Optical solitons with power-law asymptotics
Robert W. Micallef, Vsevolod V. Afanasjev, Yuri S. Kivshar, and John D. Love
Optical Sciences Centre, The Australian National University Canberra, Australian Capital Territory 0200, Australia
Received 6 July 1995
It is shown that self-guided optical beams with power-law asymptotics, i.e., algebraic optical solitons, can
be regarded as a special case of sech-type solitons i.e. solitons with exponentially decaying asymptotics in the
limit where the beam propagation constant coincides with the threshold for linear wave propagation. This leads
to the conjecture that algebraic optical solitons should be inherently unstable due to interactions with linear
waves, even in cases when the corresponding family of sech-type solitons is stable. This conjecture is verified
numerically for a wide class of optical solitons described by the generalized nonlinear Schro ¨dinger equation
with two competing nonlinearities. S1063-651X9611908-5
PACS numbers: 42.65.Tg, 42.60.Jf, 42.65.Jx
I. INTRODUCTION
There is growing interest in the subject of self-guided
nonlinear waves spatial solitons in uniform nonlinear me-
dia 1–9. Since the appearance of the classical paper by
Chiao et al. 1, attention has concentrated mainly on the
self-focusing or self-defocusing Kerr medium. However,
practical materials often display physical effects, such as
saturation, which can only be described by more generalized
models of the nonlinear refractive index. For such non-Kerr
materials theoretical predictions of new nonlinear effects, in-
cluding multistability 2 and nonlinear switching and steer-
ing 8, are very important. In particular, it has been shown
recently 9 that, for certain forms of the generalized nonlin-
ear Schro
¨
dinger NLS equation with two nonlinear terms of
opposite sign, e.g., cubic-quintic nonlinearity, there exist
weakly-localized solitary waves, i.e., solitary waves with
power-law asymptotics, the so-called algebraic solitons.
We note that the existence of different types of solitary
waves with power-law asymptotics has been already estab-
lished from other models of nonlinear physics, e.g., 10–18.
In their application to nonlinear optics, such weakly-
localized solitary waves are often treated as a class of sepa-
rate solutions contrasting with the sech-type solitons, i.e.,
solitons with exponentially decaying tails see, e.g., 9 and
discussions therein.
The main objective of our paper is threefold. First, we
show that the existence of algebraic optical solitons is a ge-
neric property of the generalized NLS models with two non-
linear terms of opposite sign but arbitrary power and we find
these solutions in an explicit form. Second, we demonstrate
that the algebraic solitons appear as the special limit of more
general, sech-type solitons when the propagation constant
coincides exactly with the threshold for periodic wave propa-
gation. Third, we point out that the origin of this kind of
algebraic solitons automatically implies that they should be
inherently unstable, but, as we show here, the character of
this instability depends on the power of the nonlinearity. As
we demonstrate numerically, if the amplitude of the alge-
braic soliton is decreased initially by a small amount, then
the perturbation grows, and finally the algebraic soliton de-
cays into diffractive linear waves. Otherwise, if the ampli-
tude of the algebraic soliton is initially increased, the alge-
braic soliton evolves into a sech-type soliton. The instability
becomes exponentially growing and manifests itself even
more strongly when the algebraic solitons belong to an un-
stable branch of the sech-type solitons.
The paper is organized as follows. Section II presents our
model, which leads to the generalized NLS equation with
two power-law nonlinearities. The different types of bright
soliton solutions to this equation, including the special case
of nonlinear periodic waves, are analyzed in Sec. III. Then,
in Sec. IV, we discuss algebraic solitons and their properties.
In particular, we demonstrate that algebraic solitons can be
regarded as a special limit of sech-type solitons, a property
which was not noted in Ref. 9. This limit corresponds ex-
actly to a threshold between solitary waves and continuous
waves. Such an observation allows us to understand the char-
acter of the instability of algebraic solitons and also to pre-
dict their behavior in collisions. Some general discussions
about the link between algebraic solitons and guided modes
of graded-index planar waveguides are presented in Sec. V,
and in Sec. VI we make some concluding remarks.
II. MODEL
We consider the propagation of a monochromatic scalar
electric field E in a medium with an intensity-dependent re-
fractive index, n =n
0
+n
nl
( | E |
), where n
0
is the uniform
refractive index of the unperturbed medium, and n
nl
( | E |
)
describes the variation in the index due to the field intensity
| E | , where is a positive constant. For small field intensi-
ties, we expand n
nl
as a power series in | E |
, and retain the
first two terms,
n
nl
a | E |
+b | E |
2
, 1
where a and b are constants. In the case of nonlinear satu-
ration, we must have ab 0. Then, within the weak-
guidance approximation, solutions of the governing Max-
well’s equation can be presented in the form
E X , Z ; t =E X , Z e
i
0
Z -i t
+c.c., 2
where c.c. denotes complex conjugate, is the source fre-
quency, and
0
=2 n
0
/ is the plane-wave propagation
constant for the uniform background medium, in terms of the
PHYSICAL REVIEW E SEPTEMBER 1996 VOLUME 54, NUMBER 3
54 1063-651X/96/543/29367/$10.00 2936 © 1996 The American Physical Society