Asymptotic Analysis - Volume 85, issue 1-2

Authors: Ghomari, Kaoutar | Messirdi, Bekkai | Vũ Ngọc, San

Article Type: Research Article

Abstract: This article reviews the Birkhoff–Gustavson normal form theorem (BGNF) near an equilibrium point of a quantum Hamiltonian. The BGNF process is thereafter used to investigate the spectrum of Schrödinger operators in the 1:1, 1:2 and 1:3 resonances. A computer program is proposed to compute the coefficients of the BGNF up to any order.

Keywords: Birkhoff normal form, harmonic oscillator, Bargmann representation, resonances, Fermi resonance

DOI: 10.3233/ASY-131186

Citation: Asymptotic Analysis, vol. 85, no. 1-2, pp. 1-28, 2013

Authors: Poliakovsky, Arkady

Article Type: Research Article

Abstract: We develop a variational technique for some wide classes of nonlinear evolutions. The novelty here is that we derive the main information directly from the corresponding Euler–Lagrange equations. In particular, we prove that not only the minimizer of the appropriate energy functional but also any critical point must be a solution of the corresponding evolutional system.

Keywords: evolution equations, variational principles

DOI: 10.3233/ASY-131171

Citation: Asymptotic Analysis, vol. 85, no. 1-2, pp. 29-74, 2013

Authors: Wu, Hao | Jiang, Jie

Article Type: Research Article

Abstract: In this paper, we study the Cauchy problem of a time-dependent drift-diffusion-Poisson system for semiconductors. Existence and uniqueness of global weak solutions are proven for the system with a higher-order nonlinear recombination-generation rate R. We also show that the global weak solution will converge to a unique equilibrium as time tends to infinity.

Keywords: drift-diffusion-Poisson system, global weak solution, uniqueness, long-time behavior

DOI: 10.3233/ASY-131176

Citation: Asymptotic Analysis, vol. 85, no. 1-2, pp. 75-105, 2013

Authors: Fila, Marek | Ishige, Kazuhiro | Kawakami, Tatsuki

Article Type: Research Article

Abstract: We consider the following initial value problem for a two-dimensional semilinear elliptic equation with a dynamical boundary condition: −Δu=up , x∈R2 + , t>0, ∂t u+∂ν u=0, x∈∂R2 + , t>0, u(x,0)=φ(x1 )≥0, x=(x1 ,0)∈∂R2 + , where u=u(x,t), ∂t :=∂/∂t, ∂ν :=−∂/∂x2 , R2 + :={(x1 ,x2 ): x1 ∈R,x2 >0} and p>1. We show that small solutions behave asymptotically like suitable multiples of the Poisson kernel. This is an extension of previous results of the authors of this paper to the two-dimensional case.

Keywords: semilinear elliptic equation in a half-plane, dynamical boundary condition, large time behavior

DOI: 10.3233/ASY-131183

Citation: Asymptotic Analysis, vol. 85, no. 1-2, pp. 107-123, 2013