Dynamics and stability of localized nonlinear waves in inhomogeneous media

Stability theory of multidimensional soliton solutions in the Nonlinear Schrodinger (NLS) equation is extended to include general lattice-type potentials. The roles of the two stability conditions, i.e., the power (slope, Vakhitov-Kolokolov) condition and the (lesser known) spectral condition are further elucidated by numerical computations. Solitons in lattice-type media, including those that possess defects, edge-dislocations and quasicrystal structure, are found by employing a spectral fixed-point computational scheme; their evolution is studied by direct numerical computation of the NLS equation. Violation of the power condition leads to a self-focusing or self-defocusing instability, whereas violation of the spectral condition is found to invoke a drift instability. Importantly, these instabilities are found to apply to inhomgenous background media as well as to general  (i.e., non-solitonic) solutions.
 

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