Long Range Scattering and Modified Wave Operators for the Wave-Schrödinger System II

We continue the study of scattering theory for the system consisting of a Schrödinger equation and a wave equation with a Yukawa type coupling in space dimension 3. In a previous paper, we proved the existence of modified wave operators for that system with no size restriction on the data and we determined the asymptotic behaviour in time of solutions in the range of the wave operators, under a support condition on the asymptotic state required by the different propagation properties of the wave and Schrödinger equations. Here we eliminate that condition by using an improved asymptotic form for the solutions. Communicated by Bernard Helffer submitted 20/02/03, accepted: 24/06/03     

Asymptotically recurrent solutions ofβ-differential equations

In the paper methods from the theory of extensions of dynamical systems are used to studyβ-differential equations whose solutions possess the uniqueness property and depend continuously on the initial data and on the right-hand side of the equation. The Zhikov-Bronshtein theorems concerning asymptotically almost periodic solutions of ordinary differential equations are extended toβ-differential equations (in particular, to total differential equations). Along with asymptotic almost periodicity, we also consider asymptotic recurrence, weak asymptotic distality, and asymptotic distality. To the equations we associate dynamical systems generated by the space of the right-hand sides and the spaces of the solutions and of the initial data of solutions of the equation. Generally, the phase semigroups of the dynamical systems are not locally compact. Translated fromMatermaticheskie Zametki, Vol. 67, No. 6, pp. 837–851, June, 2000.     

Asymptotics for solutions of periodic difference systemsSupported in part by Fondecyt 1030535, 1080034, DGI-MATH UNAP 2008 and Diufro 120521.

Using dichotomies and periodic conditions, we obtain asymptotic formulas for solutions of a difference system of Poincaré type with periodic coefficients. Some results about the theory of existence of periodic solutions for linear difference systems are presented. At the end, an open problem on the asymptotic spectral representation is proposed.     

Positive solutions for Lotka–Volterra competition system with large cross-diffusion in a spatially heterogeneous environment

In this paper, we study the stationary problem for the Lotka–Volterra competition system with cross-diffusion in a spatially heterogeneous environment. Although some sufficient conditions for the existence of positive solutions are obtained by using global bifurcation theory, the information for their structure is far from complete. In order to get better understanding of the competition system with cross-diffusion, we focus on the asymptotic behaviour of positive solutions and derive two shadow systems as the cross-diffusion coefficient tends to infinity, moreover, the structure of positive solutions of the limiting system is analysed. The result of asymptotic behaviour also reveals different phenomena from that studied in Wang and Li (2013).     

Local asymptotic attractivity for nonlinear quadratic functional integral equations

In this paper, an existence result for local asymptotic attractivity of the solutions is proved for a nonlinear quadratic functional integral equation under certain growth conditions which in turn gives the existence as well as asymptotic stability of solutions. A couple of examples are provided for indicating the natural realizations of abstract theory presented in the paper.     

Quasi-classical asymptotics of solutions to the matrix factorization problem with quadratically oscillating off-diagonal elements

The paper investigates the asymptotic behavior of solutions to the 2 × 2 matrix factorization (Riemann-Hilbert) problem with rapidly oscillating off-diagonal elements and quadratic phase function. A new approach to study such problems based on the ideas of the stationary phase method and M. G. Krein’s theory is proposed. The problem is model for investigating the asymptotic behavior of solutions to factorization problems with several turning points. Power-order complete asymptotic expansions for solutions to the problem under consideration are found. These asymptotics are used to construct asymptotics for solutions to the Cauchy problem for the nonlinear Schrödinger equation at large times.     

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.     

Two dimensional solitary waves in shear flows

In this paper we study existence and asymptotic behavior of solitary-wave solutions for the generalized Shrira equation, a two-dimensional model appearing in shear flows. The method used to show the existence of such special solutions is based on the mountain pass theorem. One of the main difficulties consists in showing the compact embedding of the energy space in the Lebesgue spaces; this is dealt with interpolation theory. Regularity and decay properties of the solitary waves are also established.     

Existence of solutions for dual singular integral equations with convolution kernels in case of non-normal type

This paper is devoted to the study of dual singular integral equations with convolution kernels in the case of non-normal type. Via using the Fourier transforms, we transform such equations into Riemann boundary value problems. To solve the equation, we establish the regularity theory of solvability. The general solutions and the solvable conditions of the equation are obtained. Especially, we investigate the asymptotic property of solutions at nodes. This paper will have a significant meaning for the study of improving and developing complex analysis, integral equations and Riemann boundary value problems.     

Steady-state Solution for Reaction-diffusion Models with Mixed Boundary Conditions

In this paper, we deal with a diffusive predator-prey model with mixed boundary conditions, in which the prey population can escape from the boundary of the domain while predator population can only live in this area and can not leave. We first investigate the asymptotic behaviour of positive solutions and obtain a necessary condition ensuring the existence of positive steady state solutions. Next, we investigate the existence of positive steady state solutions by using maximum principle, the fixed point index theory, Lpestimation, and embedding theorems, Finally, local stability and uniqueness are obtained by linear stability theory and perturbation theory of linear operators.     

