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The Hamiltonian structure of the two-dimensional dispersionless Toda hierarchy is studied, this being a particular example of a system of hydrodynamic type. The polynomial conservation laws for the system turn out, after a change of…
We extend the modeling framework of port-Hamiltonian descriptor systems to include under- and over-determined systems and arbitrary differentiable Hamiltonian functions. This structure is associated with a Dirac structure that encloses its…
Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semiclassical models with mode…
We consider the construction of the deformed Whitham system for the KdV-equation in the one-phase case and investigate the conservation of the Hamiltonian properties in this situation. It is shown then, that both the Gardner - Zakharov -…
The closed string model in the background gravity field is considered as a bi-Hamiltonian system in assumption that string model is the integrable model for particular kind of the background fields. The dual nonlocal Poisson brackets(PB),…
We discuss from a bi-Hamiltonian point of view the Hamilton-Jacobi separability of a few dynamical systems. They are shown to admit, in their natural phase space, a quasi-bi-Hamiltonian formulation of Pfaffian type. This property allows us…
We study the hydrodynamic-type system of differential equations modeling isothermal no-slip drift flux. Using the facts that the system is partially coupled and its subsystem reduces to the (1+1)-dimensional Klein--Gordon equation, we…
We consider multi-gradient fluids endowed with a volumetric internal energy which is a function of mass density, volumetric entropy and their successive gradients. We obtained the thermodynamic forms of equation of motions and equation of…
Physical systems with symmetry arise abundantly in applications, and are endowed with interesting mathematical structures. The present paper focusses on linear reciprocal and input-output Hamiltonian systems. Their characterization is…
We obtain, for a subclass of structure functions characterizing a first class Hamiltonian system, recursive relations from which the general form of the local symmetry transformations can be constructed in terms of the independent gauge…
This paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known integrable discretization of the Non-linear Schrodinger system) can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect…
We present a geometrical demonstration for persistence properties for a bi-Hamiltonian system modelling waves in a shallow water regime. Both periodic and non-periodic cases are considered and a key ingredient in our approach is one of the…
We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV…
We consider the Gibbs representation over space-time of non-equilibrium dynamics of Hamiltonian systems defined on a lattice with local interactions. We first write the corresponding action functional as a sum of local terms, defining a…
We investigate the energy transfer dynamics in a donor-acceptor model by developing a time-local master equation technique based on a variational transformation of the underlying Hamiltonian. The variational transformation allows a…
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…
We construct so-called Darboux transformations and solutions of the dynamical Hamiltonian systems with several space variables $\frac{\partial \psi}{\partial t}=\sum_{k=1}^r H_k(t)\frac{\partial \psi}{\partial \zeta_k}\,$ $( H_k(t)=…
Two identical particles driven by the same steady force through a viscous fluid may move relative to one another due to hydrodynamic interactions. The presence or absence of this relative translation has a profound effect on the dynamics of…
A rigid body in an ideal fluid is an important example of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(3) = so(3)\ltimes\mathbb R^3$. We present the bi-Hamiltonian structure and the corresponding variables of…
The time evolution governed by the Boltzmann kinetic equation is compatible with mechanics and thermodynamics. The former compatibility is mathematically expressed in the Hamiltonian and Godunov structures, the latter in the structure of…