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The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a…

The logistic function is shown to be solution of the Riccati equation, some second-order nonlinear ordinary differential equations and many third-order nonlinear ordinary differential equations. The list of the differential equations having…

In this paper we study a mixed problem for the nonlinear Schr\"odinger equation globally that have a nonlinear adding, in which the coefficient is a generalized function. Here is proved a global solvability theorem of the considered problem…

We describe a solving semi-decision method based on examination of the rational structures of the generalized integrating factors of first-order ODEs. We propose a conjecture that for some family of equations of the type…

An explicit perturbative solution to all orders is given for a general class of nonlinear differential equations. This solution is written as a sum indexed by rooted trees and uses the Green function of a linearization of the equations. The…

Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links…

In this article we study a class of generalised linear systems of difference equations with given non-consistent initial conditions and infinite many solutions. We take into consideration the case that the coefficients are square constant…

In this manuscript we prove the existence of solutions to a fully nonlinear system of (degenerate) elliptic equations of Lane-Emden type and discuss a inhomogeneous generalization.

In this paper, we study the global existence and regularity of H\"older continuous solutions for a series of nonlinear partial differential equations describing nonlinear waves.

In present paper we propose an approach based on examination of the structure of the general solution of equations of the type dy/dx=P(x,y)/Q(x,y), with P and Q polynomials only in y. Under the term structure we mean the dependency…

We fully classify completely multiplicative sequences which are given by generalised polynomial formulae, and obtain a similar result for (not necessarily completely) multiplicative sequences under the additional restriction that the…

We prove a nonlinear regularity principle in sequence spaces which produces universal estimates for special series defined therein. Some consequences are obtained and, in particular, we establish new inclusion theorems for multiple summing…

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…

We investigate the presence of localized solutions in models described by a single real scalar field with generalized dynamics. The study offers a method to solve very intricate nonlinear ordinary differential equations, and we illustrate…

Complex-linearization of a class of systems of second order ordinary differential equations (ODEs) has already been studied with complex symmetry analysis. Linearization of this class has been achieved earlier by complex method, however,…

In this article we develop and analyze novel iterative regularization techniques for the solution of systems of nonlinear ill--posed operator equations. The basic idea consists in considering separately each equation of this system and…

In this paper we propose local and global existence results for the solution of systems characterized by the coupling of ODEs and PDEs. The coexistence of distinct mathematical formalisms represents the main feature of hybrid approaches, in…

The aim of this text is to extend the theory of generalized ordinary differential equations to the setting of metric spaces. We present existence and uniqueness theorems that significantly improve previous results even when restricted back…

We present a new solution for fundamental problems in nonlinear dynamical systems: finding, verifying, and stabilizing cycles. The solution we propose consists of a new control method based on mixing previous states of the system (or the…

We prove a general theorem on the stochastic convergence of appropriately renormalized models arising from nonlinear stochastic PDEs. The theory of regularity structures gives a fairly automated framework for studying these problems but…