Related papers: Ideals of partial differential equations
We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as…
In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby modified equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the…
We consider several notions of genericity appearing in algebraic geometry and commutative algebra. Special emphasis is put on various stability notions which are defined in a combinatorial manner and for which a number of equivalent…
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…
We discuss how point transformations can be used for the study of integrability, in particular, for deriving classes of integrable variable-coefficient differential equations. The procedure of finding the equivalence groupoid of a class of…
Some ideas and remarks are presented concerning a possible Lagrangian approach to the study of internal boundary conditions relating integrable fields at the junction of two domains. The main example given in the article concerns single…
We analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution. Earlier results on…
The symmetric space sine-Gordon models arise by conformal reduction of ordinary 2-dim $\sigma$-models, and they are integrable exhibiting a black-hole type metric in target space. We provide a Lagrangian formulation of these systems by…
Motivated by applications to probability and mathematical finance, we consider a parabolic partial differential equation on a half-space whose coefficients are suitably Holder continuous and allowed to grow linearly in the spatial variable…
A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra f. Under certain conditions f-invariant systems of differential…
It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual…
Using conformal coordinates associated with projective conformal relativity we obtain a conformal Klein-Gordon partial differential equation. As a particular case we present and discuss a conformal `radial' d'Alembert-like equation. As a…
This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain H\"older-type classes in which a random field is…
In this article, we present a comprehensive framework for constructing smooth, localized solutions in systems of semi-linear partial differential equations, with a particular emphasis to the Gray-Scott model. Specifically, we construct a…
We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of…
Symmetry, which describes invariance, is an eternal concern in mathematics and physics, especially in the investigation of solutions to the partial differential equation (PDE). A PDE's nonlocally related PDE systems provide excellent…
We consider the discrete Boussinesq integrable system and the compatible set of differential difference, and partial differential equations. The latter not only encode the complete hierarchy of the Boussisesq equation, but also incorporate…
In the present work we examine the dynamics of a model for oscillons in 1-dimensional spacetime field theories with a cubic nonlinearity. We utilize a reduction of the model to first and third harmonics, which leads to a reduced partial…
The sine-Gordon model on the half-line with a dynamical boundary introduced by Delius and one of the authors is considered at quantum level. Classical boundary conditions associated with classical integrability are shown to be preserved at…
This article is focused on two related topics within the study of partial differential equations (PDEs) that illustrate a beautiful connection between dynamics, topology, and analysis: stability and spatial dynamics. The first is a property…