SINGULAR AND RAREACTIVE SOLUTIONS TO A NONLINEAR VARIATIONAL WAVE EQUATION

51. IntroductionIn this paper, we study the ealstence and regularity properties of weak solutions to thefonowillg nonlinear wave equationwhere c(.) is a given smooth, bounded, and positive function, "o(x) E Lip(B),ul(x) ELoo(R).Equation (1.1) is the Euler-Lagrange equation of the action principlefrom which comes the name variational wave equation. It is the simplest represelltative ofa large class of variational wave equatinns in classical field theories and general relativity(see [6]). We…     

ATTRACTORS FOR DISCRETIZATION OF GINZBURG-LANDAU-BBM EQUATIONS

1. IntroductionIn this paper) we consider the following periodic initial value problem for the system ofGinzburg-Landau equation coupled with BBM equationwhere e(x,t) is a complex function, n(x, t) is a real scalar function, at a, 5, 7, al, a2, FI, adZare real constants, and gi (x), g200 are given real functions.This problem describes the nonlinear interactions between Langmuir wave and ion acousticwave in plasma physics, e(x, t) denotes electric field, n(x, t) the perturbation of density (…     

STABILITY OF THE SOLUTION OF THE ITERATED EQUATION sum from i=1 to n λ_if(x)=F(x)

This note is a further work of discussion on the iterated equation λ_1f(x)+λ_2f~2(x)+…λ_nf~n(x)-P(x), following the author's conclusions of the existence and uniqueness of this equation. In this note we prove that the sufficient condition of the stability of the solution is the same one as the uniqueness of the solution (see [4]).     

Likely Limit Sets of a Class of p-order Feigenbaum's Maps

A continuous map from a closed interval into itself is called a p-order Feigenbaum's map if it is a solution of the Feigenbaum's equation fp(λx) =λf(x).In this paper,we estimate Hausdorff dimensions of...     

A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N-1Va(xi-xj) where Va(x) = a-2V (x/a). It is well known that the Gross-Pitaevskii (GP) equation is a nonlinear Schrdinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices {k ut, k ≥ 1} solves the GP hierarchy. Denote by ψN,t the solution to the N-particle Schrdinger equation. Under the assumption that a = N-ε for 0 ε 3/4, we prove that as N →∞ the limit points of the k-particle density matrices of ψN,t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫V (x) dx.     

DEEP REDUCTION OF DARBOUX TRANSFORMATION TO A3×3 SPECTRAL PROBLEM

gi. IntroductionWave propagation in optical fibers is governed by the higher order nonlinear Schrodinger(HNLS) equation. In receDt years, many authors have analyzed the HNLS equation fromdtherellt poillts of view[1'2]. How to find the exact solutions of the equation is an illterestingwork.In 13], the following higher-order nonlinear Schr6dinger equationwas discussed. It was found in 14] that when FI: PZ: P3 = 1: 6: 3, this equation can betransformed to.Consider a 3 x 3 spectral problem…     

Stability of planar diffusion wave for nonlinear evolution equation

It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f’(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.     

A Constrained Optimization Approach for LCP

In this paper, LCP is converted to an equivalent nonsmooth nonlinear equation system H(x,y) = 0 by using the famous NCP function-Fischer-Burmeister function. Note that some equations in H(x, y) = 0 are nonsmooth and nonlinear hence difficult to solve while the others are linear hence easy to solve. Then we further convert the nonlinear equation system H(x, y) = 0 to an optimization problem with linear equality constraints. After that we study the conditions under which the K-T points of the optimization problem are the solutions of the original LCP and propose a method to solve the optimization problem. In this algorithm, the search direction is obtained by solving a strict convex programming at each iterative point, However, our algorithm is essentially different from traditional SQP method. The global convergence of the method is proved under mild conditions. In addition, we can prove that the algorithm is convergent superlinearly under the conditions: M is P0 matrix and the limit point is a strict complementarity solution of LCP. Preliminary numerical experiments are reported with this method.     

MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY

In this article,we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity-△u+(λa(x)+1)u=(1/|x|α*F(u))f(u) in R~N,where N≥3,0 αmin{N,4},λ is a positive parameter and the nonnegative potential function a(x) is continuous.Using variational methods,we prove that if the potential well int(a~(-1)(0)) consists of k disjoint components,then there exist at least 2~k-1 multi-bump solutions.The asymptotic behavior of these solutions is also analyzed as λ→+∞.     

THE PATCH PROPERTY OF SELF-DUAL YANG-MILLS FIELD

SDYM-field is a holomorphic vector bandles over twistor Euclidean Space covered by two patches U_1 and U_2. In this paper, the property of the field on each patch and relationship between them are found. We point out that if the field on one patch correspond to the left SDYM-J equation, then on the other patch will be right SDYM-J equation. The Lagrangian form of the field is also found, which has relation to H_n=SL(n, c)/SU(n) nonlinear sigma model with Wess-Zumino terms.     

Generalized Extended tanh-function Metho d for Traveling Wave Solutions of Nonlinear Physical Equations

In this paper,the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations.We choose Fisher's equation,the nonlinear schr(o|¨)dinger equation to illustrate the validity and advantages of the method.Many new and more general traveling wave solutions are obtained.Furthermore,this method can also be applied to other nonlinear equations in physics.     

