Hamilton's Principle and Rankine–Hugoniot Conditions for General Motions of Mixtures

In previous papers [1, 2], we presented hyperbolic governing equations and jump conditions for barotropic fluid mixtures. Now we extend our results to the most general case of twocomponent media. We obtain governing equations for each component. This is not a system of conservation laws. Nevertheless, using Hamilton's principle we are able to obtain a complete set of Rankine–Hugoniot conditions. For the twofluid case, the jump relations do not involve the conservation of the total momentum and the total energy.Sommario. In precedenti lavori [1, 2] sono state dedotte equazioni di governo iperboliche e condizioni di salto per miscele fluide barotropiche. I risultati sono estesi al caso più generale di mezzi a due componenti, ottenendo le equazioni di governo per ciascun componente. Questo sistema non è derivabile dalle leggi di conservazione. Nondimeno, usando il principio di Hamilton è possibile ottenere un insieme completo di condizioni di Rankine–Hugoniot. Nel caso dei due fluidi, le condizioni di salto non coinvolgono la conservazione del momento e dell'energia totali.     

Generalized vorticity for bubbly liquidand dispersive shallow water equations

Received April 1, 2001 / Published online August 31, 2001     

Evolution of extinction spectra of monolayer plasmon-resonant colloidal crystals in the process of their synthesis by the moving meniscus method

Regularities of the evolution of the extinction spectra of periodic colloidal structure composed of silver nanoparticles formed from bulk hydrosol on the surface of dielectric substrate are studied using the moving meniscus method. Spectra were calculated by the coupled dipole method. The effect of individual parameters of disperse system on the pattern of extinction contour in the band of plasmon absorption of a synthesized two-dimensional crystalline structure is analyzed.     

Criterion of hyperbolicity for non‐conservative quasilinear systems admitting a partially convex conservation law

A system of conservation laws admitting an additional convex conservation law can be written as a symmetric t‐hyperbolic in the sense of Friedrichs system. However, in mathematical modeling of complex physical phenomena, it is customary to use non‐conservative hyperbolic models. We generalize the Godunov–Friedrichs–Lax approach to this new class of models. Copyright © 2011 John Wiley & Sons, Ltd.     

Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed.