Nonlinear envelope-function equation and strongly localized vibrational modes in anharmonic lattices

A nonlinear envelope-function equation and matching conditions for the short-wavelength vibrational excitations in monatomic anharmonic lattices are derived. This equation coincides with the nonlinear Klein-Gordon or Schr\"odinger equation only in the limit of relatively small oscillation amplitudes. Strongly localized soliton and soliton-kink solutions of this equation are found within the rotating-wave approximation. High-frequency intrinsic localized and propagating modes in one-, two-, and three-dimensional homogeneous hard anharmonic lattices, previously obtained by Sievers and Takeno in a discrete-lattice theory, are described within the framework of the continuous envelope-function approach. The existence of low-frequency exotic soliton-kink localized modes in homogeneous soft anharmonic lattices is predicted. Localized modes are also considered in inhomogeneous lattices with mass defect or force-constant defect to the host nearest neighbors. Large-amplitude high-frequency asymmetric localized modes are predicted to exist in hard anhmarmonic lattices near the almost immovable heavy impurities. Large-amplitude symmetric localized modes near the heavy or weakly bonded impurity, and singular and algebraic-soliton nonlinear vibrational modes, localized near the light or strongly bonded impurities, are found, which do not exist in a linear lattice.