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      Computation and stability of waves in equivariant evolution equations

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          Abstract

          Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant evolution equa- tion. In numerical computations the freezing method takes advantage of this structure by splitting the evolution of the PDE into the dynamics on the underlying Lie group and on some reduced phase space. The approach raises a series of questions which were answered to a certain degree by the project: linear stability implies non-linear (asymp- totic) stability, persistence of stability under discretisation, analysis and computation of spectral structures, first versus second order evolution systems, well-posedness of partial differential algebraic equations, spatial decay of wave profiles and truncation to bounded domains, analytical and numerical treatment of wave interactions, relation to connecting orbits in dynamical systems. A further numerical problem related to this topic will be discussed, namely the solution of non-linear eigenvalue problems via a contour method.

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          Most cited references28

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          Euclidean symmetry and the dynamics of rotating spiral waves

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            Stability of Travelling Waves

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              An integral method for solving nonlinear eigenvalue problems

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                Author and article information

                Journal
                29 October 2018
                Article
                1810.12168
                89e207d3-0f45-4df2-94eb-59b240de4219

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                37Lxx, 65J08, 74J30, 65L15, 35B40
                math.NA

                Numerical & Computational mathematics
                Numerical & Computational mathematics

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