Existence of solitary solutions in nonlinear chains

Techniques for examining the existence and stability of localized modes are presented. The methods are demonstrated in detail for a discrete nonlinear Schr\"odinger (DNS) equation, but also apply to other systems, e.g., the discrete nonlinear Klein-Gordon (DNKG) equation. Stationary states may be found via variational principles or through generating functions. The latter technique makes use of solutions of a continuous difference-equation and allows for localized modes with different symmetry properties. It is shown that several families of stationary solutions exist, and a constructive procedure to calculate the latter is presented. In the case of the DNS equation, an analytical stability criterion for symmetric solitons (N-theorem) proves that the discrete equation exhibits localization in regimes where blow-up cannot occur in the continuum analog. Stable nonlinear solutions are found for the DNKG equation. The analytical calculations are supplemented by numerical simulations. \textcopyright{} 1996 The American Physical Society.

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