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Title: Perturbed multi-symplectic systems : intersections of invariant manifolds and transverse instability
Author: Blyuss, Kostyantyn B.
Awarding Body: University of Surrey
Current Institution: University of Surrey
Date of Award: 2004
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This thesis deals with two aspects of dynamics in systems described by multi-symplectic partial differential equations. The first part is devoted to the study of heteroclinic intersections in systems which govern the dynamics of travelling waves in multi-symplectic partial differential equations with perturbations. In this study a version of the Melnikov method is developed which takes into account the symmetry of the systems under consideration. The presence of the symmetry leads to various interesting differences between the method we develop and the standard approach. In particular, a result about persistence of the fixed point of the Poincare map under perturbations has to be amended since the unperturbed fixed point is non-hyperbolic. The symmetry also results in the necessity to consider separately two cases: when the perturbation has no component in the group direction at all, or when on average it has no component in the group direction when evaluated on the unperturbed solutions. For each of those cases we discuss persistence of the fixed points of the Poincare map and persistence of invariant manifolds, where the knowledge of the symmetry in incorporated in the geometrical constructions. Finally, we derive Melnikov-type conditions in both aforementioned cases which guarantee the existence of transverse intersections of the stable and unstable manifolds. We discuss some possible areas of applications of the Melnikov-type method derived and illustrate the method on the examples of a perturbed Korteweg-de Vries equation and a perturbed nonlinear Schrodinger equation. Implications of the transverse or topological intersections of the manifolds for possible chaotic behaviour in the systems are discussed together with directions of further investigation. The second part of this thesis considers the stability of solitary waves with respect to perturbations which are transverse to the basic direction of propagation of these waves. Using various analytical and numerical techniques, we study this problem for the solitary waves of the (2+l)-dimensional Boussinesq equation and the generalised fifth-order Kadomtsev-Petviashvili equation. For both equations we use a geometric condition for transverse instability based on the multi-symplectic formulation of the equation to derive a condition for transverse instability in the long-wavelength regime. Then an Evans function approach is employed to determine the dependence of the instability growth rate on the transverse wavenumber for all possible wavenumbers. In the case of the (2+l)-dimensional Boussinesq equation this is done analytically, while for the generalised fifth-order Kadomtsev-Petviashvili equation we have to resort to numerical simulations. Finally, for the (2+l)-dimensional Boussinesq equation we perform direct numerical simulations of the full equation to investigate the nonlinear stage of the evolution of the transversely unstable solitary waves, and the result is that the instability leads to the collapse of the solitary wave. The thesis is concluded by a discussion of some open problems.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available