Publication

Existence And Stability Of High Frequency Standing Waves For A Nonlinear Schrodinger Equation

François Samer Genoud
2009
Journal paper
Abstract

This article is concerned with the existence and orbital stability of standing waves for a nonlinear Schrodinger equation (NLS) with a nonautonomous nonlinearity. It continues and concludes the series of papers [6, 7, 8]. In [6], the authors make use of a continuation argument to establish the existence in R x H-1(R-N) of a smooth local branch of solutions to the stationary elliptic problem associated with (NLS) and hence the existence of standing wave solutions of (NLS) with small frequencies. Complementary conditions on the nonlinearity are found, under which either stability of the standing waves and bifurcation of the branch of solutions from the point (0, 0) is an element of R x H-1 (R-N) occur, or instability and asymptotic bifurcation occur. The main hypotheses in [6] concern the behaviour of the nonlinearity with respect to the space variable at infinity. The paper [7] extends the results of [6] to (NLS) with more general nonlinearities. In [8], the global continuation of the local branch obtained in [6] is proved under additional hypotheses on the nonlinearity. In particular, spherical symmetry with respect to the space variable is assumed. The aim of the present work is to prove the existence and discuss the orbital stability of standing waves with high frequencies, independently of the results obtained in [6] and [8]. The main hypotheses now concern the behaviour of the nonlinearity with respect to the space variable around the origin. The methods are the same in spirit as that of [6] and permit to discuss the asymptotic behaviour of the global branch of solutions obtained in [8].

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Related concepts (32)
Bifurcation diagram
In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory.
Standing wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect to time, and the oscillations at different points throughout the wave are in phase. The locations at which the absolute value of the amplitude is minimum are called nodes, and the locations where the absolute value of the amplitude is maximum are called antinodes.
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Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around 20kHz to around 300GHz. This is roughly between the upper limit of audio frequencies and the lower limit of infrared frequencies. These are the frequencies at which energy from an oscillating current can radiate off a conductor into space as radio waves, so they are used in radio technology, among other uses.
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