相关论文: Time Dependent Recursion Operators and Symmetries

We consider the recursion operators with nonlocal terms of special form for evolution systems in (1+1) dimensions, and extend them to well-defined operators on the space of nonlocal symmetries associated with the so-called universal Abelian…

We report a class of symmetry-intergable third-order evolution equations in 1+1 dimensions under the condition that the equations admit a second-order recursion operator that contains an adjoint symmetry (integrating factor) of order six.…

We present easily verifiable sufficient conditions of time-independence and commutativity for local and nonlocal symmetries for a large class of homogeneous (1+1)-dimensional evolution systems. In contrast with the majority of known…

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…

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear PDE to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order…

We consider $u_t=u^{\alpha} u_{xxx}+n(u)u_xu_{xx}+m(u)u_x^3+ r(u)u_{xx} +p(u)u_x^2 + q(u)u_x+s(u)$ with $\alpha=0$ and $\alpha=3$, for those functional forms of $m, n, p, q, r, s$ for which the equation is integrable in the sense of an…

It is widely known that the recursion operator is a very important component of integrability. It allows one to describe in a compact form both hierarchies of the generalized symmetries and infinite series of the local conservation laws. In…

We find a sequence consisting of time dependent evolution vector fields whose time independent part corresponds to the master symmetries for the Toda equations. Each master symmetry decomposes as a sum consisting of a group symmetry and a…

The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to…

An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a…

In geometry of nonlinear partial differential equations, recursion operators that act on symmetries of an equation $\mathcal{E}$ are understood as B\"{a}cklund auto-transformations of the equation $\mathcal{TE}$ tangent to $\mathcal{E}$. We…

We study the phenomenon of composite operator renormalization and mixing in systems where time-translational invariance is broken and the evolution is out-of-equilibrium. We show that composite operators mix also through non-local memory…

We provide a concise introduction to the symmetry approach to integrability. Some results on integrable evolution and systems of evolution equations are reviewed. Quasi-local recursion and Hamiltonian operators are discussed. We further…

The problem of classification into symmetry integrable classes is solved for a family of second order nonlinear evolution equations labeled by arbitrary functions. Four nonequivalent symmetry integrable classes are thus obtained and the…

The $x$-dependence of the symmetries of (1+1)-dimensional scalar translationally invariant evolution equations is described. The sufficient condition of (quasi)polynomiality in time $t$ of the symmetries of evolution equations with constant…

Using the methods of the theory of formal symmetries, we obtain new easily verifiable sufficient conditions for a recursion operator to produce a hierarchy of local generalized symmetries. An important advantage of our approach is that…

This article studies the solutions of time-dependent differential inclusions which is motivated by their utility in the modeling of certain physical systems. The differential inclusion is described by a time-dependent set-valued mapping…

Non-relativistic quantum theory of non-interacting particles in the spacetime containing a region with closed time-like curves (time-machine spacetime) is considered with the help of the path-integral technique. It is argued that, in…

We find direct and inverse recursion operators for integrable cases of the rmdKP and rdDym equations. Also, we study actions of these operators on the contact symmetries and find shadows of nonlocal symmetries of these equations.

It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate them usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient…