Multi-component integrable generalizations of the Fokas-Lenells equation, associated with each irreducible Hermitian symmetric space are formulated. Description of the underlying structures associated to the integrability, such as the Lax representat
ion and the bi-Hamiltonian formulation of the equations is provided. Two reductions are considered as well, one of which leads to a nonlocal integrable model. Examples with Hermitian symmetric spaces of all classical series of types A.III, BD.I, C.I and D.III are presented in details, as well as possibilities for further reductions in a general form.
The matter creation model of Prigogine--Geheniau--Gunzig--Nardone is revisited in terms of a redefined creation pressure which does not lead to irreversible adiabatic evolution at constant specific entropy. With the resulting freedom to choose a part
icular gas process, a flat FRWL cosmological model is proposed based on three input characteristics: (i) a perfect fluid comprising of an ideal gas, (ii) a quasi-adiabatic polytropic process, and (iii) a particular rate of particle creation. Such model leads to the description of the late-time acceleration of the expanding Universe with a natural transition from decelerating to accelerating regime. Only the Friedmann equations and the laws of thermodynamics are used and no assumptions of dark energy component is made. The model also allows the explicit determination as functions of time of all variables, including the entropy, the non-conserved specific entropy and the time the accelerating phase begins. A form of correspondence with the dark energy models (quintessence, in particular) is established via the $Om$ diagnostics. Parallels with the concordance cosmological $Lambda$CDM model for the matter-dominated epoch and the present epoch of accelerated expansion are also established via slight modifications of both models.
The present paper is dedicated to integrable models with Mikhailov reduction groups $G_R simeq mathbb{D}_h.$ Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on
the realization of the $G_R$-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with $mathbb{D}_h$ symmetries are presented.
The cosmological dynamics of a quintessence model based on real gas with general equation of state is presented within the framework of a three-dimensional dynamical system describing the time evolution of the number density, the Hubble parameter, an
d the temperature. Two global first integrals are found and examples for gas with virial expansion and van der Waals gas are presented. The van der Waals system is completely integrable. In addition to the unbounded trajectories, stemming from the presence of the conserved quantities, stable periodic solutions (closed orbits) also exist under certain conditions and these represent models of a cyclic Universe. The cyclic solutions exhibit regions characterised by inflation and deflation, while the open trajectories are characterised by inflation in a fly-by near an unstable critical point.
A two-layer fluid system separated by a pycnocline in the form of an internal wave is considered. The lower layer is infinitely deep, with a higher density than the upper layer which is bounded above by a flat surface. The fluids are incompressible a
nd inviscid. A Hamiltonian formulation for the fluid dynamics is presented and it is shown that an appropriate scaling leads to the integrable Benjamin-Ono equation.
A single incompressible, inviscid, irrotational fluid medium bounded by a free surface and varying bottom is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the surface w
aves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types taking into account the effect of the slowly varying bottom. The arising KdV equation with variable coefficients is studied numerically when the initial condition is in the form of the one soliton solution for the initial depth.
