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In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of time-dependent symmetries. In particular we describe how the finite dimensional Hamiltonian structure of the reduced system is obtained…
We study the character of dependence on the data and the nonlinear structure of the equation for the solutions of the homogeneous Dirichlet problem for the evolution $p(x,t)$-Laplacian with the nonlinear source \[…
In this paper we consider a class of $p$-evolution equations of arbitrary order with variable coefficients depending on time and space variables $(t,x)$. We prove necessary conditions on the decay rates of the coefficients for the…
Several classes of systems of evolution equations with one or two vector unknowns are considered. We investigate also systems with one vector and one scalar unknown. For these classes all equations having the simplest higher symmetry are…
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 survey provides classification results for integrable one-field evolution equations of orders 2, 3 and 5 with the constant separant. The classification is based on necessary integrability conditions following from the existence of the…
A closed expression for the harmonic oscillator wave function after the passage of a linear signal with arbitrary time dependence is derived. Transition probabilities are simple to express in terms of Laguerre polynomials. Spontaneous…
We extend a contraction mapping argument for ordinary state-dependent delay differential equations to evolutionary partial differential equations in the sense of R. Picard, that is, to equations of the form $\bigl(\partial_{t}…
In this article the time evolution operator of two interacting quantum oscillators, whose Hamiltonian is an element of the complex $\left\{ h(1) \oplus h(1) \right\} \uplus u(2)$ algebra, is analyzed using the Feynman time ordering operator…
Solvable structures are exploited in order to find families of explicit solutions to evolution PDEs admitting suitable differential constraints. The effectiveness of the method is verified on several explicit examples.
The paper studies a PDE model for the growth of a tree stem or a vine, having the form of a differential inclusion with state constraints. The equations describe the elongation due to cell growth, and the response to gravity and to external…
We present a list of (1+1)-dimensional second-order evolution equations all connected via a proposed generalised hodograph transformation, resulting in a tree of equations transformable to the linear second-order autonomous evolution…
The time evolution of correlation functions in statistical systems is described by an exact functional differential equation for the corresponding generating functionals. This allows for a systematic discussion of non-equilibrium physics…
The first evolution equation is derived under the Variation Evolving Method (VEM) that seeks optimal solutions with the variation evolution principle. To improve the performance, its compact form is developed. By replacing the states and…
A discrete analogue of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear…
We solve the time evolution of a nonlinear optomechanical Hamiltonian with arbitrary time-dependent mechanical displacement, mechanical single-mode squeezing and a time-dependent optomechanical coupling up to the solution of two…
The present study gives a mathematical framework for self-evolution within autonomous problem solving systems. Special attention is set on universal abstraction, thereof generation by net block homomorphism, consequently multiple order…
This article is the third in a series the aim of which is to use Lie group theory to obtain exact analytic solutions of Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable…
Inspired by the works of \cite{baz2} and \cite{kian}, this study develops an abstract framework for analyzing differential equations with space-dependent fractional time derivatives and bounded operators. Within this framework, we establish…
We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrised family of unbounded linear operators, which are…