Symmetry Group of Tzitzeica Surfaces PDE

A new symmetry criterion based on the distance function and applications to PDE's

We prove that, if $Ω\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial Ω$ of the function $$φ(y) = \int_0^{λ(y)} \prod_{j=1}^{n-1}[1-t κ_j(y)]\, dt$$ implies that $Ω$ is a ball. Here $k_j(y)$ and $λ(y)$ denote respectively the principal curvatures and the cut value of a boundary point $y \in \partial Ω$. We apply this geometric result to different symmetry questions for PDE's: an overdetermined system of Monge-Kantorovich type equations (which can be viewed as the limit as $p \to + \infty$ of Serrin's symmetry problem for the $p$-Laplacian),…

Symmetries of PDEs Systems in Solar Physics and Contact Geometry

One considers a special class of PDEs systems and one determines the associated symmetry group. Particulary, for the Blair system, one finds the symmetry group. A solutions of the Blair system gives a conformally flat contact metric structure and also it defines a "force-free" model of solar physics. By using the symmetry groups theory, one shows that the known solutions are group-invariant solutions and one gives new solutions.

Integrable dispersionless PDE in 4D, their symmetry pseudogroups and deformations

We study integrable non-degenerate Monge-Ampere equations of Hirota type in 4D and demonstrate that their symmetry algebras have a distinguished graded structure, uniquely determining the equations. This is used to deform these heavenly type equations into new integrable PDE of the second order with large symmetry pseudogroups. We classify the obtained symmetric deformations and discuss self-dual hyper-Hermitian geometry of their solutions, which encode integrability via the twistor theory.

Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system

We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discre…

$\mathfrak{sl}_3$型的外在几何与线性微分方程

Extrinsic geometry and linear differential equations of $\mathfrak{sl}_3$-type

本文应用外在几何一般理论，对与$\mathfrak{sl}(3)$伴随表示相关的齐次结构作完整局部分类，发现7类3维接触流形上的新型二阶线性PDE系统。

An analysis of symmetry groups of generalized $m$-quasi-Einstein manifolds

In this paper emphasis is placed on how the behavior of the solutions of a PDE is affected by the geometry of the generalized $m$-quasi-Einstein manifold, and vice versa. Considering a $n$-dimensional generalized $m$-quasi-Einstein manifold which is conformal to a pseudo-Euclidean space, we prove the most general symmetry group of maximal dimension. Moreover, we demonstrate that there is no different low dimensional invariant on a generalized $m$-quasi-Einstein manifold. As an application, we use the invariant structure of the metric to provide an example of shrinking $m$-quasi-Einstein manifo…

Moment maps, nonlinear PDE, and stability in mirror symmetry

We study the deformed Hermitian-Yang-Mills (dHYM) equation, which is mirror to the special Lagrangian equation, from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold $\mathcal{H}$ mirror to Solomon's space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with $C^{1,α}$ regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on col…

偏微分方程不变性缩减。II：一般框架

Invariant Reduction for Partial Differential Equations. II: The General Framework

本文提出一个框架，用于系统计算PDE系统的不变形式，展示Noether定理如何继承到缩减系统，并通过实例展示点和高阶对称性缩减的应用。

Application of Symmetry Groups to the Observability Analysis of Partial Differential Equations

Symmetry groups of PDEs allow to transform solutions continuously into other solutions. In this paper, we use this property for the observability analysis of nonlinear PDEs with input and output. Based on a differential-geometric representation of the nonlinear system, we derive conditions for the existence of special symmetry groups that do not change the trajectories of the input and the output. If such a symmetry group exists, every solution can be transformed into other solutions with the same input and output trajectories but different initial conditions, and this property can be used to …

pde2path中的微分几何分岔问题——算法与教程示例

Differential geometric bifurcation problems in pde2path -- algorithms and tutorial examples

本文介绍如何利用MATLAB工具箱pde2path处理微分几何分岔问题，通过求解浸入曲面法向位移PDE并进行延拓、分岔检测与分支切换，在极小曲面、常平均曲率曲面及生物膜模型中发现了对称性破缺分岔。