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Bifurcations of travelling wave solutions in the discrete NLS equations. (English) Zbl 1086.34059
The paper is concerned with changes in behaviour of solutions to nonlinear discrete Schrödinger equations. The paper is motivated particularly by the need to clarify some contradictions in the existing literature which are identified by the authors in the introduction. The initial analysis is based on a reduction of the discrete NLS equation to a linear mixed advance-delay differential equation. One can derive an equation for the resonances of the travelling wave solutions and the authors explain why one should investigate repeated roots of this equation. The later analysis is based on a discrete dynamical systems approach and the derivation of a centre manifold and normal form reduction. In conclusion, the authors have shown that they have cleared up some of the earlier contradictory results, and have identified two open problems.
34K18Bifurcation theory of functional differential equations
34K06Linear functional-differential equations
37D10Invariant manifold theory