A numerical solution of a two-dimensional transport equation

In this paper we present a variational method for approximating solutions of the Dirichlet problem for the neutron transport equation in the stationary case. Error estimates from numerical examples are used to evaluate an approximation of the solution with respect to the steps of two grids.     

Submanifolds of constant sectional curvature in Pseudo-Riemannian manifolds

The generalized equation and the intrinsic generalized equation are considered. The solutions of the first one are shown to correspond to Riemannian submanifolds M ⁿ (K) of constant sectional curvature of psedo-Riemannian manifolds % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaamaaDaaaleaaieGacaWFZbaabaGaaGOmaiaad6gacqGHsisl% caaIXaaaaOGaaiikamaanaaabaGaam4saaaacaGGPaaaaa!3D97!\[\overline M _s^{2n - 1} (\overline K )\] of index s, with % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgc% Mi5oaanaaabaGaam4saaaaaaa!3965!\[K \ne \overline K \], flat normal bundle and such that the normal principal curvatures are different from % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgk% HiTmaanaaabaGaam4saaaaaaa!388B!\[K - \overline K \]. The solutions of the intrinsic generalized equation correspond to Riemannian metrics defined on open subsets of R ⁿ which have constant sectional curvature. The relation between solutions of those equations is given. Moreover, it is proven that the submanifolds M under consideration are determined, up to a rigid motion of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \], by their first fundamental forms, as solutions of the intrinsic generalized equation. The geometric properties of the submanifolds M associated to the solutions of the intrinsic generalized equation, which are invariant under an (n – 1)-dimensional group of translations, are given. Among other results, it is shown that such submanifolds are foliated by (n – 1)-dimensional flat submanifolds which have constant mean curvature in M. Moreover, each leaf of the foliation is itself foliated by curves of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \] which have constant curvatures.     

Global solutions of the Navier-Stokes equation with strong viscosity

Following Ebin and Marsden the Navier-Stokes equation is viewed as a perturbation of a geodesic flow on the group of volume preserving diffeomorphisms on a compact Riemannian manifold. Existence and uniqueness of bounded solutions for all position time is shown by taking a higher order diffusion term.Partial y supported by Alexander von Humboldt Foundation     

Non-linear partial differential equations via vector fields on homogeneous Banach manifolds

The vector field formulation of and the Sato-Segal-Wilson approach to soliton equations are related to each other in this paper. From Banach Lie groups associated with the MKdV hierarchy of differential equations, we derive homogeneous Banach manifolds of solutions on which these equations are realized by vector fields. In the same way, we obtain homogeneous Banach manifolds of solutions to the sine-Gordon equation. The scattering and inverse scattering maps in this set-up are also discussed.     

On the two-point boundary-value problem for the Riccati matrix differential equation

Constructive sufficient conditions for univalent resolvability of a two-point boundary value problem for nonlinear Riccati equation are obtained. An illustrative example is given.     

Oscillations of linear integro-differential equations

Sufficient conditions which guarantee that certain linear integro-differential equation cannot have a positive solution are established.     

On stabilizability of evolution systems of partial differential equations on ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×[0,+∞) by time-delayed feedback controlsby time-delayed feedback controls

In this work we obtain sufficient conditions for stabilizability by time-delayed feedback controls for the system

$\frac{{\partial w\left( {x,t} \right)}}{{\partial t}} = A(D_x )w(x,t) - A(D_x )u(x,t), x \in \mathbb{R}^n , t > h, $

where D _x =(-i?/?x ₁,...-i?/?x _n ), A(σ) and B(σ) are polynomial matrices (m×m), det B(σ)≡0 on ? ⁿ , w is an unknown function, u(·,t)=P(D _x )w(·,t?h) is a control, h>0. Here P is an infinite differentiable matrix (m×m), and the norm of each of its derivatives does not exceed Γ(1+|σ|²)^γ for some Γ, γ∈? depending on the order of this derivative. Necessary conditions for stabilizability of this system are also obtained. In particular, we study the stabilizability problem for the systems corresponding to the telegraph equation, the wave equation, the heat equation, the Schrödinger equation and another model equation. To obtain these results we use the Fourier transform method, the Lojasiewicz inequality and the Tarski—Seidenberg theorem and its corollaries. To choose an appropriate P and stabilize this system, we also prove some estimates of the real parts of the zeros of the quasipolynomial det {Iλ-A(σ)+B(σ)P(σ)e ^-hλ.

