Bifurcations of completely integrable 2-variable first-order partial differential equations

In this paper, we consider an implicit 2-variable first-order partial differential equation with complete integral. As an application of the Legendrian singularity theory, we give a generic classification of bifurcations of such differential equations with respect to the equivalence relation which is given by the group of point transformations following S. Lie?s view. Since two one-parameter unfoldings of such differential equations are equivalent if and only if induced one-parameter unfoldings of integral diagrams are equivalent for generic equations, our normal forms are represented by one-parameter integral diagrams.     

Techniques for searching first integrals by lie group and application to gyroscope system

In the paper, the methods of finding first integrals of an autonomous system using one-parameter Lie groups are discussed. A class of nontrivial one-parameter Lie groups admitted by the classical gyroscope system is found, and based on the properties of first integral determined by the one-parameter Lie group, the fourth first integral of the gyroscope system in Euler case, Lagrange case and Kovalevskaya case can be obtained in a uniform idea. An error on the fourth first integral in general Kovalevskaya case (A = B = 2C, ZG =0), which appeared in literature is found and corrected.     

Perestroikas of Convexified Apparent Contours and Global Phase Diagrams in Thermodynamics

We obtain the classification of singularities occurring in families of convex hulls of apparent contours up to codimension 3. The results for codimension 2 singularities allow us to supplement Varchenko's classification of local singularities of thermodynamic phase diagrams of binary mixtures. Singularities of three-parameter families specify so-called global phase diagrams in three-dimensional parameter spaces and define all local perestroikas of phase diagrams in generic one-parameter families of binary mixtures.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for bi-parameter and bilinear Fourier integral operators

Fourier integral operators play an important role in Fourier analysis and partial differential equations. In this paper, we deal with the boundedness of the bilinear and bi-parameter Fourier integral operators, which are motivated by the study of one-parameter FIOs and bilinear and bi-parameter Fourier multipliers and pseudo-differential operators. We consider such FIOs when they have compact support in spatial variables. If they contain a real-valued phase φ(x, ξ, η) which is jointly homogeneous in the frequency variables ξ, η, and amplitudes of order zero supported away from the axes and the antidiagonal, we can show that the boundedness holds in the local-L² case. Some stronger boundedness results are also obtained under more restricted conditions on the phase functions. Thus our results extend the boundedness results for bilinear and one-parameter FIOs and bilinear and bi-parameter pseudo-differential operators to the case of bilinear and bi-parameter FIOs.     

Several results on the families of diffeomorphisms onS 1

In this paper, we give a necessary and sufficient condition for the one-parameter families of diffeomorphisms onS ¹ to be stable and a necessary condition for the multi-parameter families to be stable; and, moreover, we prove that phase-locking is a generic property of the one-parameter families of diffeomorphisms onS ¹. We also get a necessary and sufficient condition of phase-locking for the one-parameter families of integral diffeomorphisms onS ¹ which strengthens a result in [2].     

Interaction of the folded Whitney umbrella and swallowtail in slow-fast systems

The graph of a first integral of a smooth slow-fast system with two slow variables is a singular surface in the three-dimensional space; the variation of an external parameter on which the system depends gives rise to perestroikas (=transitions) of this surface. We find a normal form and present figures of the perestroika that describes the interaction between the swallowtail and folded Whitney umbrella on the graph of a first integral of a generic one-parameter family of such systems.     

On some systems of nonlinear integral Hammerstein-type equations on the semiaxis

We prove the existence for a one-parameter family of solutions of a system of nonlinear integral Hammerstein-type equations on the positive semiaxis and study the asymptotic behavior of the obtained solutions at infinity.     

本刊英文版Vol.33(2017),No.2论文摘要（英文）

<正>L~p Estimates for Bi-parameter and Bilinear Fourier Integral Operators Qing HONG Lu ZHANG Abstract Fourier integral operators play an important role in Fourier analysis and partial differential equations.In this paper,we deal with the boundedness of the bilinear and biparameter Fourier integral operators,which are motivated by the study of one-parameter FIOs and bilinear and bi-parameter Fourier multipliers and pseudo-differential operators.We consider such FIOs when they have compact support in spatial variables.If they contain a real-valued phaseφ(x,ξ,η)which is jointly homogeneous in the frequency variablesξ,η,and amplitudes of order zero supported away from the axes and the anti-diagonal,we can show that the boundedness holds in the local-L~2 case.Some stronger boundedness results are also obtained under more     

Floating Bodies of Equilibrium

A long cylindrical body of circular cross-section and homogeneous density may float in all orientations around the cylinder axis. It is shown that there are also bodies of non-circular cross-sections which may float in any direction. Apart from those found by Auerbach for  ρ= 1/2  , there are one-parameter families of cross-sections for  ρ≠ 1/2  which have a p -fold rotation symmetry. For given p they have this property for   p − 2  different densities ρ. The differential equation governing the non-circular boundary curve is derived. Its solution is expressed in terms of an elliptic integral.     

