Landau-Lifshitz方程派生的球面锥对称族及其演化

直至2008年,尚未见到讨论同时具有外磁场和各向异性场的多维Landau-Lifshitz方程的显式动态解的文献.当缺少外加磁场或各向异性场时,许多作者曾经尝试过Hirota的著名方法,Lie代数结构与Backlund变换,双线性变换等,但精确解仍然没有构造出来.本文首先给出了一种求解具有外磁场和各向异性场的Landa...     

Some exact solutions to multidimensional Landau-Lifshitz equation with uprush external field and anisotropy field

In this paper we give a method that can construct some exact solutions to a multidimensional Landau-Lifshitz equation with uprush external field and anisotropy field.     

The Landau-Lifshitz model for shape optimization of MRAM memories

We study a generalized Landau-Lifshitz (LL) equation with space-variable coefficients. We extend the results on existence and regularity of the smooth solutions for the case of the LL equation with constant coefficients. Further we study the sensitivity of the solution and provide continuous dependence on the coefficients. As a motivation we mention optimization of a magnetic core of MRAM and we perform real computations determining the optimal shape by maximizing the writing efficiency and minimizing the side-effects. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some new nonlinear wave solutions and their convergence for the (2+1)‐dimensional Broer–Kau–Kupershmidt equation

We use the bifurcation method of dynamical systems to study the (2+1)‐dimensional Broer–Kau–Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow‐up wave solutions, periodic smooth wave solutions, periodic blow‐up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow‐up wave solutions converge to the solitary wave solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical methods for a coupled system of differential equations arising from a thermal ignition problem

This article is concerned with monotone iterative methods for numerical solutions of a coupled system of a first‐order partial differential equation and an ordinary differential equation which arises from fast‐igniting catalytic converters in automobile engineering. The monotone iterative scheme yields a straightforward marching process for the corresponding discrete system by the finite‐difference method, and it gives not only a computational algorithm for numerical solutions of the problem but also the existence and uniqueness of a finite‐difference solution. Particular attention is given to the “finite‐time” blow‐up property of the solution. In terms of minimal sequence of the monotone iterations, some necessary and sufficient conditions for the blow‐up solution are obtained. Also given is the convergence of the finite‐difference solution to the continuous solution as the mesh size tends to zero. Numerical results of the problem, including a case where the continuous solution is explicitly known, are presented and are compared with the known solution. Special attention is devoted to the computation of the blow‐up time and the critical value of a physical parameter which determines the global existence and the blow‐up property of the solution. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Painlev\'{e} Analysis and Auto-B\"{a}cklund Transformation for a General Variable Coefficient Burgers Equation with Linear Damping Term

This paper investigates a general variable coefficient (gVC) Burgers equation with linear damping term. We derive the Painlev\''{e} property of the equation under certain constraint condition of the coefficients. Then we obtain an auto-B\"{a}cklund transformation of this equation in terms of the Painlev\''{e} property. Finally, we find a large number of new explicit exact solutions of the equation. Especially, infinite explicit exact singular wave solutions are obtained for the first time. It is worth noting that these singular wave solutions will blow up on some lines or curves in the $(x,t)$ plane. These facts reflect the complexity of the structure of the solution of the gVC Burgers equation with linear damping term. It also reflects the complexity of nonlinear wave propagation in fluid from one aspect.     

A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution

In this paper, we establish a version of the Feynman–Kac formula for multidimensional stochastic heat equation driven by a general semimartingale. This Feynman–Kac formula is then applied to study some nonlinear stochastic heat equations driven by nonhomogeneous Gaussian noise: first, an explicit expression for the Malliavin derivatives of the solutions is obtained. Based on the representation we obtain the smooth property of the density of the law of the solution. On the other hand, we also obtain the Hölder continuity of the solutions.     

Blow up problem for Landau-Lifshitz equations in two dimensions

In 1986, Zhou and Guo in [1] proved the global existence of weak solution for generalized Landau-Lifshitz equations without Gilbert term in multi-dimensions. They consider thehomogeneous boundary problemwith the initial value conditionfor the system of ferromagnetic chain with several variableswhere j(x, t, Z) is a given 3-dimensional vector funC-non in x e R", t E R+, Z e R', W(x)is a given 3-dimensional initial value function on fi, fi is a bounded domain in n-dimensionalEuclidean space…     

Non-constant stable solutions to Landau-Lifshitz equation

We prove the existence of solutions to the static Landau-Lifshitz equation corresponding to the homotopy classes of continuous functions from . Here is a non-simply connected bounded domain in . The stability of the solutions to the time-dependent Landau-Lifshitz equation is also obtained. Received June 29, 1996 / in revised form June 18, 1997 / Accepted July 13, 1997     

