Mechano-transduction in tumour growth modelling.

The evolution of biological systems is strongly influenced by physical factors, such as applied forces, geometry or the stiffness of the micro-environment. Mechanical changes are particularly important in solid tumour development, as altered stromal-epithelial interactions can provoke a persistent increase in cytoskeletal tension, driving the gene expression of a malignant phenotype. In this work, we propose a novel multi-scale treatment of mechano-transduction in cancer growth.

Dispersion relations and wave operators in self-similar quasicontinuous linear chains.

We construct self-similar functions and linear operators to deduce a self-similar variant of the Laplacian operator and of the D'Alembertian wave operator. The exigence of self-similarity as a symmetry property requires the introduction of nonlocal particle-particle interactions. We derive a self-similar linear wave operator describing the dynamics of a quasicontinuous linear chain of infinite length with a spatially self-similar distribution of nonlocal interparticle springs.

Two approaches to study essentially nonlinear and dispersive properties of the internal structure of materials.

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.

Propagation of localized longitudinal strain waves in a plate in the presence of cubic nonlinearity.

Possible propagating longitudinal strain solitary waves in a plate are shown to be seriously altered when physical cubic nonlinearity is taken into account in the modeling. This also affects an amplification of the wave due to the transverse instability of plane-localized waves and due to the plane-wave interaction.

Nonlinear dynamics of incommensurate surface layers.

We describe analytically the nonlinear dynamics of the incommensurate surface layer ("self-modulated" system) with a spatially periodical structure. In the framework of the Frenkel-Kontorova model the nonlinear excitations of the periodic soliton lattice, such as moving additional kinks and gap solitons, are investigated.

Solitary Rayleigh waves in the presence of surface nonlinearities.

The propagation of Rayleigh waves is investigated in a solid substrate of linear material covered by a film consisting of a material with large nonlinear elastic moduli. For this system, a nonlinear evolution equation is derived that may be regarded as a special case in a wider class of evolution equations with a specific type of nonlocal nonlinearity. Periodic pulse train solutions are computed.