lar to that in
the
4
model arises. When
increases, a fractal structure unlike
4
s gradually emerges. But when
is
greater than one, this fractal structure disappears. Analytically, we explain these collision behaviors by a
variational model that qualitatively reproduces the main features of these collisions. This variational model
helps to establish that these window sequences and fractal structures are caused entirely or partially by a
resonance mechanism between the translational motion and width oscillations of vector solitons. Next, we
investigate collision dependence on initial polarizations of vector solitons. We discovered a sequence of
reection windows that is phase induced rather than resonance induced. Analytically, we have derived a simple
formula for the locations of these phase-induced windows, and this formula agrees well with the numerical
data. Last, we discuss collision dependence on relative amplitudes of initial vector solitons. We show that when
vector solitons have different amplitudes, the collision structure simplies. Feasibility of experimental obser-
vation of these results is also discussed at the end of the paper.
DOI: 10.1103/PhysRevE.64.056616
PACS number s : 42.65.Tg, 05.45.Yv, 42.81.Dp
I. INTRODUCTION
Collision of solitary waves is an important problem for
both physical and mathematical reasons. Physically, solitary-
wave collisions are common phenomena in science and en-
gineering. For instance, a popular technology in ber com-
munication systems is wavelength-division multiplexing
WDM . In a WDM system, optical pulses in different fre-
quency channels collide with each other all the time 13 .
In waveguides such as crystals, photorefractives, and air,
beam collision is being utilized to achieve instant beam
steering and control 410 . Water-wave collisions in the
ocean and on the beach are even more familiar 11 . Math-
ematically, solitary-wave collision is a major branch of non-
linear waves. In integrable systems, solitary waves collide
elastically. But if the system is nonintegrable, this collision
may be highly nontrivial. Much work has been done on
solitary-wave collisions in a large array of physical systems.
Various collision scenarios such as transmission, reection,
annihilation, trapping, creation of solitary waves and even
mutual spiraling have been reported 1220 . In particular,
for kink-antikink collisions in the
4
and related models, an
interesting sequence of resonant reection windows has been
discovered 2125 . Near the edge of each resonance win-
dow, other sequences of resonant windows with more
bouncing have also been revealed 22,23,26 . This phe-
nomenon is the so-called fractal structure in
4
-related sys-
tems. In two recent articles, we found a somewhat different
fractal structure in vector-soliton collisions in the noninte-
grable coupled nonlinear Schro�dinger
NLS
equations
27,28 . When we zoom in to various collision-velocity win-
dows, we obtain a copy, a horizontal reection, or a vertical
reection of the original graph with all the major geometric
features preserved. Unlike the
4
fractal, zooming opera-
tions for the vector-soliton fractal are performed not at edges
of a resonant window. More importantly, the basic structure
of this coupled NLS fractal is a sequence of multipass and
multibounce windows, while the basic structure of the
4
fractal is a collection of two-pass windows. Very recently,
breather interactions in a weakly discrete sine-Gordon equa-
tion have been studied, and a fractal structure has been iden-
tied there as well 20 .
The fractal dependence is probably the most vivid mani-
festation of both complexity and regularity in solitary-wave
collisions. Such dependence in the coupled NLS equations is
particularly signicant since those equations directly govern
pulse propagation in birefringent bers and WDM systems
1,2,29 . Those equations are also closely related to beam
propagation in crystals and photorefractive waveguides 10 .
In 27,28 , this fractal structure was shown only for one
cross-phase modulational XPM coefcient
2/3 that cor-
responds to linear ber birefringence. In addition, only col-
lisions of vector solitons with equal amplitudes and orthogo-
nal polarizations were studied. In engineering applications,
elliptical ber birefringence is not uncommon 30 . Further-
more, initial solitons are not always in orthogonal polariza-
tions or with equal amplitudes. Thus, how this fractal struc-
ture changes when the XPM coefcient, initial polarizations,
and relative amplitudes vary is obviously a pressing ques-
tion. In addition, in 27,28 , the mechanism for this fractal
dependence was argued as a resonance between the transla-
tional motion and internal oscillations of vector solitons on
numerical and heuristic grounds. A more quantitative ana-
lytical theory to explain this fractal structure is clearly called
upon. All these important questions will be addressed in the
present paper.
