Related papers: Fr\"olicher-Nijenhuis geometry and integrable matr…
This is a generalization of the procedure presented in [3] to construct semisimple bi-flat $F$-manifolds $(M,\nabla^{(1)},\nabla^{(2)},\circ,*,e,E)$ starting from homogeneous solutions of degree -1 of Darboux-Egorov-system. The Lam\'e…
Let $\omega$ be a plane autonomous system and C its configuration of algebraic integral curves. If the singularities of C are quasi-homogeneous we we present new criteria that guarantee Darboux integrability. We use this to construct…
We use a binary Darboux transformation to obtain exact multisoliton solutions of the principal chiral model and its noncommutative generalization. We also show that the exact multisolitons of the noncommutative principal chiral model in two…
The study of symmetries of partial differential equations (PDEs) has been traditionally treated as a geometrical problem. Although geometrical methods have been proven effective with regard to finding infinitesimal symmetry transformations,…
Local M-smoothers are interesting and important signal and image processing techniques with many connections to other methods. In our paper we derive a family of partial differential equations (PDEs) that result in one, two, and three…
From a given Fokker-Planck equation, a multi-parameter deformed partner Fokker-Planck equation is constructed. This is done by first deleting a set of eigenstates of the original FPE by the multi-step Darboux-Crum transformation, and then…
The first part of the book is devoted to the symmetry approach to classification of scalar integrable evolution PDEs with two independent variables. In the second part systems of evolution equations with polynomial homogeneous right-hand…
Solutions of partial differential equations (PDEs) on manifolds have provided important applications in different fields in science and engineering. Existing methods are majorly based on discretization of manifolds as implicit functions,…
As the third of our series of papers on differential geometry of microlinear Frolicher spaces, this paper is devoted to the Frolicher-Nijenhuis calculus of their named bracket. The main result is that the Frolicher-Nijenhuis bracket…
The elliptic 2-Hessian equation is a fully nonlinear partial differential equation (PDE) that is related to intrinsic curvature for three dimensional manifolds. We introduce two numerical methods for this PDE: the first is provably…
Frobenius companion matrices arise when we write an $n$-th order linear ordinary differential equation as a system of first order differential equations. These matrices and their transpose have very nice properties. By using the powers of…
Partial differential equations (PDEs) play a central role in describing many physical phenomena. Various scientific and engineering applications demand a versatile and differentiable PDE solver that can quickly generate solutions 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…
Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of heavenly equations of Plebanski that govern heavenly gravitational metrics. In this paper, we present a class of scalar second-order PDEs…
We consider a system of PDEs of Monge-Kantorovich type arising from models in granular matter theory and in electrodynamics of hard superconductors. The existence of a solution of such system (in a regular open domain…
We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities.
The core object of this paper is a pair $(L, e)$, where $L$ is a Nijenhuis operator and $e$ is a vector field satisfying a specific Lie derivative condition, i.e., $Lie_{e}L=\operatorname{Id}$. Our research unfolds in two parts. In the…
We study integrable non-degenerate Monge-Ampere equations of Hirota type in 4D and demonstrate that their symmetry algebras have a distinguished graded structure, uniquely determining the equations. This is used to deform these heavenly…
An integrable field theory, due to path-independence on the space-time plane, should yield together with an infinite set of independent conserved charges also similar dual charges determining the boundary and defect contributions. On the…
We give sufficient conditions for three- or four-dimensional truncated Poincare-Dulac normal forms of resonance degree two to be meromorphically nonintegrable when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues…