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Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry

Bridges, Thomas J. and Derks, Gianne (1999) Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry Proceedings of the Royal Society of London A, 455. pp. 2427-2469.


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The linear stability of solitary-wave or front solutions of Hamiltonian evolutionary equations, which are equivariant with respect to a Lie group, is studied. The organizing centre for the analysis is a multisymplectic formulation of Hamiltonian PDEs where a distinct symplectic operator is assigned for time and space. This separation of symplectic structures leads to new characterizations of the following components of the analysis. The states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure. The momentum of the connecting orbit, or shape of the solitary wave, considered as a heteroclinic orbit in a phase-space representation, is given a new characterization as a one-form on the tangent space to the heteroclinic manifold and this one-form is a restriction of the temporal symplectic structure. For the linear stability analysis, a new symplectic characterization of the Evans function and its derivatives are obtained, leading to an abstract geometric proof of instability for a large class of solitary-wave states of equivariant Hamiltonian evolutionary PDEs. The theory sheds new light on several well-known models: the gKdV equation, a Boussinesq system and a nonlinear wave equation. The generalization to solitary waves associated with multidimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Bridges, Thomas
Date : 8 July 1999
Uncontrolled Keywords : stability, Evans function, multisymplectic structure, wave equations, Hamiltonian structure, solitary waves
Additional Information : Published in Proceedings of the Royal Society of London A, 455, 2427-2469. © 1999 The Royal Society.
Depositing User : Mr Adam Field
Date Deposited : 27 May 2010 14:41
Last Modified : 16 Jan 2019 16:22

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