On the localized invariant solutions of some non-local hydrodynamic-type models. (English) Zbl 1100.35512

Nikitin, A.G.(ed.) et al., Proceedings of the fifth international conference on symmetry in nonlinear mathematical physics, Kyïv, Ukraine, June 23–29, 2003. Part 3. Kyïv: Institute of Mathematics of NAS of Ukraine (ISBN 966-02-3227-6). Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine. Mathematics and its Applications. 50(3), 1510-1517 (2004).
The authors analyze conditions assuring existence of invariant wave patterns in non-local hydrodynamic models of structured media. To prove existence of nonlinear wave patterns, a combination of symmetry reduction and qualitative analysis is used. A family of invariant solutions of some modelling systems of PDE with account of non-local effects is considered. These effects are manifested when an intense pulse loading (impact, explosion etc.) is applied to media, possessing an internal structure on mesoscale. The models studied in the paper apply when the continual approach is still valid, but presence of the internal structure cannot be ignored. Two non-local models are presented describing long wave propagation in the media with internal structure. The main result obtained is that the hydrodynamic-type systems, with non-local effects taken into account, possess periodic and soliton-like travelling wave solutions. Presence of these solutions is pointed out to be a direct consequence of the non-local effects, since an arbitrary local hydrodynamic-type model would not possess such solutions.
For the entire collection see [Zbl 1088.17002].


35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35L70 Second-order nonlinear hyperbolic equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems