DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

ON NONLINEAR SCHRÖDINGER EQUATIONS INDUCED
FROM NEARLY BICHROMATIC WAVES
S. KANAGAWA, B. T. NOHARA, A. ARIMOTO, AND K. TCHIZAWA
We consider a bichromatic wave function u b (x,t) defined by the Fourier transformation
and show that it satisfies a kind of nonlinear Schrödinger equation under some conditions
for the spectrum function S(k) and the angular function ω(ξ,η).
Copyright © 2006 S. Kanagawa et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
We consider a wave function defined by the following Fourier transformation:
∫ ∞
u m (x,t) = S m (k)e i{kx−ω(k)t} dk, x ∈ R, t ≥ 0, (1.1)
−∞
where i = √ −1, k is a frequency number, S m (k) is a spectrum function, and ω(k) isan
angular frequency. From the definition, we can see that u m (x,t) is a mixture of some
waves with different frequencies on some bandwidth controlled by the spectrum function
S m (k). When S m (k) is a delta function δ k0 (·) concentrated on a frequency k 0 ,thewave
function u m (x,t) is called the (purely) monochromatic wave u 1 (x,t), that is,
∫ ∞
u 1 (x,t) = δ k0 (k)e i{kx−ω(k)t} dk
−∞
= cos { k 0 x − ω(k)t } + isin { k 0 x − ω(k)t } .
(1.2)
On the other hand, u m (x,t) is called a nearly monochromatic wave function if S m (k)
is a unimodal function with a small compact support. As to some application of nearly
monochromatic waves, see, for example [6]. In this paper, we focus on the envelope function
defined by
A m (x,t) = u m(x,t)
u 1 (x,t) , (1.3)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 477–485