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. 77102.

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Abstract
Partial differential equations in one space dimension and time, which are gradientlike 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 LazutkinTreschev invariant. A new relation between the two has been discovered and is moreover linked to transversal construction of homoclinic orbits: the sign of the LazutkinTreschev 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 timedependent 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 LazutkinTreschev 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 reactiondiffusion equations and pattern formation are presented.
Item Type:  Article  

Divisions :  Faculty of Engineering and Physical Sciences > Mathematics  
Authors : 


Date :  1 January 2015  
Identification Number :  10.1088/09517715/28/1/77  
Uncontrolled Keywords :  Science & Technology, Physical Sciences, Mathematics, Applied, Physics, Mathematical, Mathematics, Physics, solitary waves, symplectic, Maslov index, homoclinc orbits, transversality, LazutkinTreschev invariant, Evans function, SOLITARY WAVES, HAMILTONIANSYSTEMS, 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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