Blow‐up solutions to Landau–Lifshitz–Maxwell systems

The Landau–Lifshitz–Gilbert equation describes the evolution of spin fields in continuum ferromagnetics. The present paper consists of two parts. The first one is to prove the local existence of smooth solution to the Landau–Lifshitz–Maxwell systems in dimensions three. The second is to prove the finite time blow up of solutions for these systems. It states that for suitably chosen initial data, the short time smooth solutions to the Landau–Lifshitz–Maxwell equations do blow up at finite time. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical Study of Blow‐Up Mechanisms for Davey–Stewartson II Systems

We present a detailed numerical study of various blow‐up issues in the context of the focusing Davey–Stewartson II equation. To this end, we study Gaussian initial data and perturbations of the lump and the explicit blow‐up solution due to Ozawa. Based on the numerical results it is conjectured that the blow‐up in all cases is self‐similar, and that the time‐dependent scaling behaves as in the Ozawa solution and not as in the stable blow‐up of standard L ² critical nonlinear Schrödinger equation. The blow‐up profile is given by a dynamically rescaled lump.     

Sharp Markov brothers type inequality in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> on a closed interval

A recent modification of a classic Landau-Lifshitz equation that includes the socalled spin-transfer torque is widely recognized in physics community as a model of magnetization dynamics in certain nanodevices. Motivated by some experimental evidence, we introduce a generalization of this model, coupled Landau-Lifshitz equations with spin-transfer torque terms, and analyze it from dynamical systems standpoint. An explicit stability criterion for the critical points in terms of all parameters of the system is derived and illustrated with stability diagrams. Our analysis provides certain guidelines for the design of magnetic nanodevices with optimized response to control parameters.     

On the Cauchy problem for the generalized Camassa–Holm equation

We establish the local well-posedness for the generalized Camassa–Holm equation. We also prove that the equation has smooth solutions that blow up in finite time.     

ON INITIAL BOUNDARY VALUE PROBLEMS FOR NONLINEAR SCHRDINGER EQUATIONS

ＯＮＩＮＩＴＩＡＬ　ＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＳＦＯＲＮＯＮＬＩＮＥＡＲＳＣＨＲＤＩＮＧＥＲＥＱＵＡＴＩＯＮＳ￥ＬｉＹｏｎｇｓｈｅｎｇ（李用声）ＣｈｅｎＱｉｎｇｙｉ（陈庆益）（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＨｕａｚｈｏｎｇＵｎｖ．ｏｆＳｃｉ...     

Evolution of hypersurfaces by the mean curvature minus an external force field

In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.     

ASYMPTOTICS FOR POSITIVE SOLUTIONS OF THE WEIGHTED LAPLACE EQUATIONS INVOLVING CRITICAL GROWTH

1IntroductionConsiderthefollowingproblemwl1ereflCR",n23,isaboundeddomainwithsmoothboundarycontainingtheorigin.TheexponentscrjP,rrandpsatisfyItisknownthatthefollowillgS(i1)'llf'v-Hardyinequalityholdsifandonlyif(1.2)issatisfied(cL[4])foralluECj(R"),theconstantS(cr,P)seebelow(1.5).Weshallbeconcernedwiththecasesp5o,p 1>2andcrta>P-2.DenotebyHt(fl)thespaceofcompletionofCi(fl)undertheinnerproduct(u,v)=jnlxld7u'7l)dx.uEH:(fl)iscalledaweaksolutionof(1-1)iff0rallvEHt(fl).Ithasbeeushownin[5]thata…     

Adiabatic approximations for Landau-Lifshitz equations

The asymptotics with respect to a small parameter for solutions of a system of Landau-Lifshitz equations with slowly varying coefficients and small dissipative terms is investigated. These equations are a mathematical model of a uniaxial ferromagnet in a time-dependent magnetic field. The asymptotics constructed make it possible to describe the magnetization reversal effect and to reveal the influence of the parameters of the external magnetic field and dissipation on the stability of this process.     

On the equivalence of the electromagnetic problem of diffraction by an inhomogeneous bounded dielectric body to a volume singular integro-differential equation

The paper is concerned with the smoothness of the solutions to the volume singular integrodifferential equations for the electric field to which the problem of electromagnetic-wave diffraction by a local inhomogeneous bounded dielectric body is reduced. The basic tool of the study is the method of pseudo-differential operators in Sobolev spaces. The theory of elliptic boundary problems and field-matching problems is also applied. It is proven that, for smooth data of the problem, the solution from the space of square-summable functions is continuous up to the boundaries and smooth inside and outside of the body. The results on the smoothness of the solutions to the volume singular integro-differential equation for the electric field make it possible to resolve the issues on the equivalence of the boundary value problem and the equation.     

Simple explicit formulae for finite time blow up solutions to the complex KdV equation

Simple explicit formulae for finite time blow up solutions to the complex KdV equation are obtained via a Darboux transformation. Diffusions induced by perturbations are calculated.