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It is shown that essentially nonlinear models for solids with complex internal structure may be studied using phenomenological and proper structural approaches. It is found that both approaches give rise to the same nonlinear equation for traveling longitudinal macrostrain waves. However, presence of the connection between macro- and microfields in the proper structural model prevents a realization of some important solutions. First, it is obtained that simultaneous existence of compression and tensile waves is impossible in contrast to the phenomenological approach. Then, it is found that correlation between macro- and internal strains gives rise to the crucial influence of the velocity of the macrostrain wave on the existence of either compression or tensile localized strain waves. Also, it is shown that similar profiles of the macrostrain solitary waves may be accompanied by distinct profiles of the microstrain waves. Finally, the dispersion curves for the waves belong to the different branches. This is important for internal structural deviations caused by the dynamical loading due to the localized macrostrain wave propagation and demonstrates a need in the development of the proper structural approaches relative to the phenomenological ones.

We obtain exact solutions U (x, y, z, t) of the three-dimensional sine-Gordon equation in a form that Lamb previously proposed for integrating the two-dimensional sine-Gordon equation. The three-dimensional solutions depend on arbitrary functions F (α) and Φ(α, β), whose arguments are some functions α (x, y, z, t) and β (x, y, z, t). The ansatzes must satisfy certain equations. These are an algebraic system of equations in the case of one ansatz. In the case of two ansatzes, the system of algebraic equations is supplemented by first-order ordinary differential equations. The resulting solutions U (x, y, z, t) have an important property, namely, the superposition principle holds for the function tan(U/4). The suggested approach can be used to solve the generalized sine-Gordon equation, which, in contrast to the classical equation, additionally involves first-order partial derivatives with respect to the variables x, y, z, and t, and also to integrate the sinh-Gordon equation. This approach admits a natural generalization to the case of integration of the abovementioned types of equations in a space with any number of dimensions.

A new model of a double continuum with variable local topology is used to develop an essentially nonlinear theory of a medium with a cardinally rearranged periodic structure. This theory is based on a continualization of the periodic structure of a complex crystalline lattice consisting of two sublattices. In the long-wave approximation, the standard linear theory of acoustic and optic oscillations of the complex lattice is generalized. In this generalization, an internal translational symmetry of relative shear of the sublattices is taken into account. As a result, the interaction between the sublattices is expressed in terms of a nonlinear periodic force described, in particular, as a sine of the relative shear of two atoms belonging to an elementary cell. The corresponding equations describe elastic and inelastic catastrophic deformations due to the structural instability which accompanies phase transitions, twinning, defect formation, etc. Some static and dynamic problems are analyzed.

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