Geometric Interpretation of Second Elliptic Integrable System

Geometries and Symmetries of Soliton equations and Integrable Elliptic equations

We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation for harmonic maps from the plane to a compact Lie group, for primitive maps from the plane to a $k$-symmetric space, and constant mean curvature surfaces and isothermic surfaces in space forms are integrable elliptic systems. We also give a survey of (i) construction of solution…

Darboux可积椭圆系统的几何学

The Geometry of Darboux Integrable Elliptic Systems

本文研究了通过复Pfaffian系统H的全纯解构造实椭圆微分系统，并证明其具有Darboux可积性。

Elliptic Integrable Systems: a Comprehensive Geometric Interpretation

We give a geometric interpretation of all the $m$-th elliptic integrable systems associated to a $k'$-symmetric space $N=G/G_0$ (in the sense of C.L. Terng). It turns out that we have to introduce the integer $m_{k'}$ defined by m_{1}=0 and m_{k'}= [(k'+1)/2]. Then the general problem splits into three cases : the primitive case ($m < m_{k'}$), the determined case ($m_{k'}\leq m \leq k'-1$) and the underdetermined case ($m \geq k'$). We prove that we have an interpretation in terms of a sigma model with a Wess-Zumino term. Moreover we prove that we have a geometric interpretation in terms of t…

带可积边界条件的椭圆sinh-Gordon方程 II——Baker-Akhiezer理论

On the elliptic sinh-Gordon equation with integrable boundary conditions II -- Baker-Akhiezer theory

研究带Durham边界条件的椭圆sinh-Gordon方程有限型解，推导边界条件成立的必要有理准则及谱曲线4点附加条件，明确互补边界下准则的充分性，为相关解的构建奠定基础。

Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the governing equation in terms of elliptic integrals. We show that the instances when the elliptic int…

Quadratic integrals of geodesic flow, webs, and integrable billiards

We present a geometric interpretation of integrability of geodesic flow by quadratic integrals in terms of the web theory and construct integrable billiards on surfaces admitting such integrals.