Nonlinear elastodynamics of materials with strong ellipticity condition: Carroll-type solutions

• Nonlinear Klein–Gordon equation in nonlinear elastodynamics: compact-like waves and waves with oscillatory spatial dependence.
• Derivation of the nonlinear Schrödinger (NLS) equation from the nonlinear Klein–Gordon equation.
• Modulated NLS-type equation: compactons and standing waves.

Classes of deformations in nonlinear elastodynamics with origin in pioneering work of Carroll are investigated for an isotropic elastic solid subject to body forces corresponding to a nonlinear substrate potential. Exact solutions are obtained which, inter alia, are descriptive of the propagation of compact waves and motions with oscillatory spatial dependence. It is shown that a description of slowly modulated waves leads to a novel class of generalized nonlinear Schrödinger equations. The latter class, in general, is not integrable. However, a procedure is presented whereby integrable Hamiltonian subsystems may be isolated for a broad class of deformations.