Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework

The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on...

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Bibliographic Details
Published in:Physica. D Vol. 172; no. 1; pp. 190 - 216
Main Authors: Bridges, Thomas J., Derks, Gianne, Gottwald, Georg
Format: Journal Article
Language:English
Published: Elsevier B.V 15.11.2002
Subjects:
Evans function
Fifth-order KdV
Linear stability
Numerical exterior algebra
Evans function
Fifth-order KdV
Linear stability
Numerical exterior algebra
ISSN:0167-2789, 1872-8022
Online Access:Get full text
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Summary:The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several new features, including a rigorous numerical algorithm for choosing starting values, a new method for numerical analytic continuation of starting vectors, the role of the Grassmannian G 2( C 5) in choosing the numerical integrator, and the role of the Hodge star operator for relating ⋀ 2( C 5) and ⋀ 3( C 5) and deducing a range of numerically computable forms for the Evans function. The algorithm is illustrated by computing the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity.
ISSN:0167-2789
1872-8022
DOI:10.1016/S0167-2789(02)00655-3