     

A necessary and sufficient condition for the existence of an exponential attractor

We give a necessary and sufficient condition for the existence of an exponential attractor. The condition is formulated in the context of metric spaces. It also captures the quantitative properties of the attractor, i.e., the dimension and the rate of attraction. As an application, we show that the evolution operator for the wave equation with nonlinear damping has an exponential attractor.     

Weak solutions to a nonlinear variational wave equation and some related problems

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L ^p Young measure theory and related compactness results, in the first section. Then we use the L ^p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.     

Radial-type complete solutions for a class of partial differential equations

We give some fundamental solutions of a class of iterated elliptic equations including Laplace equation and its iterates.     

Stability of the analytic and numerical solutions for impulsive differential equations

This paper deals with the stability analysis of the analytic and numerical solutions of impulsive differential equations. In particular, the linear equation with variable coefficients and the nonlinear equation are considered. The stability conditions of the analytic solutions of these impulsive differential equations and the numerical solutions of the θ-methods are obtained. Finally, some numerical experiments are given.     

Popular ansatz methods and solitary wave solutions of the Kuramoto-Sivashinsky equation

Some methods to look for exact solutions of nonlinear differential equations are discussed. It is shown that many popular methods are equivalent to each other. Several recent publications with “new” solitary wave solutions for the Kuramoto-Sivashinsky equation are analyzed. We demonstrate that all these solutions coincide with the known ones.      

Solutions to the XXX type Bethe ansatz equations and flag varieties

We consider a version of the A _N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl _N +1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B _N and C _N . Populations of B _N and C _N type are isomorphic to the flag varieties of C _N and B _N types respectively.     

Oscillation results for second order nonlinear differential equations

In this paper, the authors present some new results for the oscillation of the second order nonlinear neutral differential equations of the form

一类推广的KdV方程的新行波解

本文研究了一类推广的Kd V方程的行波解求解的问题.利用新的G展开法,并借助Mathematica计算软件,获得了该方程的含有多个任意参数的新的行波解,分别为三角函数解、双曲函数解、有理函数解和指数函数解,扩大了该类方程的解的范围.     

New Exact Solutions of One Nonlinear Equation in Mathematical Biology and Their Properties

The classical Lie approach and the method of additional generating conditions are applied to constructing multiparameter families of exact solutions of the generalized Fisher equation, which is a simplification of the known coupled reaction–diffusion system describing spatial segregation of interacting species. The exact solutions are applied to solving nonlinear boundary-value problems with zero Neumann conditions. A comparison of the analytic results and the corresponding numerical calculations shows the importance of the exact solutions obtained for the solution of the generalized Fisher equation.     

Bifurcations of travelling wave solutions for the mK(n, n) equation

Using the method of planar dynamical systems to the mK(n, n) equation, the existence of uncountably infinite many smooth and non-smooth periodic wave solutions, solitary wave solutions and kink and anti-kink wave solutions is proved. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All possible exact explicit parametric representations of smooth and non-smooth travelling wave solutions are obtain.     

Partial decay on simple manifolds

Existence, uniqueness and weighted regularity of solutions of linear and nonlinear second-order uniformly elliptic differential equations on complete punctured compact N-manifolds, N > 2. Application to prescribed curvature problems: scalar curvature in a quasi-isometry class (including a contribution to the Lichnérowicz-York equation of General Relativity); Ricci curvature in a weighted Kähler class (with a related result in equiaffine geometry). A new asymptonic behaviour is allowed throughout, called partial decay, which requires its own maximum principle.Current support: CNRS; partial support; CEE contract GADGET # SC1-0105-C.     

On the concentration properties for the nonlinear Schrödinger equation with a Stark potential

In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.     

First-Order and Second-Order,Chaos-Free,Finite Difference Schemes for Fisher Equation

A new class of finite difference schemes is constructed for Fisher partial differential equation i.e. the reaction-diffusion equation with stiff source term: $au(1-u)$. These schemes have the properties that they reduce to high fidelity algorithms in the diffusion-free case namely in which the numerical solutions preserve the properties in the exact solutions for arbitrary time step-size and reaction coefficient α＞0 and all nonphysical spurious solutions including bifurcations and chaos that normally appear in the standard discrete models of Fisher partial differential equation will not occur. The implicit schemes so developed obtain the numerical solutions by solving a single linear algebraic system at each step. The boundness and asymptotic behaviour of numerical solutions obtained by all these schemes are given. The approach constructing the above schemes can be extended to reaction-diffusion equations with other stiff source terms.