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An energy-based stability criterion for solitary travelling waves in Hamiltonian lattices. (English) Zbl 1402.37086
Summary: In this work, we revisit a criterion, originally proposed in [G. Friesecke and R. L. Pego, Nonlinearity 17, No. 1, 207–227 (2004; Zbl 1103.37049)] for the stability of solitary travelling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-travelling frame, as well as at that of the Floquet problem arising when considering the travelling wave as a periodic orbit modulo shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the travelling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed.

##### MSC:
 37K60 Lattice dynamics; integrable lattice equations 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 35C07 Traveling wave solutions 35C08 Soliton solutions
##### Keywords:
solitary travelling wave; Hamiltonian lattice; stability
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