Related papers: The AB equations and the $\bar\partial$-dressing m…

In this paper, we study the Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the…

We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the…

We formulate the soliton equations on the lattice in terms of the reduced Moyal algebra which includes one parameter. The vanishing limit of the parameter leads to the continuous soliton equations.

We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the…

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}_\varepsilon$ with periodic coefficients depending on ${\mathbf x}/ \varepsilon$, $\varepsilon>0$. We study the…

Dunajski generalization of the second heavenly equation is studied. A dressing scheme applicable to Dunajski equation is developed, an example of constructing solutions in terms of implicit functions is considered. Dunajski equation…

In this paper we present a numerical scheme for solving a system of Boussinesq-type equations. This can correspond to longitudinal displacements in a multi-layered elastic bar with delamination, with conditions on the interface between the…

The work is devoted to the development of numerical methods for computing "formal solutions" of interval systems of linear algebraic equations. These solutions are found in Kaucher interval arithmetic, which extends and completes the…

A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical…

In this paper we present the unification of two existing numerical methods for the construction of solutions of the Korteweg-de Vries (KdV) equation. The first method is used to solve the Cauchy initial-value problem on the line for rapidly…

We study regularity and decay properties for the solutions of the Cauchy problem for time-fractional partial differential equations, with tempered initial data, belonging to suitable (weighted) Sobolev spaces, associated with a differential…

We consider a quasilinear parabolic Cauchy problem with spatial anisotropy of orthotropic type and study the spatial localization of solutions. Assuming the initial datum is localized with respect to a coordinate having slow diffusion rate,…

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m)x U(n)). The simplest representative of the corresponding integrable hierarchy is given by a multi-component Kaup-Newell derivative nonlinear…

Two discretizations of the vector nonlinear Schrodinger (NLS) equation are studied. One of these discretizations, referred to as the symmetric system, is a natural vector extension of the scalar integrable discrete NLS equation. The other…

In this paper we study the construction of a discrete solution for a hyperbolic system of partial differentials of the strongly coupled type. In its construction, the discrete separation of matricial variable method was followed. Two…

We consider $N$-wave type equations related to the orthogonal algebras obtained from the generic ones via additional reductions. The first $\mathbb{Z}_2$-reduction is the canonical one. We impose a second $\mathbb{Z}_2$-reduction and…

This paper is a survey on universal algorithms for solving the matrix Bellman equations over semirings and especially tropical and idempotent semirings. However, original algorithms are also presented. Some applications and software…

In this work, we study the Cauchy problem of the Elastic Beam equation with initial value in weighted Sobolev space $H^{1,1}(\mathbb{R})$ via the $\bar{\partial}$-steepset descent method. Begin with the Lax pair of the Elastic Beam…

Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…

In this paper a special semi-smooth equation associated to the second order cone is studied. It is shown that, under mild assumptions, the semi-smooth Newton method applied to this equation is well-defined and the generated sequence is…