Periodic families of solutions for difference equations with a first integral

In the following paper we establish that a one-parameter family of N- periodic solutions out of the origin is guaranteed to exist when the dimension of the N- periodic solution space of the corresponding linear problem is unity. When this dimension is greater than unity we establish that one parameter families generically exist. These results are obtained by adapting the method of Hale3 to a N-periodic difference equation with a N-periodic first integral     

Vector-Valued Functions Generated by the Operator of Finite Order and Their Application to Solving Operator Equations in Locally Convex Spaces

This work is devoted to solving some classes of operator equations, based on the solution of auxiliary one-parameter family of equations, which is obtained fromthe original operator equation by formal replacement of the operator of the integrated parameter. Solutions are vector-valued functions represented by power series or integral. We investigate some properties of these solutions, namely, growth characteristics, the domain of analyticity. The investigation is realized by means of order and type of operator, operator order and operator type of the vector relative to the operator.     

A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields

We introduce a hierarchy of integrable partial differential equations in 2+1 dimensions arising from the commutation of one-parameter families of vector fields, and we construct the formal solution of the associated Cauchy problems using the inverse scattering method for one-parameter families of vector fields. Because the space of eigenfunctions is a ring, the inverse problem can be formulated in three distinct ways. In particular, one formulation corresponds to a linear integral equation for a Jost eigenfunction, and another formulation is a scalar nonlinear Riemann problem for suitable analytic eigenfunctions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 147–156, July, 2007.     

Solvability of a Nonlinear Integral Equation in Dynamical String Theory

We investigate an integral equation of the convolution type with a cubic nonlinearity on the entire real line. This equation has a direct application in open-string field theory and in p-adic string theory and describes nonlocal interactions. We prove that there exists a one-parameter family of bounded monotonic solutions and calculate the limits of solutions constructed at infinity.     

Modeling of an Asymptotically Central Markov Process on 3D Young Graph

The paper discusses the asymptotics of path probabilities in Markov processes close to a central one on 3D Young graph. We consider a one-parameter family of such processes and compute the parameter value in such a way that the centrality condition is satisfied with the greatest possible accuracy. We defined a normalized dimension of paths in the 3D Young graph. We study the growth and oscillations of these normalized dimensions along greedy and random trajectories of the processes using large computer experiments involving 3D Young diagrams with millions of boxes.     

Bifurcation of common levels of first integrals of the kovalevskaya problem

The structure of integral manifolds in the Kovalevskaya problem of a heavy solid about a fixed point is considered. An analytic definition of a bifurcation set is obtained, and bifurcation diagrams are constructed. The number of two-dimensional toruses that appear in the composition of the integral manifold is indicated for each connected component, additional to the bifurcation set in the space of first integral constants.     

Steady Nonsymmetrical Gravity Waves with Shear in Water

A new type of steady two-dimensional inviscid gravity wave with shear is computed numerically. These waves appear at relatively low amplitudes and lack symmetry with respect to any crest or trough. A boundary integral formulation is used to obtain a one-parameter family of nonsymmetrical solutions through a symmetry-breaking bifurcation.     

Basin of attraction of a cusp–fold singularity in 3D piecewise smooth vector fields

Generic one-parameter families of piecewise smooth vector fields on R³${\mathbb{R}}^3$ presenting the so-called cusp–fold singularity are studied. The bifurcation diagrams are exhibited and the asymptotic and structural stabilities are discussed.     

One-parameter family of integrable solutions of a system of nonlinear integral equations of the Hammerstein–Volterra type in the supercritical case

We study a system of nonlinear integral equations of the Hammerstein–Volterra type on a half-line in the supercritical case. We show that this system has a one-parameter family of positive integrable bounded solutions. We describe the structure of each solution in this family. The monotone dependence of the solutions on the parameter is proved.     

Knot theory for self-indexed graphs

We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We call a self-indexed graph with integral flow a comte. The analogy with links allows us to define transformations of comtes generalizing the Reidemeister moves on link diagrams. We show that many invariants of links can be generalized to comtes, most notably the linking number, the Alexander polynomials, the link group, etc. We also discuss finite type invariants and quandle cocycle invariants of comtes.