Singularités de solutions d'équations aux dérivées partielles

We develop the general mathematical setting necessary to study the singularities of local solutions of the quasi-linear first-order systems of PDEs with a free initial condition. In the good cases, it is possible to describe these singularities as a function of the free initial conditions satisfied by the solutions. Using the transversality theorems, it is then possible to describe the singularities of generic solutions, and of generic families of solutions under deformation of the initial conditions. We apply this study by giving classifications of an important classe of hyperbolic quasi-linear first-order systems in the plane, the reducible systems, and of an almost general class of hyperbolic quasi-linear second-order equations in the plane.     

Timelike Constant Mean Curvature Surfaces with Singularities

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behavior of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized surfaces are cuspidal edges, swallowtails, and cuspidal cross caps.     

Generic singularities of implicit systems of first order differential equations on the plane

For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well-known singularities of generic vector fields on the plane and the singularities described by a generic first order implicit differential equations, there exists only one generic singularity described by the implicit first order equation supplied by Whitney umbrella surface generically embedded to the space of directions on the plane.     

On a Class of Systems of Linear and Nonlinear Fredholm Integral Equations of the Third Kind with Multipoint Singularities

Based on a new approach, we show that finding solutions for a class of systems of linear (respectively, nonlinear) Fredholm integral equations of the third kind with multipoint singularities is equivalent to finding solutions of systems of linear (respectively, nonlinear) Fredholm integral equations of the second kind with additional conditions. We study the existence, nonexistence, uniqueness, and nonuniqueness of solutions for this class of systems of Fredholm integral equations of the third kind with multipoint singularities.     

从R^n到R^3的光滑映射的generic奇点

本文的目的在于用微分拓扑的方法讨论从Ｒ＾ｎ到Ｒ＾３的光滑映射的ｇｅｎｅｒｉｃ奇点的分类问题,我们证明这样的映射只有三类奇点：折点,尖点和燕尾点,并给出了它们的标准型。它推广了Ｊ．Ｍａｒｔｉｎｅｔ（１）的结果。     

A method for calculating the Painlevé transcendents

A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.     

The Quenching Set of a MEMS Capacitor in Two-Dimensional Geometries

The formation of finite time singularities in a nonlinear parabolic fourth order partial differential equation (PDE) is investigated for a variety of two-dimensional geometries. The PDE is a variant of a canonical model for Micro–Electro Mechanical systems (MEMS). The singularities are observed to form at specific points in the domain and correspond to solutions whose values remain finite but whose derivatives diverge as the finite time singularity is approached. This phenomenon is known as quenching. An asymptotic analysis reveals that the quenching set can be predicted by simple geometric considerations suggesting that the phenomenon described is generic to higher order parabolic equations which exhibit finite time singularity.     

A method for the numerical solution of the Painlevé equations

A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.     

Rational ODEs with Movable Algebraic Singularities

A class of second-order rational ordinary differential equations, admitting certain families of formal algebraic series solutions, is considered. For all solutions of these equations, it is shown that any movable singularity that can be reached by analytic continuation along a finite-length curve is an algebraic branch point. The existence of these formal series expansions is straightforward to determine for any given equation in the class considered. We apply the theorem to a family of equations, admitting different kinds of algebraic singularities. As a further application we recover the known fact for generic values of parameters that the only movable singularities of solutions of the Painlevé equations   P_II – P   _VI   are poles.     

Remarks on ellipticity and regularity for non-inhibited shells and other constrained systems

We examine ellipticity properties for three examples of constrained systems with a Lagrange multiplier. The system of incompressible elasticity is in the restricted context of the Grubb-Geymonat elliptic systems. The Reissner-Mindlin system of plates is in the Agmon-Douglis-Nirenberg context. The system of thin shells which are not geometrically rigid is elliptic or hyperbolic at elliptic or hyperbolic points of the surface, respectively; moreover, this property is only concerned with equations, without boundary conditions. Some properties of the wave fronts and propagation of singularities are given.     

Painlevé Classification Problems Featuring Essential Singularities

In this article we construct and solve all Painlevé-type differential equations of the second order and second degree that are built upon, in a natural well-defined sense, the "sn-log" equation of Painlevé, the general integral of which admits a movable essential singularity (elliptic function of a logarithm). This equation (which was studied by Painlevé in the years 1893–1902) is frequently cited in the modern literature to elucidate various aspects of Painlevé analysis and integrability of differential equations, especially the difficulty of detecting essential singularities by local singularity analysis of differential equations. Our definition of the Painlevé property permits movable essential singularities, provided there is no branching. While the essential singularity presents no serious technical problems, we do need to introduce new techniques for handling "exotic" Painlevé equations, which are Painlevé equations whose singular integrals admit movable branch points in the leading terms. We find that the corresponding full class of Painlevé-type equations contains three, and only three, equations, which we denote SD-326-I, SD-326-II, and SD-326-III, each solvable in terms of elliptic functions. The first is Painlevé's own generalization of his sn-log equation. The second and third are new, the third being a 15-parameter exotic master equation. The appendices contain results (in general, without uniqueness proofs) of related Painlevé classification problems, including full generalizations of two other second-degree equations discovered by Painlevé, additional examples of exotic Painlevé equations and Painlevé equations admitting movable essential singularities, and third-order equations featuring sn-log and other essential singularities.     

