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15
petition in Nonlinear
Schr¨odinger Models
. Gaididei
. Christiansen
. Mingaleev
ABSTRACT Three types of nonlinear Schr¨odinger models with multiple
length scales are considered. It is shown that the length-petition
universally gives rise to new localized stationary states. Multistability phe-
nomena with a controlled switching between stable states e possible.
1 Introduction
The basic dynamics of deep water and plasma waves,light pulses in nonlin-
ear optics and charge and energy transport in condensed matter and bio-
physics [3,29,32,12] is described by the fundamental nonlinear Schr¨ odinger
(NLS) equation
∂
i ψ+ L2∂2ψ+ V |ψ|2ψ+ f(x)ψ=0, ()
∂t x
where ψ(x, t) is plex amplitude of quasi-monochromatic wave trains
or the wave function of the carriers. The second term represents the dis-
persion and L is the dispersion length (. in the theory of charge (energy)
transfer L2 =¯h2/2m with m being an effective mass). The nonlinear term,
V |ψ|2ψ,describes a self-interaction of the quasiparticle caused either by
its interaction with low-frequency excitations (phonons,plasmons,etc.) [7]
or by the intensity dependent refractive index of the material (Kerr effect)
[23]. The function f(x) is a parametric perturbation which can be a local-
ized impurity potential,a disorder potential,a periodic refractive index,an
external electric field,etc. It is well known that as a result petition
between dispersion and nonlinearity nonlinear waves with properties of par-
ticles,solitons,arise. One may also say that petition leads to the√
appearance of the new length-scale: the width of the soliton, ζ= L/ V .
The presence of the parametric perturbation f(x) introduces additional
interplays between nonlinearity,dispersion and perturbations. In the re-
cent paper by Bishop et al. [1] the concept peting length-scales and
. Christiansen, . Sørensen, and . Scott (Eds.): LNP 542, pp. 307−321