Related papers: Integrable 7-point discrete equations and evolutio…
A large class of variational equations for geometric objects is studied. The results imply conformal monotonicity and Liouville theorems for steady, polytropic, ideal flow, and the regularity of weak solutions to generalized Yang-Mills and…
In this paper we give a description of the asymptotic behavior, as $\epsilon\to 0$, of the $\epsilon$-gradient flow in the finite dimensional case. Under very general assumptions we prove that it converges to an evolution obtained by…
We propose a new analyzing method, which is called the tautological flow method, to analyze the integrability of partial difference equations (P$\Delta$Es) based on that of partial differential equations (PDEs). By using this method, we…
It is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations, members of which are compatible between one another on a higher dimensional lattice. There turn out to be two such families, a…
Lamb has identified a certain class of moving space curves with soliton equations. We show that there are two other classes of curve evolution that may be so identified. Hence three distinct classes of curve evolution are associated with a…
We give a complete point-symmetry classification of all third-order evolution equations of the form $u_t=F(t,x,u,u_x, u_{xx})u_{xxx}+G(t,x,u,u_x, u_{xx})$ which admit semi-simple symmetry algebras and extensions of these semi-simple Lie…
We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of…
In this paper, we develop discrete versions of Darboux transformations and Crum's theorems for two second order difference equations. The difference equations are discretised versions (using Darboux transformations) of the spectral problems…
We consider a general formulation of gradient flow evolution for problems whose natural framework is the one of metric spaces. The applications we deal with are concerned with the evolution of {\it capacitary measures} with respect to the…
A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence, uniqueness and path-continuity of infinite-time solutions is proved by an extension of the Ovsyannikov method. This…
One of the difficulties encountered when studying physical theories in discrete space-time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of the ways of addressing this difficulty is to…
The Yang-Mills gradient flow and the observable E(t), defined by the square of the field strength tensor at t>0, are calculated at finite lattice spacing and tree-level in the gauge coupling. Improvement of the flow, the gauge action and…
Riccati's differential equation is formulated as abstract equation in finite or infinite dimensional Banach spaces. Since the Riccati's differential equation with the Cole-Hopf transform shows a relation between the first order evolution…
An iterative scheme is presented to solve analytically the relativistic fluid dynamics equations. The scheme is applied to longitudinal expansion, transversal symmetric and transversal asymmetric (triaxial) expansion as well. Within this…
In this paper, we develop an algebraic approach to classifying contact symmetries of the second-order nonlinear evolution equations. Up to contact isomorphisms, all inequivalent PDEs admitting semi-simple algebras, solvable algebras of…
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal…
We present the explicit formulae, describing the structure of symmetries and formal symmetries of any scalar (1+1)-dimensional evolution equation. Using these results, the formulae for the leading terms of commutators of two symmetries and…
An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of first degree for systems of discrete evolution equations and an answer to why there exist…
We establish unique continuation for various discrete nonlinear wave equations. For example, we show that if two solutions of the Toda lattice coincide for one lattice point in some arbitrarily small time interval, then they coincide…
Methods for the computation of invariants and symmetries of nonlinear evolution, wave, and lattice equations are presented. The algorithms are based on dimensional analysis, and can be implemented in any symbolic language, such as…