Higher-Order Differential Equations Having a Singularity in an Interior Point

Ordinary differential equations of an arbitrary order having a non-integrable singularity inside the interval are considered under additional matching conditions for solutions at the singular point. We construct special fundamental systems of solutions for this class of differential equations, study their asymptotical, analytical and structural properties and the behavior of the corresponding Stokes multipliers. These fundamental systems of solutions are used in spectral analysis of differential operators with singularities.     

Deformationen isolierter Kurvensingularitäten mit eingebetteten Komponenten

We study deformations of isolated curve singularities of arbitrary embedding dimension which may have embedded components. The existence of an embedded component has the effect that the generic fibre consists of isolated (fat) points and disjoint 1-dimensional components. We introduce for isolated curve singularities and for fat points a Milnor number μ and a σ-invariant which generalize the well known invariants for reduced curve singularities. The invariants μ, σ and the length of the embedded component control the topology of the generic fibre in a way which is surprisingly similar to the case of reduced curves, although many new phenomena occur. This article was processed by the author using the Springer-Verlag TEX mamath macro package 1990.     

Boundary-contact problems for domains with conical singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have conical singularities. Such problems represent continuous operators between weighted Sobolev spaces and subspaces with asymptotics. Ellipticity is formulated in terms of extra transmission conditions along the interfaces with a control of the conormal symbolic structure near conical singularities. We show regularity and asymptotics of solutions in weighted spaces, and we construct parametrices. The result will be illustrated by a number of explicit examples.     

Integrability in a discrete world

We review our findings on integrable discrete systems with emphasis on the discrete integrability detector we have proposed under the name of singularity confinement. We have indeed shown, in a host of examples, that it is possible, by studying the structure of the singularities of discrete systems, to identify the integrable ones. A most important result of this approach is the discovery of discrete Painlevé equations of which a lengthy list exists today. These equations, being integrable systems, are characterised by particularly rich properties which are under active investigation. We present here an overview of these properties and stress the similarities and differences that exist between discrete and continuous Painlevé equations.     

Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems

Multi-valued solutions are constructed for 2 × 2 first-order systems using a generalization of the hodograph transformation. The solution is found as a complex analytic function on a complex Riemann surface for which the branch points move as part of the solution. The branch point singularities are envelopes for the characteristics and thus move at the characteristic speeds. We perform an analysis of stability of these singularities with respect to perturbations of the initial data. The generic singularity types are folds, cusps, and nondegenerate umbilic points with non-zero 3-jet. An isolated singularity is generically a square root branch point corresponding to a fold. Two types of collisions between singularities are generic: At a “tangential” collision between two singularities moving at the same characteristic speed, a cube root branch point is formed, corresponding to a cusp. A “non-tangential” collision, between two square root branch points moving at different characteristic speeds, remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point. These results are also valid for a diagonalizable n-th order system for which there are exactly two speeds. © 1993 John Wiley & Sons, Inc.     

Generic bifurcations of framed curves in a space form and their envelopes

The generic singularities and bifurcations are classified for one-parameter families of curves with frames in the space forms En+1,Sn+1,Hn+1. Two kinds of frames are considered; adapted frames and osculating frames. In particular, we give the classification results on the singularities of envelopes associated to framed curves.The associated envelopes and their singularities are classified. characterised in term of geometric invariants of framed curves. We apply to the global problem of framed curves and to the extension problem of surfaces with boundaries in three space. generalising the results obtained in Ishikawa (2010) [12].     

Global solution curves for several classes of singular periodic problems

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for three classes of periodically forced equations with singularities, including the equations arising in micro-electro-mechanical systems (MEMS), the ones in condensed matter physics, as well as A.C. Lazer and S. Solimini’s (Lazer and Solimini, 1987) problem.     

Stability of local transitivity of a generic control system on a surface with boundary

A classification of generic singularities of local transitivity of smooth control systems on surfaces with boundary is obtained. The stability of these singularities and of the entire set of points with identical properties of local transitivity with respect to small perturbations of a generic system is proved.