Soliton-like solutions and other wave patterns in the nonlocal model of structured media

Stability and dynamical features of solitary wave solutions for a hydrodynamic-type system taking into account nonlocal effects

“…Let us stress that existence of a one-parameter family of solitary wave solutions is connected with the employment of the dynamic equation of state (4), taking account of the effects of pure spatial non-locality. The presence of one-parametric families of localized TW solutions is rather not inherent to another known hydrodynamic-type non-local models [29], accounting for temporal [50][51][52] or spatio-temporal [53] non-localities. Note that throughout the text we do not use the analytical description of the solitary wave solutions.…”

Peculiarities of wave fields in nonlocal media

“…If we need to describe the relaxing processes in more detail, then we can incorporate the terms with the third temporal derivatives in DES (14). In this case DES has the form 3…”

Compacton-like solutions to some nonlocal hydrodynamic-type model

“…Let us note that DES with higher derivatives were employed in many papers dealing with the structured media. Qualitative and numerical investigations undertaken in [22,23,24] show that the modeling system taking into account both types of non-localities possesses a rich set of the TW solutions, including periodic, quasi-periodic, multi-periodic, soliton-like and chaotic regimes.…”

“…Now, comparing the RHS of second equation of system (24) with partial derivative of ( 25) with respect to R, we get the system…”

“…The Hamiltonian function ( 26) is a first integral of the system (24), so every phase trajectory can be presented in the form…”