Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

We consider the periodic solutions of a semilinear variable coefficient wave equation arising from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media.The variable coefficient characterizes the inhomogeneity of media and its presence usually leads to the destruction of the compactness of the inverse of the linear wave operator with periodic-Dirichlet boundary conditions on its range.In the pioneering work of Barbu and Pavel(1997),they gave the existence and regularity of the periodic solution for Lipschitz,nonresonant and monotone nonlinearity under the assumptionηu>0(see Section 2 for its definition)on the coefficient u(x)and left the caseηu=0 as an open problem.In this paper,by developing the invariant subspace method and using the complete reduction technique and Leray-Schauder theory,we obtain the existence of periodic solutions for such a problem when the nonlinear term satisfies the asymptotic nonresonance conditions.Our result needs neither requirements on the coefficient except the natural positivity assumption(i.e.,u(x)>0)nor the monotonicity assumption on the nonlinearity.In particular,when the nonlinear term is an odd function and satisfies the global nonresonance conditions,there is only one(trivial)solution to this problem in the invariant subspace.

EXTENSION AND APPLICATION OF NEWTON'S METHOD IN NONLINEAR OSCILLATION THEORY

In this paper we suggest and prove that Newton's method may calculate the asymptotic analytic periodic solution of strong and weak nonlinear nonautonomous systems, so that a new analytic method is offered for studying strong and weak nonlinear oscillation systems. On the strength of the need of our method, we discuss the existence and calculation of the periodic solution of the second order nonhomogeneous linear periodic system. Besides, we investigate the application of Newton's method to quasi-linear systems. The periodic solution of Duffing equation is calculated by means of our method.

Unb ounded Motions in Asymmetric Oscillators Dep ending on Derivatives

In this paper, we consider the unboundedness of solutions for the asymmetric equation x''+ax^+-bx^-+(x)ψ(x')+f(x)+g(x')=p(t),where x^+= max{x, 0}, x^-= max{-x, 0}, a and b are two different positive constants,f(x) is locally Lipschitz continuous and bounded, (x), ψ(x), g(x) and p(t) are continuous functions, p(t) is a 2π-periodic function. We discuss the existence of unbounded solutions under two classes of conditions: the resonance case 1/a^(1/2)+1/b^(1/2)∈Q and the nonresonance case 1/a^(1/2)+1/b^(1/2)?

A Note on a Singular Dirichlet Problem

The Dirichlet problem to a second order differential equation with some singularities is studied. Some existence results are established to the problem which generalize some results recently obtained by D. O’Regan by eliminating some superfluous constrains to the problem. Also, some new results have been proven which may provide more useful information for the study of the problem.

Femtosecond strong-field coherent control of nonresonant ionization with shaped pulses

The strong-field coherent control of the nonresonant ionization of nitrous oxide using shaped pulses is investigated.We study the dependence of periodic coherent oscillation of the total ionization yield on the variation of laser phase parameters. The physical mechanism of the strong-field coherent control is investigated experimentally and theoretically by the nonresonant spectral phase interferences in the frequency domain. We show that the intense shaped pulses with broadband and off-resonance can be used as a robust strong-field coherent control method.

Preparation of entangled simultaneous nonresonant atomic states through atom-field interaction

A scheme is proposed for generating a three-atom maximal entanglement W state. It is based on the simultaneous nonresonant interaction of atoms with a single-mode cavity field. Our scheme is insensitive to the cavity field, so the cavity field in our scheme can be initially in thermal states.

Highly selective population of two excited states in nonresonant two-photon absorption

A nonresonant two-photon absorption process can be manipulated by tailoring the ultra-short laser pulse. In this paper, we theoretically demonstrate a highly selective population of two excited states in the nonresonant two- photon absorption process by rationally designing a spectral phase distribution. Our results show that one excited state is maximally populated while the other state population is widely tunable from zero to the maximum value. We believe that the theoretical results may play an important role in the selective population of a more complex nonlinear process comprising nonresonant two-photon absorption, such as resonance-mediated （2-~l）-three-photon absorption and （2q-1）-resonant multiphoton ionization.

Nonresonant and Resonant Nonlinear Absorption of CdSe-Based Nanoplatelets

We present a comprehensive understanding of the nonHneer absorption characteristics of CdSe- based nanoplatelets （NPLs） synthesized by the solution-phase method and the colloidal atomic layer deposition approach through Z- scan techniques at 532nm with picosecond pulses. The CdSe NPLs exhibit strong two-photon induced free carrier absorption （effective three-photon absorption） upon the nonresonant excitation, resulting in a remarkable optical limiting behavior with the limiting threshold of approximately 75 GW/cm2. A nonlinear optical switching from saturable absorption （SA） to reverse saturable absorption （RSA） with increasing the laser intensity is observed when coating CdSe NPLs with a monolayer of CdS shell to realize the resonant absorption. The SA behavior originates from the ground state bleaching and the RSA behavior is attributed to the free carrier absorption. These findings explicitly demonstrate the potential applications of CdSe-based NPLs in nonlinear optoelectronics such as optical limiting devices, optical pulse compressors and optical switching devices.

Electrically Driven Nonresonant Single Molecular Switches

Electrical switching of a single-molecule junction provides a practical module to perform sophisticated operations in electronic devices.However,designing an all-electrically-driven molecular switch is a great challenge.Here,we experimentally and theoretically investigated the charge transport characteristics of isoindigo(ISO)-molecules at the single-molecule level using the scanning tunneling microscope break junction technique.We find that the single-molecule junctions of ISO-molecules display bias voltage-driven switching characteristics.These switches are realtime,reversible,and nondestructive under low-bias voltages.Experimental results show that the mechanism of the switch is not the transition from nonresonant charge transport to resonant charge transport,but it is the shift of the frontier orbital energy levels of ISO-molecules and the change of the interfacial electronic coupling with bias voltage.Our results will advance the design of high-performance bias voltage-driven molecular switches.

MULTIPLE POSITIVE SOLUTIONS OF NONRESONANT SINGULAR BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

This paper studies the existence of multiple positive solutions of nonresonant singular boundary value problem of second order ordinary differential equations. A sufficient condition for the existence of C[0,1] multiple positive solutions as well as C1[0, 1] multiple positive solutions is given by means of the fixed point theorems on cones.

SECOND BOUNDARY VALUE PROBLEM FOR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

In this paper we prove a uniqueness theorem of solutions for second boundary value problem of a class of second order nonlinear differential equation. In the proof the spectral theory of second order differential operators and Schauder's fixed point theorem are used.

NONLINEAR THEORY OF 3-dB HYBRIDCOUPLED POWER COMBINER

A complete nonlinear model of the 3-dB hybrid-coupled power combiner is proposed forthe first time and a set of state equations describing this nonlinear system are derived.Nonlinear analysis on the hybrid-coupled power combiner is performed by using the Routh-Hurwitz stability criterion. All the operation parameters, such as the output power, thecombining efficiency, the operating frequency and the bandwidth, are calculated correctly.Therefore, it is possible to fulfil the computer-aided analysis (CAA) and the computer-aideddesign (CAD) of complex corporation and tandem power-combining systems.

Generation of broadly tunable coherent VUV/XUV radiations in Xe and Kr

Broadly tunable coherent VUV/XUV radiations from 92 to 145 nm were generated by nonresonant four-wave sum-mixing (FWSM) and third-harmonic generation (THG) in pulsed jet of Xe or Kr. The dependence of phase-matching on gas density, atomic dispersion was observed. THG enhanced by third-photon resonant with the high Rydberg states in Xe was observed, and the resolved principal quantum number was up to 34.