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On the non-local hydrodynamic-type system and its soliton-like solutions. (English) Zbl 1237.76049

Summary: We consider a hydrodynamic system of balance equations for mass and momentum. This system is closed by the dynamic equation of state, taking into account the effects of spatio-temporal non-localities. Using group theory reduction, we obtain a system of ODEs, describing a set of approximate traveling wave solutions to the source system. The factorized system, containing a small parameter, proves to be Hamiltonian when the parameter is zero. Using Melnikov’s method, we show that the factorized system possesses, in general, a one-parameter family of homoclinic loops, corresponding to the approximate soliton-like solutions of the source system.

MSC:

76E99 Hydrodynamic stability
35Q51 Soliton equations
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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