Attractors and transients for a perturbed periodic KdV equation: A nonlinear spectral analysis

Summary In this paper we rigorously show the existence and smoothness inε of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameterε. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions.     

半线性摄动电报方程的渐近理论及应用

对二阶半线性摄动电报方程的初值问题.本交给出了一个渐近方法.证明了渐近理论及形式近似解的合理性都在时间变量无穷大时(即0≤t≤O(|ε|-1)成立.作为浙近理论的应用,我们对一个带初问题的特殊电报方程进行了研究,得到了两个|ε|-1阶渐近近似解.     

Stabilization to a positive equilibrium for some reaction–diffusion systems

The aim of this paper is to study the asymptotic behavior of solutions for some reaction–diffusion systems in biology. First, we establish a Liouville type theorem for entire solutions of these reaction–diffusion systems. Based on this theorem, we derive the stabilization of the solutions of the reaction–diffusion system to the unique positive constant state, under the condition that this positive constant state is globally stable in the corresponding kinetic systems. Two specific examples about spreading phenomena from ecology and epidemiology are given to illustrate the application of this theory.     

Long range scattering for the modified Schrödinger map in two space dimensions

We study the asymptotic behaviour in time of solutions and the theory of scattering for the modified Schrödinger map in two space dimensions. We solve the Cauchy problem with large finite initial time, up to infinity in time, and we determine the asymptotic behaviour in time of the solutions thereby obtained. As a by product, we obtain global existence for small data in Hk∩FHk with k>1. We also solve the Cauchy problem with infinite initial time, namely we construct solutions defined in a neighborhood of infinity in time, with prescribed asymptotic behaviour of the previous type.     

Convergence in almost periodic cooperative systems with a first integral

This paper is to investigate the asymptotic dynamics in almost periodic cooperative systems with a first integral. By appealing to the theory of skew-product semiflows we establish the asymptotic almost periodicity of bounded solutions to such systems, which extends the existing convergence results for time independent and periodic cooperative systems with a first integral and proves a conjecture of B. Tang, Y. Kuang and H. Smith in SIAM J. Math. Anal., 24 (1993), 1331-1339.

     

Viscous-Inviscid Interactions in a Boundary-Layer Flow Induced by a Vortex Array

In this paper, we investigate the asymptotic validity of boundary-layer theory. For a flow induced by a periodic row of point-vortices, we compare Prandtl’s boundary-layer solution to Navier-Stokes solutions with different Reynolds numbers. We show how Prandtl’s solution develops a finite-time separation singularity. On the other hand, the Navier-Stokes solutions are characterized by the presence of two distinct types of viscous-inviscid interactions that can be detected by the analysis of the enstrophy and of the pressure gradient on the wall. Moreover, we apply the complex singularity-tracking method to Prandtl and Navier-Stokes solutions and analyze the previous interactions from a different perspective.     

Two-periodic waves and asymptotic property for generalized 2D Toda lattice equation

In this paper, two-periodic wave solutions are constructed for the (2 + 1)-dimensional generalized Toda lattice equation by using Hirota bilinear method and Riemann theta function. At the same time, we analyze in details asymptotic properties of the two-periodic wave solutions and give their asymptotic relations between the periodic wave solutions and the soliton solutions.     

Existence and uniqueness of generalized monopoles in six-dimensional non-Abelian gauge theory

In this paper, we establish the existence and uniqueness of the spherically symmetric monopole solutions in SO(5) gauge theory with Higgs scalar fields in the vector representation in six-dimensional Minkowski space-time and obtain sharp asymptotic estimates for the solutions. Our method is based on a dynamical shooting approach that depends on two shooting parameters which provides an effective framework for constructing the generalized monopoles in six-dimensional Minkowski space-time.     

Long Range Scattering and Modified Wave Operators for the Maxwell–Schrödinger System II. The General Case

We study the theory of scattering for the Maxwell–Schr?dinger system in space dimension 3, in the Coulomb gauge. We prove the existence of modified wave operators for that system with no size restriction on the Schr?dinger and Maxwell asymptotic data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators. The method consists in partially solving the Maxwell equations for the potentials, substituting the result into the Schr?dinger equation, which then becomes both nonlinear and nonlocal in time. The Schr?dinger function is then parametrized in terms of an amplitude and a phase satisfying a suitable auxiliary system, and the Cauchy problem for that system, with prescribed asymptotic behaviour determined by the asymptotic data, is solved by an energy method, thereby leading to solutions of the original system with prescribed asymptotic behaviour in time. This paper is the generalization of a previous paper with the same title. However it is entirely self contained and can be read without any previous knowledge of the latter. Submitted: November 7, 2006. Accepted: November 14, 2006.