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

Abstract

We analyze the conditions, which guarantee the existence of periodic and soliton-like traveling wave solutions in the non-local hydrodynamic model of structured media.

Traveling Wave Solutions of the Nonlocal Brenner Hydrodynamic Models

“…The above general relation can be reduced to a weaker nonlocal case by expanding the integral term into the power series (cf. with [Peerling, 1999;Vladimirov et al, 2012]):…”

Bifurcation Approach to Analysis of Travelling Waves in Nonlocal Hydrodynamic-Type Models

“…A hydrodynamic system of balance equations for mass and momentum is considered by Vladimirov et al [1]:…”

“…This system is closed by the dynamic equation of state, taking into account the effects of spatiotemporal nonlocalities. Using group theory reduction, the authors of [1] obtained the following system of the ordinary differential equations:…”

“…which describes a set of approximate travelling wave solutions to the source system (1) (see the initial system (1) of PDEs in [1]), where , and 1 , are constant parameters; ≪ 0. When = 0, it immediately obtains that = 1 = const, and the system (2) reduces to the following two-dimensional system:…”

“…1 , then system (3) possesses a one-parameter family of periodic solutions, localized around the critical point 2 ( 2 , 0) in a bounded set . The boundary of this set is formed by the homoclinic intersection of separatrices of the saddle point 1 ( 1 , 0).…”