Generalized Wilczynski invariants for non-linear ordinary differential equations

Integrable systems in projective differential geometry

Some of the most important classes of surfaces in projective 3-space are reviewed: these are isothermally asymptotic surfaces, projectively applicable surfaces, surfaces of Jonas, projectively minimal surfaces, etc. It is demonstrated that the corresponding projective "Gauss-Codazzi" equations reduce to integrable systems which are quite familiar from the modern soliton theory and coincide with the stationary flows in the Davey-Stewartson and Kadomtsev-Petviashvili hierarchies, equations of the Toda lattice, etc. The corresponding Lax pairs can be obtained by inserting a spectral parameter in …

Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry

It is demonstrated that the stationary Veselov-Novikov (VN) and the stationary modified Veselov-Novikov (mVN) equations describe one and the same class of surfaces in projective differential geometry: the so-called isothermally asymptotic surfaces, examples of which include arbitrary quadrics and cubics, quartics of Kummer, projective transforms of affine spheres and rotation surfaces. The stationary mVN equation arises in the Wilczynski approach and plays the role of the projective "Gauss-Codazzi" equations, while the stationary VN equation follows from the Lelieuvre representation of surface…

Fundamental invariants of systems of ODEs of higher order

We find the complete set of fundamental invariants for systems of ordinary differential equations of order $\ge 4$ under the group of point transformations generalizing similar results for contact invariants of a single ODE and point invariants of systems of the second and the third order. It turns out that starting from systems of order $(k+1)\ge 4$, the complete set of fundamental invariants is formed by $k$ generalized Wilczynski invariants coming from the linearized system and an additional invariant of degree 2.

论 C 类次最大对称向量常微分方程的唯一性

On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class

本文对 C 类向量常微分方程在固定一个非零 C 类不变量、其余不变量为零的条件下给出次最大对称情形的局部分类，推广了 Sophus Lie 的经典标量结果，并利用调和理论识别最低权向量为调和 2-上链。

The point equivalence problem for ordinary differential equations of the second order

We use E. Cartan's method to solve the problem of equivalence of the second order ordinary differential equations with respect to the pseudogroup of point transformations.

Rational curves and ordinary differential equations

The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn out to provide an infinite dimensional family of integrable systems.