooth solution. In Chapter 4 , we consider an integrable compacton equation introduced by Rosenau in Rosenau [1996]. This equation has been previously studied numerically in an unpublished work by Hyman and Rosenau cited in Rosenau [2006]. We present an independent numerical study, confirming the claim of Rosenau [2006] that travelling compacton equations to this equation do not contribute to the initial value problem. Besides, we analyse the local conservation laws of this equation and show that most of them are violated by any solution having a compact, dynamically evolving support. We confirm numerically that such solutions, which had not been described before, do indeed exist. Finally, in Chapter 5 we present our conclusions and discuss open problems related to this work.