Related papers: Time-Dependent Symmetries of Variable-Coefficient …
A {\it Lie system} is a nonautonomous system of first-order differential equations admitting a {\it superposition rule}, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants.…
It was recently demonstrated that time-dependent PDE problems can numerically be solved with a fully pseudospectral scheme, i.e. using spectral expansions with respect to both spatial and time directions (Hennig and Ansorg, 2009 [15]). This…
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…
This work represents a PhD thesis concerning three main topics. The first one deals with the study and applications of Lie systems with compatible geometric structures, e.g. symplectic, Poisson, Dirac, Jacobi, among others. Many new Lie…
We perform enhanced Lie symmetry analysis of generalized fifth-order Korteweg-de Vries equations with time-dependent coefficients. The corresponding similarity reductions are classified and some exact solutions are constructed.
We consider a second order linear evolution equation with a dissipative term multiplied by a time-dependent coefficient. Our aim is to design the coefficient in such a way that all solutions decay in time as fast as possible. We discover…
The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order…
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…
We study a class of infinite dimensional Lie algebras called generalized Witt algebras (in one variable). These include the classical Witt algebra and the centerless Virasoro algebra as important examples. We show that any such generalized…
In this paper, we study Whittaker modules for graded Lie algebras. We define Whittaker modules for a class of graded Lie algebras and obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules and the set…
We associate an square to any two dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behaviour of the algebra. We determine the identities of degrees…
Can we effectively learn a nonlinear representation in time comparable to linear learning? We describe a new algorithm that explicitly and adaptively expands higher-order interaction features over base linear representations. The algorithm…
The normal form for a system of ode's is constructed from its polynomial symmetries of the linear part of the system, which is assumed to be semi-simple. The symmetries are shown to have a simple structure such as invariant function times…
One-dimensional nonrelativistic systems are studied when time-independent potential interactions are involved. Their supersymmetries are determined and their closed subsets generating kinematical invariance Lie superalgebras are pointed…
Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning…
The properties of nonlinear PDEs that generate filtered solutions are explored with particular attention given to the constraints on the residual term. The analysis is carried out for nonlinear PDEs with an emphasis on evolution problems…
The relationship is made between matrix integrals, Toda master-symmetries, Virasoro constraints and orthogonal polynomials.
We develop some calculation schemes to determine dynamics of a wide class of integrable quantum-optical models using their symmetry adapted reformulation in terms of polynomial Lie algebras $su_{pd}(2)$. These schemes, based on "diagonal"…
Twisted symmetries, widely studied in the last decade, proved to be as effective as standard ones in the analysis and reduction of nonlinear equations. We explain this effectiveness in terms of a Lie-Frobenius reduction; this requires to…
We develop a gradient-flow theory for time-dependent functionals defined in abstract metric spaces. Global well-posedness and asymptotic behavior of solutions are provided. Conditions on functionals and metric spaces allow to consider the…