Symmetries of Quantum Lax Equations for the Painlevé Equations

Based on the fact that the Painlevé equations can be written as Hamiltonian systems with affine Weyl group symmetries, a canonical quantization of the Painlevé equations preserving such symmetries has been studied recently. On the other hand, since the Painlevé equations can also be described as isomonodromic deformations of certain second-order linear differential equations, a quantization of such Lax formalism is also a natural problem. In this paper, we introduce a canonical quantization of Lax equations for the Painlevé equations and study their symmetries. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.     

On the Free Boundary to the Incompressible Euler Equations

In this paper, we consider an inviscid, incompressible, irrotational fluid in a region of R^3 with free boundary. Motivated by [1], we find that in this particular case, we do not need the complicated energy functional in [1], instead we can use a simpler replacement and get the a priori energy estimate for a positive time, which depends only on the initial data.     

On the Oscillation of the Fourth Order Differential Equations

61.IntroductionWeconsiderthefourthorderdifferentialequationx(`)(t)-p(t)lx(t)I'sgnx(t)=O,(1.l)wherea>l,isconstant,andp(t)ispositivecontinuousfunctionon[t,,oo),to>o.Werestrictourattentiontothosenontrivialsolutionsx(t)of(1.1)whichexistonsomehalf-[ine[T:,co)cy[t,,oo),suchasolutioniscalledapropersolutionof(1.l).ApropersolutioniscalledoscillatoryifithasarbitrarilylargezerosFotherwiseitiscallednonoscillatory.Wecon-sidertheoscillationandnonoscillationofpropersolutionsof(l,l).WeletEdenotethesetofall…     

Integration of the Gauss–Codazzi Equations

The Gauss–Codazzi equations imposed on the elements of the first and the second quadratic forms of a surface embedded in are integrable by the dressing method. This method allows constructing classes of Combescure-equivalent surfaces with the same rotation coefficients. Each equivalence class is defined by a function of two variables (master function of a surface). Each class of Combescure-equivalent surfaces includes the sphere. Different classes of surfaces define different systems of orthogonal coordinates of the sphere. The simplest class (with the master function zero) corresponds to the standard spherical coordinates.     

On the Solvability of Some Monge-Ampere Equations

This paper considers: (Ⅰ)M(u)≡det (D~2u) = f(x, u, Du) in W, u|w = g|δwHere f and g are smooth on W ×R×R~n and W respectively, f is positive, and gconvex. W is a uniformly convex domain.     

Convergence of the Vlasov-Poisson-Boltzmann System to the Incompressible Euler Equations

In this paper we establish the convergence of the Vlasov-Poisson-Boltzmann system to the incompressible Euler equations in the so-called quasi-neutral regime. The convergence is rigorously proved for time intervals on which the smooth solution of the Euler equations of the incompressible fluid exists. The proof relies on the relative-entropy method.     

On the Stationary Navier–Stokes Equations in the Half-Plane

We consider the stationary incompressible Navier–Stokes equation in the half-plane with inhomogeneous boundary condition. We prove the existence of strong solutions for boundary data close to any Jeffery–Hamel solution with small flux evaluated on the boundary. The perturbation of the Jeffery–Hamel solution on the boundary has to satisfy a nonlinear compatibility condition which corresponds to the integral of the velocity field on the boundary. The first component of this integral is the flux which is an invariant quantity, but the second, called the asymmetry, is not invariant, which leads to one compatibility condition. Finally, we prove the existence of weak solutions, as well as weak–strong uniqueness for small data and provide numerical simulations.     

Linear Equations for the Number of Intervals Which are Isomorphic with Boolean Lattices and the Dehn–Sommerville Equations

Let P be a finite poset. Let L: = J(P) denote the lattice of order ideals of P. Let b _i (L) denote the number of Boolean intervals of L of rank i.

We construct a simple graph G(P) from our poset P. Denote by f _i (P) the number of the cliques K _i+1, contained in the graph G(P).

Our main results are some linear equations connecting the numbers f _i (P) and b _i (L).

We reprove the Dehn–Sommerville equations for simplicial polytopes.

In our proof, we use free resolutions and the theory of Stanley–Reisner rings.     

Dynamical Behavior for the Three-dimensional Generalized Hasegawa-Mima Equations

We consider the following generalized three-dimensional (3-D) dissipative Hasegawa-Mima equations:     

The Initial-Boundary Value Problem for the Boussinesq Equations

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Well-posedness and Global Attractors for the Burgers-Ginzburg-Landau Equations

In this paper we consider the Burger-Ginzburg-Landau equations, and prove the existence of the global attractor in with finite Hausdorff and fractal dimensions.     

Behavior of the Solutions for the Chemotaxis Equations with Saturation Growth

In this paper,we study the behavior of the solutions to a kind of chemotaxis equations that     

A Class of the Quasilinear Hyperbolic Equations with the Inhomogenous Terms

In this paper,we discuss a class of the quasillinear hyperbolic equations with the inhomogeneous terms:ur σ(v)x 2α(t)u=0,vr-ux=0 Under the certain of hypothesis,we prove the globally existence theorems of the smooth solutions for is cauchy problem.     

On the Geometrical Meaning of the Dynamic Equations in Analytical Mechanics

In this paper,the modern geometrical structure of analytical mechanics,the exteriordifferential forms and the geometrical meaning of dynamic equations are briefly discussed.     

On the Suitable Weak Solutions for the Cauchy Problem of the Boussinesq Equations

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Different Behaviour for the Solutions of Some Chemotaxis Equations

Inbiology,itisveryimportanttoinvestigatethemovementofsomecellsororganismsinsomegivenbiologica1system(cf.[1,2]).Inordertounderstandhowthemovementrulesareaffectedbytheeffectofthechemo-attractant,OthmerandStevensintroducedin[1]severalgeneralclassesofpartialdifferentialequations.Inoneoftheirmodels,theyconsideredamasterequation,i.e.barrierandnearestneighborlatticemodel.Followingalimitingprocessthemodelisdescribedbythefollowingsystemofpartialdifferentialequations:wherep(x,t)istheparticledensityofap…     

Energy Scattering for the Generalized Davey—Stewartson Equations

Abstract Considring the generalized Davey-Stewartson equation i-△u+λ│u│~pu+μE(│u│~q)│u│~(q-2)u=0,where λ>0,μ≥0,E=F~(-1)(ξ_1~2│ξ│~2)F,we obtain the existence of scattering operator in ∑(R~n):u{u∈H~1(R~n):│x│u∈L~2(R~n)}.     

On the Integrability of the Abel and of the Extended Liénard Equations

Acta Mathematicae Applicatae Sinica, English Series - We present some exact integrability cases of the extended Liénard equation y″ + f(y)(y′)n + k(y)(y′)m + g(y)y′ +...     

Sufficient Conditions for the Oscillation of Non-autonomous Difference Equations

Abstract In this paper, we obtain sufficient conditions for the oscillation of the non-autonomous differenceequations x(n 1)-x(n) sum from i=1 to m p_i(n)x(n-r_i(n))=0,which are the discrete analog of the delay differential equations considered in[1].     

Remarks on the Scattering Result of Nonlinear Schroedinger Equations

0 Introduction 　　In this paper, we give another proof of the scattering result of the following Cauchy problem in the energy space