A discussion is presented, within a simple unifying scheme, about different types of symmetry of PDE's, with the introduction and a precise characterization of the notions of "standard" and "weak" conditional symmetries, together with their relationship with exact and partial symmetries. An extensive use of "symmetry-adapted" variables will be made, and some clarifying examples are provided.
Two essential methods, the symmetry analysis and of the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations are discussed. The main similarities and differences of these two different methods are given.
A New Non-Linear Density Fluctuations Stochastic Partial Differential Equation With a Singular Coefficient of Relevance to Polymer Dynamics and Rheology: Discussions, Proofs of Solution Existence, Uniqueness, and a Conjecture
This paper describes the notion of σ-symmetry, which extends the one of λ-symmetry, and its application to reduction procedures of systems of ordinary differential equations and of dynamical systems as well. We also consider orbital symmetries, which give rise to a different form of reduction of dynamical systems. Finally, we discuss how dynamical systems can be transformed into higher-order ordinary differential equations, and how these symmetry properties of the dynamical systems can be transferred into reduction properties of the corresponding ordinary differential equations. Many examples …
It is known that the notion of graded differential algebra coincides with the notion of monoid in the monoidal category of complexes. By using the monoidal structure introduced by M. Kapranov for the category of $N$-complexes we define the corresponding generalization of graded differential algebras as the monoids of this category. It turns out that this generalization coincides with the notion of graded $q$-differential algebra which has been previously introduced and studied.
When we consider a differential equation $Δ=0$ whose set of solutions is ${\cal S}_Δ$, a Lie-point exact symmetry of this is a Lie-point invertible transformation $T$ such that $T({\cal S}_Δ)={\cal S}_Δ$, i.e. such that any solution to $Δ=0$ is tranformed into a (generally, different) solution to the same equation; here we define {\it partial} symmetries of $Δ=0$ as Lie-point invertible transformations $T$ such that there is a nonempty subset ${\cal P} \subset {\cal S}_Δ$ such that $T({\cal P}) = {\cal P}$, i.e. such that there is a subset of solutions to $Δ=0$ which are transformed one into t…
In this paper, we introduce the notion of a super tangent bundle of a manifold, and extend the basic notions of differential geometry such as differential forms, exterior derivation, connection, metric and divergence on manifolds that equipped with a super tangent bundle.
We develop the notions of multiplicative Lie conformal and Poisson vertex algebras, local and non-local, and their connections to the theory of integrable differential-difference Hamiltonian equations. We establish relations of these notions to $q$-deformed $W$-algebras and lattice Poisson algebras. We introduce the notion of Adler type pseudodifference operators and apply them to integrability of differential-difference Hamiltonian equations.
This note provides an overview of the notion of observable within the setting of multisymplectic geometry. We essentially follow the ideas described by F. Hélein and J. Kouneiher [17] [18] [19] and in particular in keeping with the approach developed in [16]. The following discussion should be considered as a synthesis of the principal motives and results presented in those writings. This note is a contribution to the Frontiers of Fundamental Physics (FFP11) Conference Proceedings.
In this paper we introduce a notion of integrability in the non autonomous sense. For the cases of 1 + 1/2 degrees of freedom and quadratic homogeneous Hamiltonians of 2 + 1/2 degrees of freedom we prove that this notion is equivalent to the classical complete integrability of the system in the extended phase space. For the case of quadratic homogeneous Hamiltonians of 2 + 1/2 degrees of freedom we also give a reciprocal of the Morales-Ramis result. We classify those systems by terms of symplectic change of frames involving algebraic functions of time, and give their canonical forms.
We construct a class of exponential type solutions for the linear, delayed heat equation. These representations may be used to provide a priori ansatzes for certain boundary and/or initial-value problems arising in heat transfer. Several of the important mathematical properties of the representations are examined, including a discussion of the dependence on the delay parameter.
We recall the notion of a differential operator over a smooth map (in linear and non-linear settings) and consider its versions such as formal $\hbar$-differential operators over a map. We study constructions and examples of such operators, which include pullbacks by thick morphisms and quantization of symplectic micromorphisms.
Every second order system of autonomous differential equations can be described by an autonomous holonomic dynamical system with a Lagrangian part, an effective potential and a set of generalized forces. The kinematic part of the Lagrangian defines the kinetic metric which subsequently defines a Riemannian geometry in the configuration space. We consider the generic function $I=K_{ab}(t,q^{c})\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q^{c})\dot{q}% ^{a}+K(t,q^{c})$ and require the quadratic first integral condition $dI/dt=0$ without involving any type of symmetry Lie or Noether. Condition $dI/dt=0$ leads…
In this work we introduce the notion of differential-algebraic ansatz for the heat equation and explicitly construct heat equation and Burgers equation solutions given a solution of a homogeneous non-linear ordinary differential equation of a special form. The ansatz for such solutions is called the $n$-ansatz, where $n+1$ is the order of the differential equation.
We introduce the new notion of epsilon-graded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras. We define and study the associated notion of epsilon-derivation-based differential calculus, which generalizes the derivation-based differential calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of epsilon-graded algebras, in particular some graded matrix algebras and the Moyal algebra. This last example permits also t…
In this paper we use the Riemann-Hilbert problem, with jumps supported on appropriate curves in the complex plane, for matrix biorthogonal polynomials and apply it to find Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions as well. For this aim, Sylvester type differential Pearson equations for the matrix of weights are shown to be instrumental. Several applications are given, in order of increasing complexity. First, a general discussion of non-Abelian Hermite biorthogonal polynomials in the real line, understood as those whose matrix of w…
