Geometry of Darboux-Manakov-Zakharov systems and its application

The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov systems of semilinear partial differential equations \label{GDMZabstract} \frac{\partial^2 u}{\partial x_i\partial x_j}=f_{ij}\Big(x_k,u,\frac{\partial u}{\partial x_l}\Big), 1\leq i<j\leq n, k,l\in\{1,...,n\} for a real-valued function $u(x_1,...,x_n)$ are studied with particular reference to the linear systems in this equation class. System (\ref{GDMZabstract}) will not generally be involutive in the sense of Cartan: its coefficients will be constrained by complicated nonlinear integrability conditions. We derive geometric tools for explicitly constructing involutive systems of the form (\ref{GDMZabstract}), essentially solving the integrability conditions. Specializing to the linear case provides us with a novel way of viewing and solving the multi-dimensional $n$-wave resonant interaction system and its modified version as well as constructing new examples of semi-Hamiltonian systems of hydrodynamic type. The general theory is illustrated by a study of these applications

Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space $X$ which is acted on by any continuous semigroup $\{S(t)\}_{t \geq 0}$. Suppose that $\S(t)\}_{t \geq 0}$ possesses a global attractor $\mathcal{A}$. We show that, for any generalized Banach limit $\underset{T \rightarrow \infty}{\rm{LIM}}$ and any distribution of initial conditions $\mathfrak{m}_0$, that there exists an invariant probability measure $\mathfrak{m}$, whose support is contained in $\mathcal{A}$, such that $$\int_{X} \phi(x) d\mathfrak{m} (x) = \underset{T\to \infty}{\rm{LIM}} \frac{1}{T}\int_0^T \int_X \phi(S(t) x) d \mathfrak{m}_0(x) d t,$$ for all observables $\phi$ living in a suitable function space of continuous mappings on $X$. This work is based on a functional analytic framework simplifying and generalizing previous works in this direction. In particular our results rely on the novel use of a general but elementary topological observation, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of compact sets. In the case when $\{S(t)\}_{t \geq 0}$ does not possess a compact absorbing set, this lemma allows us to sidestep the use of weak compactness arguments which require the imposition of cumbersome weak continuity conditions and limits the phase space $X$ to the case of a reflexive Banach space. Two examples of concrete dynamical systems where the semigroup is known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic

Semilinear Hyperbolic Equations in Curved Spacetime

This is a survey of the author's recent work rather than a broad survey of the literature. The survey is concerned with the global in time solutions of the Cauchy problem for matter waves propagating in the curved spacetimes, which can be, in particular, modeled by cosmological models. We examine the global in time solutions of some class of semililear hyperbolic equations, such as the Klein-Gordon equation, which includes the Higgs boson equation in the Minkowski spacetime, de Sitter spacetime, and Einstein & de Sitter spacetime. The crucial tool for the obtaining those results is a new approach suggested by the author based on the integral transform with the kernel containing the hypergeometric function.\\ {\bf Mathematics Subject Classification (2010):} Primary 35L71, 35L53; Secondary 81T20, 35C15.\\ {\bf Keywords:} \small {de Sitter spacetime; Klein-Gordon equation; Global solutions; Huygens' principle; Higuchi bound}Comment: arXiv admin note: text overlap with arXiv:1206.023

On the Mathematical and Geometrical Structure of the Determining Equations for Shear Waves in Nonlinear Isotropic Incompressible Elastodynamics

Using the theory of $1+1$ hyperbolic systems we put in perspective the mathematical and geometrical structure of the celebrated circularly polarized waves solutions for isotropic hyperelastic materials determined by Carroll in Acta Mechanica 3 (1967) 167--181. We show that a natural generalization of this class of solutions yields an infinite family of \emph{linear} solutions for the equations of isotropic elastodynamics. Moreover, we determine a huge class of hyperbolic partial differential equations having the same property of the shear wave system. Restricting the attention to the usual first order asymptotic approximation of the equations determining transverse waves we provide the complete integration of this system using generalized symmetries.Comment: 19 page

Nonlinear Dirac and diffusion equations in 1 + 1 dimensions from stochastic considerations

We generalize the method of obtaining the fundamental linear partial differential equations such as the diffusion and Schrodinger equation, Dirac and telegrapher's equation from a simple stochastic consideration to arrive at certain nonlinear form of these equations. The group classification through one parameter group of transformation for two of these equations is also carried out.Comment: 18 pages, Latex file, some equations corrected and group analysis in one more case adde