Related papers: A Bilinear Approach to Discrete Miura Transformati…
We show how to derive noncommutative versions of integrable partial difference equations using Darboux transformations. As an illustrative example, we use the nonlinear Schr\"odinger (NLS) system. We derive a noncommutative nonlinear…
For certain class of perturbations of the equation $u_t=f(u) u_x$, we prove the existence of change of coordinates, called quasi-Miura transformations, that reduce these perturbed equations to the unperturbed ones. As an application, we…
This work introduces a methodology to solve ordinary differential equations using the Schur decomposition of the linear representation of the differential equation. This is done by first transforming the system into an upper triangular…
We present a method for the resolution of (oscillatory) nonlinear problems. It is based on the application of the Linear Delta Expansion to the Lindstedt-Poincar\'e method. By applying it to the Duffing equation, we show that our method…
In the context of linear inverse problems, we propose and study a general iterative regularization method allowing to consider large classes of regularizers and data-fit terms. The algorithm we propose is based on a primal-dual diagonal…
Analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is discussed. It is based on the generalized Hirota identity. This approach allows to represent generalized hierarchies of…
Using the bilinear formalism, we consider multicomponent and matrix modified KP hierarchies. The main tool is the bilinear identity for the tau-function which is realized as an expectation value of a Clifford group element composed from…
In this paper we construct nonlinear partial differential equations in more than 3 independent variables, possessing a manifold of analytic solutions with high, but not full, dimensionality. For this reason we call them ``partially…
In this work we present a new approach for the implementation of operational Tau method for the solutions of linear differential and integral equations. In our approach we use the three terms relation of an orthogonal polynomial basis to…
The principal aim of this article is to establish an iteration method on the space of resurgent functions. We discuss endless continuability of iterated convolution products of resurgent functions and derive their estimates developing the…
Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this…
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…
We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.
In [1], the non-linear space-time Hasegawa-Mima plasma equation is formulated as a coupled system of two linear PDE's, a solution of which is a pair (u, w). The first equation is of hyperbolic type and the second of elliptic type.…
The discrete Painlev\'e property is precisely defined, and basic discretization rules to preserve it are stated. The discrete Painlev\'e test is enriched with a new method which perturbs the continuum limit and generates infinitely many…
In this paper we develop a new approach to the design of direct numerical methods for multidimensional problems of the calculus of variations. The approach is based on a transformation of the problem with the use of a new class of…
A method of integrable discretization of the Liouville type nonlinear partial differential equations is suggested based on integrals. New examples of discrete Liouville type models are presented.
We propose a method for transformating linear and nonlinear hypersingular integral equations into ordinary differential equations. Linear and nonlinear polyhypersingular integral equations are transformed into partial differential…
This paper deals with non-Lipschitz nonlinear systems. Such systems can be approximated by a linear map of so-called signatures, which play a crucial role in the theory of rough paths and can be interpreted as collections of iterated…
Two main algorithmic approaches are known for making Hironaka's proof of resolution of singularities in characteristic zero constructive. Their main difference is the use of different notions of transforms during the resolution process and…