LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE -DIMENSIONAL TORUS

We consider the nonlinear wave equation (NLW) on the $d$ -dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on $\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when $d=1$ while our strategy applies in any dimension.

Long wavelength superluminal pulse propagation in a defect slab doped with GaAs/AlGaAs multiple quantum well nanostructure

In this paper, long wavelength superluminal and subluminal properties of pulse propagation in a defect slab medium doped with four-level GaAs/AlGaAs multiple quantum wells (MQWs) with 15 periods of 17.5 nm GaAs wells and 15 nm [Formula: see text] barriers is theoretically discussed. It is shown that exciton spin relaxation (ESR) between excitonic states in MQWs can be used for controlling the superluminal and subluminal light transmissions and reflections at different wavelengths. We also show that reflection and transmission coefficients depend on the thickness of the slab for the resonance and nonresonance conditions. Moreover, we found that the ESR for nonresonance condition lead to superluminal light transmission and subluminal light reflection.

Nonlinear Stability of the Triangular Libration Points for Radiating and Oblate Primaries in CR3BP in Nonresonance Condition

This paper investigates the existence of resonance and nonlinear stability of the triangular equilibrium points when both oblate primaries are luminous. The study is carried out near the resonance frequency, satisfying the conditionsω1=ω2, ω1=2ω2, andω1=3ω2in circular cases by the application of Kolmogorov-Arnold-Moser (KAM) theory. The study is carried out for the various values of radiation pressure and oblateness parameters in general. It is noticed that the system experiences resonance atω1=2ω2, ω1=3ω2for different values of radiation pressures and oblateness parameter. The caseω1=ω2corresponds to the boundary region of the stability for the system. It is found that, except for some values of the radiation pressure, and oblateness parameters and forμ≤μc=0.0385209, the triangular equilibrium points are stable.

Existence of Solutions of a Discrete Fourth-Order Boundary Value Problem

Leta,bbe two integers withb-a≥5and let𝕋2={a+2,a+3,…,b-2}. We show the existence of solutions for nonlinear fourth-order discrete boundary value problemΔ4u(t-2)=f(t,u(t),Δ2u(t-1)),t∈𝕋2,u(a+1)=u(b-1)=Δ2u(a)=Δ2u(b-2)=0under a nonresonance condition involving two-parameter linear eigenvalue problem. We also study the existence and multiplicity of solutions of nonlinear perturbation of a resonant linear problem.

A Nonresonance Condition for Boundary Value Problems

We consider the boundary value problemu″ + g(u)u = h(t, u, u′)u(0) = 0, u(π) = 0with g continuous, periodic, and positive and h continuous and bounded. It is shown that there is a solution if the mean value of g is not the square of an integer.

On a boundary value problem arising in elastic deflection theory

In this paper, a finite-difference method for the determination of an approximate solution of a fourth-order two-point boundary value problem is presented under the nonresonance condition. The solution of this linear problem can be used to find approximate solutions of a broad range of nonlinear problems in applications.