A Semi-Algorithmic Search for Lie Symmetries

Lie symmetries of semi-linear Schrödinger equations and applications

Conditional Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra conf_3. The corresponding representations of the parabolic and almost-parabolic subalgebras of conf_3 are classified and the complete list of conditionally invariant semi-linear Schrödinger equations is obtained. Applications to the phase-orderi…

Symmetries and invariance properties of stochastic differential equations driven by semimartingales with jumps

Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general càdlàg semimartingales taking values in Lie groups are defined and investigated. In order to enlarge the class of possible symmetries of SDEs, the new concepts of gauge and time symmetries for semimartingales on Lie groups are introduced. Markovian and non-Markovian examples of gauge and time symmetric processes are provided. The considered set of SDEs includes affine and Marcus type SDEs as well as smooth SDEs driven by Lévy processes. Non trivial invariance results concerning a class of itera…

Combinatorial theory of the semiclassical evaluation of transport moments II: Algorithmic approach for moment generating functions

Electronic transport through chaotic quantum dots exhibits universal behaviour which can be understood through the semiclassical approximation. Within the approximation, transport moments reduce to codifying classical correlations between scattering trajectories. These can be represented as ribbon graphs and we develop an algorithmic combinatorial method to generate all such graphs with a given genus. This provides an expansion of the linear transport moments for systems both with and without time reversal symmetry. The computational implementation is then able to progress several orders highe…

Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov--Kuznetsov equations

In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical Lie--Ovsiannikov algorithm of construction of symmetries of differential equations, describe the group classification algorithm and discuss the process of reduction of (systems of) PDEs to (systems of) equations with smaller number of independent variables in order to construct invariant solutions. The group classification algorithm and reduction process are illustr…

Weak symmetries of stochastic differential equations driven by semimartingales with jumps

Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Levy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as symmet…

具有对称性的半经典磁薛定谔算子的隧穿

Tunneling for a semi-classical magnetic Schrödinger operator with symmetries

本文研究带对称性的半经典磁薛定谔算子基态在势阱外的衰减估计，主要针对大磁参数情形，并指出Agmon估计在隧穿极限下的应用。

On contact symmetries of evolution equations

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 dimension $n\leq4$, and algebras having nontrivial Levi factors, belonging to the class under consideration, and corresponding contact symmetries they admitted are listed.

近似 Lie 对称性与奇异摄动理论

Approximate Lie symmetries and singular perturbation theory

本文从解的 Lie 对称性展开这一新视角重新表述了微分方程的摄动理论，提出“隐藏标度对称性”的概念，统一了多种奇异摄动方法，并阐明了其成功与失败的机制。

Analytic solutions of initial-boundary-value problems of transient conduction using symmetries

Lie symmetry method is applied to find analytic solutions of initial-boundary-value problems of transient conduction in semi-infinite solid with constant surface temperature or constant heat flux condition. The solutions are obtained in a manner highlighting the systematic procedure of extending the symmetry method for a PDE to investigate BVPs of the PDE. A comparative analysis of numerical and closed form solutions is carried out for a physical problem of heat conduction in a semi-infinite solid bar made of AISI 304 stainless steel.

Helmholtz conditions and symmetries for the time dependent case of the inverse problem of the calculus of variations

We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Frölicher-Nijenhuis theory on the first jet bundle, $J^1π$. We prove that a system of time dependent SODE, identified with a semispray $S$, is Lagrangian if and only if a special class, $Λ^1_S(J^1π)$, of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class $Λ^1_S(J^1π)$ of semi-basic 1-forms. Each such class contains the Poincaré-Cartan 1-form of some Lagrangian function. We prove that i…

Symmetries of vector exterior differential systems and the inverse problem in second-order Ostrohrads'kyj mechanics

Symmetries of variational problems are considered as symmetries of vector bundle valued exterior differential systems. This approach is then applied to third order ordinary variational equations of motion of the semi-classical spinning particle.