Related papers: Hodge-Frobenius equations and the Hodge-Backlund t…
A geometric interpretation is given for certain elliptic-hyperbolic systems in the plane. Among several examples, one which reduces in the elliptic region to the equations for harmonic 1-forms on the projective disc is studied in detail. A…
Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the Crocco transformation, with a first-order partial derivative taken as a new independent variable and a second-order…
A gauge-invariant form of the nonlinear Hodge equations is studied.
We develop a framework that systematically casts the solvability and uniqueness conditions of linearized geometric boundary-value problems into cohomological terms. The theory is designed to be applicable without assumptions on the…
We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks,…
The Hodge equations for 1-forms are studied on Beltrami's projective disc model for hyperbolic space. Ideal points lying beyond projective infinity arise naturally in both the geometric and analytic arguments. An existence theorem for…
Duality methods are used to generate explicit solutions to nonlinear Hodge systems, demonstrate the well-posedness of boundary value problems, and reveal, via the Hodge-B\"acklund transformation, underlying symmetries among superficially…
This paper is a survey of the subject of variations of Hodge structure (VHS) considered as exterior differential systems (EDS). We review developments over the last twenty-six years, with an emphasis on some key examples. In the penultimate…
An alternative approach - nonequilibrium evolution thermodynamics, is compared with classical Landau approach. A statistical justification of the approach is carried out with help of probability distribution function on an example of a…
Backlund transformations are used to search for solutions, particularly soliton solutions, of non-linear differential equations. In this paper we present an invariant geometrical theory of Backlund transformations for second order evolution…
This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line,…
The objective of this work is to establish a systematic study of boundary value problems within the framework of differential forms and variable exponent spaces. Specifically, we investigate the Hodge Laplacian and related first order…
We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work…
In this paper, we first get a criterion formula for whether a differential form is holomorphic with respect to the generalized complex structure induced by $\epsilon$. Next, we get the local extensions of $\overline\partial$-closed forms on…
We investigate a relationship between a particular class of two-dimensional integrable non-linear $\sigma$-models and variations of Hodge structures. Concretely, our aim is to study the classical dynamics of the $\lambda$-deformed $G/G$…
A generalized KdV equation is formulated as an exterior differential system, which is used to determine the prolongation structure of the equation. The prolongation structure is obtained for several cases of the variable powers, and…
This work develops a procedure to find classes of Lagrangian densities that describe generalizations of the Abelian Maxwell-Higgs, the Chern-Simons-Higgs and the Maxwell-Chern-Simons-Higgs models. The investigation focuses on the…
A class of variable coefficient (1+1)-dimensional nonlinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u^nu_x)_x+h(x)u^m$ is investigated. Different kinds of equivalence groups are constructed including ones with…
Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the von Mises transformation, with the unknown function taken as the new independent variable and an appropriate partial…
Alternative approach for description of the non-equilibrium phenomena arising in solids at a severe external loading is analyzed. The approach is based on the new form of kinetic equations in terms of the internal and modified free energy.…