射影空间中的二阶拟线性偏微分方程与共形结构
Second order quasilinear PDEs and conformal structures in projective space
摘要：
本文研究二阶拟线性方程f_{ij} u_{x_ix_j} =0，其中u是n个自变量x_1，.的函数，x_n，并且系数f_{ij}是一阶导数p^1= u_{x_1}，>.的函数，p^n=仅u_{x_n}。我们证明了该问题的自然等价群同构于SL（n+1，R），SL（n+1，R）通过射影变换作用在空间P^n上，坐标为p^1，. p^n.系数矩阵f_{ij}在P^n上定义了一个共形结构f_{ij} dp^idp^j。本文主要讨论n=3的情形，尽管有些结果在任何维数下都成立。利用流体动力学约化方法导出了这类方程可积的充要条件。这些条件构成了一个复杂的超定偏微分方程系统，其中系数f_{ij}是对合的。证明了可积方程的模空间是20维的。在此基础上，我们证明了满足可积条件的方程必然是守恒的，并且具有无色散的拉克斯对。在微分几何项中重新表述，可积性条件意味着共形结构f_{ij} dp^idp^j是共形平坦的，并且具有无穷多个3-共轭零坐标系。可积方程提供了丰富的明确的例子，这种共形结构参数化的初等函数，椭圆函数和模形式。
英文摘要：
We investigate second order quasilinear equations of the form f_{ij} u_{x_ix_j}=0 where u is a function of n independent variables x_1, ..., x_n, and the coefficients f_{ij} are functions of the first order derivatives p^1=u_{x_1}, >..., p^n=u_{x_n} only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n+1, R), which acts by projective transformations on the space P^n with coordinates p^1, ..., p^n. The coefficient matrix f_{ij} defines on P^n a conformal structure f_{ij} dp^idp^j. In this paper we concentrate on the case n=3, although some results hold in any dimension. The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reductions are derived. These conditions constitute a complicated over-determined system of PDEs for the coefficients f_{ij}, which is in involution. We prove that the moduli space of integrable equations is 20-dimensional. Based on these results, we show that any equation satisfying the integrability conditions is necessarily conservative, and possesses a dispersionless Lax pair. Reformulated in differential-geometric terms, the integrability conditions imply that the conformal structure f_{ij} dp^idp^j is conformally flat, and possesses an infinity of 3-conjugate null coordinate systems. Integrable equations provide an abundance of explicit examples of such conformal structures parametrized by elementary functions, elliptic functions and modular forms.
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。