A FINITE DIFFERENCE SCHEME FOR THE GENERALIZED NONLINEAR SCHRDINGER EQUATION WITH VARIABLE COEFFICIENTS

1. Generalized Nonlinear Schrsdinger EquationThe Schr6dinger equation has been extensively used in physics research, particularlyin the modeling of nonlinear dispersion waves [8]. Numerical methods for solving theSchr6dinger equation have been discussed in the literature. In this article, we considera generalized nonlinear Schr6dinger equation with variable coefficientsi: ~ g(A(x)Z) iF(t)u B(x) lulp~' u = 0, iZ ~ ~l, P > 1, (1)where u(x, 0) ~ of (x). The coefficients A(x), F(t) and, …     

RESEARCHANNOUNCEMENTS——TheCauchyProblemsandGlobalSolutionstoDeybeSystemandBurgers''''EquationsinPseudomeasureSpaces

In this paper we consider the construction of solutions to the Cauchy problem of Burgers' equationsut-γ△u + u·▽u = 0, t∈R+,x∈R3, (1)u(0,x)= u0(x), x ∈ R3, (2)in pseudomeasure spaces, where γ(?) 0 is a small parameter that plays the role of the viscosity and u = u(t,x) is a velocity-like vector field defined on R+×R3. The initial datum u0(x) is a vector-valued function defined on R3, In one space dimension Burgers' equation is     

Convergence of ground state solutions for nonlinear Schrdinger equations on graphs

We consider the nonlinear Schr¨odinger equation-?u +(λa(x) + 1)u = |u|~(p-1) u on a locally finite graph G =(V, E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ 1, the equation admits a ground state solution uλ. Moreover, as λ→∞, the solution uλconverges to a solution of the Dirichlet problem-?u + u = |u|~(p-1) u which is defined on the potential well ?. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.     

Homogenization of Incompressible Euler Equations

In this paper, we perform a nonlinear multiscale analysis for incompressible Euler equations with rapidly oscillating initial data. The initial condition for velocity field is assumed to have two scales. The fast scale velocity component is periodic and is of order one.One of the important questions is how the two-scale velocity structure propagates in time and whether nonlinear interaction will generate more scales dynamically. By using a Lagrangian framework to describe the propagation of small scale solution, we show that the two-scale structure is preserved dynamically. Moreover, we derive a well-posed homogenized equation for the incompressible Euler equations. Preliminary numerical experiments are presented to demonstrate that the homogenized equation captures the correct averaged solution of the incompressible Euler equation.     

ON THE MODIFIED NONLINEAR SCHRDINGER EQUATION IN THE SEMICLASSICAL LIMIT:SUPERSONIC,SUBSONIC,AND TRANSSONIC BEHAVIOR

The purpose of this paper is to present a comparison between the modified nonlinear Schro¨dinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schr¨odinger (NLS) equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation (in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data), or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.     

SINGULAR PERTURBATIONS OF BOUNDARY VALUE PROBLEMS FOR THIRD ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

In this paper we obtain some results about the convergence of aolutions of the boundary value problems of the third order nonlinear ordinary differential equation with a small parameter ε>0: (i=0, 1, 2) to a solution of their reduced problem as ε→0, hero z=ψ(t, x, y) is a root of the equation f(t, x, y, z, 0)=0, and about the existence of solutions of the reduced problem. In addition, under certain conditions we prove the existence of solutions of the boundary value problems (1), (2_i) (i=1, 2), and give their asymptotic estimations.     

Existence of Nondecreasing and Continuous Solutions of an Integral Equation with Linear Modification of the Argument

We use a technique associated with measures of noncompactness to prove the existence of nondecreasing solutions to an integral equation with linear modification of the argument in the space C[0, 1]. In the last thirty years there has been a great deal of work in the field of differential equations with a modified argument. A special class is represented by the differential equation with affine modification of the argument which can be delay differential equations or differential equations with linear modifications of the argument. In this case we study the following integral equation x（t） = a（t） ＋ （Tx）（t） ∫0^σ（t） u（t, s, x（s）, x（λs））ds 0 〈 λ 〈 1 which can be considered in connection with the following Cauchy problem x＇（t） = u（t, s, x（t）, x（λt））, t ∈ [0, 1], 0 〈 λ 〈 1 x（0） = u0.     

COMPREHENSIVE CONSIDERATION OF THE NEW PERIODIC SOLUTION OF THE NONLINEAR SCHRDINGER EQUATION

For the nonlinear schrodinger equation it_t u_(xx) 2|u|~u=0, the writer once obtained a new periodic solution (x, t). In this paper, we will discuss some properties of (x, t), get all the main spectrum and the auxiliary spectrum of (x, t), and analyse motion forms of the spectra. We will also prove that (x, t) is an one-band potential.     

Homogenization of a nonlinear degenerate parabolic differential equation

This paper is devoted to the homogenization of a nonlinear degenerate parabolic problem ɑtu∈-div(D(x/∈, u∈,▽u∈)+ K(x/∈, u∈))= f(x) with Dirichlet boundary condition. Here the operator D(y, s,s) is periodic in y and degenerated in ▽s. In the paper, under the two-scale convergence theory, we obtain the limit equation as ∈→ 0 and also prove the corrector results of ▽u∈ to strong convergence.