Solitons in a nonlinear elastic medium

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Abstract
The influence of nonlinearity on shear horizontal elastic (bulk and surface) waves in crystals is considered on the basis of a nonlinear equation derived from a microscopic scalar model. The continuum limit is compared with the equation obtained in the framework of nonlinear elasticity theory. Using the equation, it is demonstrated that nonlinear surface waves of the shear horizontal polarization are modulationally unstable, and, therefore, they have to produce surface solitary waves. The solitons are studied in the case of a nonlinear elastic plate when the effective asymptotic technique may be elaborated. In the main approximation of the asymptotic approach, the effective nonlinear Schrödinger equation is derived. We predict that in the nonlinear elastic plate it is possible to observe bright or dark envelope solitons that propagate without distortion. The properties of the solitons depend on the carrier wave number and the thickness of the plate. We demonstrate also that there is a discrete set of the solitons produced by linear modes of the elastic plate.