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Transversality of homoclinic orbits, the Maslov index and the symplectic Evans function

Chardard, F and Bridges, TJ (2015) Transversality of homoclinic orbits, the Maslov index and the symplectic Evans function NONLINEARITY, 28 (1). pp. 77-102.

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Abstract

Partial differential equations in one space dimension and time, which are gradient-like in time with Hamiltonian steady part, are considered. The interest is in the case where the steady equation has a homoclinic orbit, representing a solitarywave. Such homoclinic orbits have two important geometric invariants: a Maslov index and a Lazutkin-Treschev invariant. A new relation between the two has been discovered and is moreover linked to transversal construction of homoclinic orbits: the sign of the Lazutkin-Treschev invariant determines the parity of the Maslov index. A key tool is the geometry of Lagrangian planes. All this geometry feeds into linearization about the homoclinic orbit in the time-dependent system, which is studied using the Evans function. A new formula for the symplectification of the Evans function is presented, and it is proven that the derivative of the Evans function is proportional to the Lazutkin-Treschev invariant. A corollary is that the Evans function has a simple zero if, and only if, the homoclinic orbit of the steady problem is transversely constructed. Examples from the theory of gradient reaction-diffusion equations and pattern formation are presented.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
AuthorsEmailORCID
Chardard, FUNSPECIFIEDUNSPECIFIED
Bridges, TJUNSPECIFIEDUNSPECIFIED
Date : 1 January 2015
Identification Number : 10.1088/0951-7715/28/1/77
Uncontrolled Keywords : Science & Technology, Physical Sciences, Mathematics, Applied, Physics, Mathematical, Mathematics, Physics, solitary waves, symplectic, Maslov index, homoclinc orbits, transversality, Lazutkin-Treschev invariant, Evans function, SOLITARY WAVES, HAMILTONIAN-SYSTEMS, STABILITY ANALYSIS, INSTABILITY, EQUATIONS, FRONTS, MATRIX, SPACE
Related URLs :
Additional Information : © 2015 IOP Publishing Ltd & London Mathematical Society Printed
Depositing User : Symplectic Elements
Date Deposited : 18 Aug 2015 15:54
Last Modified : 01 Jan 2016 02:08
URI: http://epubs.surrey.ac.uk/id/eprint/808291

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