Complicated travelling wave solutions of a modelling system describing media with memory and spatial nonlocality

Traveling Wave Solutions of the Nonlocal Brenner Hydrodynamic Models

“…5(b)], demonstrates the hysteretic features within the values of the parameters for which the second chaotic attractor appears. This phenomenon is rather typical for nonlinear dynamical systems [Vladimirov et al, 1999;Leonov & Kuznetsov, 2013]. …”

On the non-local hydrodynamic-type system and its soliton-like solutions

“…As it is shown in [18,19], the properties of traveling wave solutions to a non-local hydrodynamic-type model depend in essential way on the values of the parameters.…”

“…In the following sections we analyze in detail the system (19) for ǫ = 0 and formulate the conditions of the existence of periodic solutions as well as the homoclinic regimes, corresponding to soliton-like travelling wave solutions of the initial system of PDEs.…”

“…The purpose of this section is to analyze whether and when the perturbed system (19) possesses the homoclinic solution. Our analysis is based on the generalized Melnikov theory presented in [23].…”

“…Since we are going to use the formalism developed in [23], we should pass in (19) to the independent variable T , for which the unperturbed system becomes Hamiltonian. The standard representation for our system in this case will be as follows:…”