On nonlinear partial differential equations with an infinite-dimensional conditional symmetry

Roman Cherniha, Malte Henkel

备注：: Latex2e, 14 pages, no figures; final form
发表信息：: J. Math. Anal. Appl. 298, 487-500 (2004)
DOI：: 10.1016/j.jmaa.2004.05.038

英文摘要：

The invariance of nonlinear partial differential equations under a certain infinite-dimensional Lie algebra A_N(z) in N spatial dimensions is studied. The special case A_1(2) was introduced in J. Stat. Phys. {\bf 75}, 1023 (1994) and contains the Schrödinger Lie algebra sch_1 as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of A_N(z). However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of A_N(z) such that the well-known Monge-Ampere equation is the required additional condition. New exact solutions are found for some representatives of this class.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Quantum conditional entropy for infinite-dimensional systems

In this paper a general definition of quantum conditional entropy for infinite-dimensional systems is given based on recent work of Holevo and Shirokov arXiv:1004.2495 devoted to quantum mutual and coherent informations in the infinite-dimensional case. The properties of the conditional entropy such as monotonicity, concavity and subadditivity are also generalized to the infinite-dimensional case.

2010-04-27math-phmath.MPquant-ph

Approximate Q-conditional symmetries of partial differential equations

Following a recently introduced approach to approximate Lie symmetries of differential equations which is consistent with the principles of perturbative analysis of differential equations containing small terms, we analyze the case of approximate $Q$--conditional symmetries. An application of the method to a hyperbolic variant of a reaction--diffusion--convection equation is presented.

2021-08-01math-phmath.MP

Stochastic Free Energies, Conditional Probability and Legendre Transform for Ensemble Change

This work extends a recently developed mathematical theory of thermodynamics for Markov processes with, and more importantly, without detailed balance. We show that the Legendre transform in connection to ensemble changes in Gibbs' statistical mechanics can be derived from the stochastic theory. We consider the joint probability $p_{XY}$ of two random variables X and Y and the conditional probability $p_{X|Y=y^*}$, with $y^*=<Y>$ according to $p_{XY}$. The stochastic free energies of the XY system (fluctuating Y ensemble) and the $X|Y$ system (fixed Y ensemble) are related by the chain rule fo…

2015-10-27cond-mat.stat-mechmath-phmath.MP

A precise definition of reduction of partial differential equations

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in (1+3) dimensions. The conditional symmetry approach when applied to the equation in question yi…

2002-07-18math-phmath.APmath.MP

The Schmidt number and partially entanglement breaking channels in infinite dimensions

A definition of the Schmidt number of a state of an infinite dimensional bipartite quantum system is given and properties of the corresponding family of Schmidt classes are considered. The existence of states with a given Schmidt number such that any their countable convex decomposition does not contain pure states with finite Schmidt rank is established. Partially entanglement breaking channels in infinite dimensions are studied. Several properties of these channels well known in finite dimensions are generalized to the infinite dimensional case. At the same time, the existence of partially e…

2011-12-15math-phmath.FAmath.MP

On the notion of conditional symmetry of differential equations

Symmetry properties of PDE's are considered within a systematic and unifying scheme: particular attention is devoted to the notion of conditional symmetry, leading to the distinction and a precise characterization of the notions of ``true'' and ``weak'' conditional symmetry. Their relationship with exact and partial symmetries is also discussed. An extensive use of ``symmetry-adapted'' variables is made; several clarifying examples, including the case of Boussinesq equation, are also provided.

2006-03-09math-phmath.MP

更多相关论文收起更多论文

Semi-infinite forms and topological vertex operator algebras

Semi-infinite forms on the moduli spaces of genus-zero Riemann surfaces with punctures and local coordinates are introduced. A partial operad for semi-infinite forms is constructed. Using semi-infinite forms and motivated by a partial suboperad of the partial operad for semi-infinite forms, topological vertex partial operads of type $k<0$ and strong topological vertex partial operads of type $k<0$ are constructed. It is proved that the category of (locally-)grading-restricted (strong) topological vertex operator algebras of type $k<0$ and the category of (weakly) meromorphic ${\Bbb Z}\times {\…

1999-06-04math-phmath.MPmath.QA

偏微分方程的不变离散化：允许无限维对称群的方程

Invariant Discretization of Partial Differential Equations Admitting Infinite-Dimensional Symmetry Groups

SUNY at New Paltz（纽约州立大学新帕尔茨分校）

研究了无限维对称伪群下偏微分方程的不变离散化问题，提出通过离散化对称伪群作用来构造不变数值方案，并通过计算验证该方法能提高数值结果的质量。

2015-02-19cs.NAmath-phmath.MP