Automorphisms of flag systems

We first discuss the problems in the theory of ordinary differential equations that gave rise to the concept of a flag system and illustrate these with the Cartan criterion for Monge equations (1st order) as well as the Cartan statement concerning the local equivalence of Monge–Ampère type equations (2nd order). Next, we describe a prolongation functor operating on the infinitesimal symmetries (automorphisms) of the Darboux flag and extending these, isomorphically, to all the symmetries of any other flag. Hence, flag systems cannot be distinguished by their symmetry algebras and the local classification of these objects is approached by considering higher order isotropies of these algebras as well as the groupoids of k-th order formal equivalences since the differential equations defining the latter provide precious information for the application of flag systems to differential equations (e.g., Cartan's criterion for non-linear Monge equations). In examining the behavior of the isotropy algebras, that can either diminish or remain the same, when passing from a derived system $S_{\nu }$ to the consecutive system $S_{\nu +1}$, we obtain a full set of numerical invariants for the elementary flag systems that moreover specifies the local models.     

Pseudo‐spherical normal Darboux images of curves on a lightlike surface

We introduce the notions of the pseudospherical normal Darboux images for the curve on a lightlike surface in Minkowski 3‐space and study these Darboux images by using technics of the singularity theory. Furthermore, we give a relation between these Darboux images and Darboux frame from the viewpoint of Legendrian dualities.     

Remarks on Neumann boundary problems involving Jacobians

In this note, we explore the validity of Wente-type estimates for Neumann boundary problems involving Jacobians. We show in particular that such estimates do not in general hold under the same hypotheses on the data for Dirichlet boundary problems.     

Rogue wave solutions for a generalized nonautonomous nonlinear equation in a nonlinear inhomogeneous fiber

In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable–coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.     

Monodromy and nonintegrability in complex Hamiltonian systems

This paper investigates the monodromy representation of the normal variational equation along a phase curve of a two-dimensional complex analytic Hamiltonian system. Geometrical conditions are presented which guarantee reducibility, together with additional hypotheses to ensure complete reducibility. Symmetries in the equations are treated in detail. Applications to establishing the nonintegrability of specific systems are presented.     

若干高维可积和不可积模型的严格解

可积和不可积模型可以描述自然科学中的诸多现象, 寻找高维非线性模型的严格解已成为可积系统的一个重要研究内容. 结合达布变换法和多线性分离变量法, 可以得到多个(2+1)维非线性模型包含任意函数的严格解, 通过选取不同的任意函数, 构造这些非线性模型新的相互激发模式. 进一步推广了形变映射理论, 建立了变系数 场和sine-Gordon以及双sine-Gordon场的形变映射关系, 从而得到高维不可积模型包含任意函数的新严格解. 对任意函数的不同选择, 构造了sine-Gordon和双sine-Gordon可积模型丰富的局域解和周期解, 如多solitoff解及其周期波推广、周期形变的蛇形孤波解以及变模的拟周期解等.