The results of this paper may be summarized as follows.
We rst investigate the collision dependence of orthogonally
polarized and equal-amplitude vector solitons on the XPM
coefcient
(
0). We nd that when
is very small, vec-
tor solitons pass through each other at collision velocities V
0
above a critical value V
c
, and trap each other at collision
PHYSICAL REVIEW E, VOLUME 64, 056616
1063-651X/2001/64 5 /056616 17 /$20.00
�2001 The American Physical Society
64
056616-1 velocities V
0
V
c
. When
increases, a sequence of reec-
tion windows similar to that in the
4
model gradually
emerges just below the critical velocity V
c
. In all these win-
dows, the two solitons pass each other only twice, but num-
bers of width oscillations of vector solitons or equivalently,
collision time between the two passes differ. When
in-
creases further, a fractal structure unlike
4
s emerges 27 .
As
approaches one, this fractal structure simplies. When
1 the Manakov model , the collision is elastic and com-
plexity disappears. When
1, the collision is not elastic,
but the collision structure remains simple. Theoretically, we
explain the above intricate collision structure and dynamics
by a simple variational model. With this model, we suc-
ceeded in qualitatively reproducing the key features of
vector-soliton collisions in the original partial differential
equations PDEs . This success helps to establish that the
mechanism for window sequences and fractal structures in
vector-soliton collisions may be attributed to a resonance
between the translational motion and width oscillations
breathing of colliding solitons. Next, we investigate colli-
sion dependence on initial polarizations of vector solitons. In
this case, we discovered a sequence of reection windows
that is quite different from that in the collision of orthogo-
nally polarized vector solitons. Collisions in these windows
are all simple and also similar to each other. We will show
that this window sequence is not induced by a resonance
mechanism. Rather, it is phase induced, i.e., it is caused by
the collisions dependence on vector solitons relative
phases. Analytically we derived a simple formula for the
window locations in this sequence, and it agrees well with
the numerical data. Last, we study collision dependence on
relative amplitudes of initial solitons. We nd that when vec-
tor solitons have different amplitudes, the collision structure
simplies.
Complexity of vector-soliton collisions is remarkable
enough. Equally remarkable is the clear pattern and regular-
ity in these collisions. In addition to the fractal structure that
exhibits clear patterns amidst complicated collisions, we also
nd that window locations in a sequence are given by simple
formulas as well. Specically, in the window sequence for
collisions of orthogonally polarized vector solitons at small
values, quantity (V
c
2
V
n
2
)
1/2
, where V
c
is the critical
velocity and V
n
is the center of the n</i>th window, is a linear
function of the window index n. In the window sequence for
collisions of nonorthogonally polarized vector solitons, V
n
1
is a linear function of n. We also nd that in the window
sequence for orthogonally polarized vector solitons, the col-
lision time T
n
depends linearly on n. These surprisingly
simple formulas testify to the high regularity of vector-
soliton collisions. It is noted that the linear dependence of
quantity (V
c
2
V
n
2
)
1/2
and collision time T
n
on window in-
dex n occurs in the
4
-related models as well 2125 .
Hence, this dependence appears to be universal in resonance-
induced window sequences.
The structure of this paper is organized as follows. In Sec.
II, we study the collision dependence of orthogonally polar-
ized and equal-amplitude vector solitons on XPM coefcient
(
0). In Sec. III, we present a variational model that
qualitatively explains the collision structure and dynamics of
Sec. II. In Secs. IV and V, we examine the collision depen-
dence on polarizations and relative amplitudes, respectively.
In Sec. VI, we summarize our main results, and argue that
the experimental observation of our results is feasible.
II. COLLISIONS OF ORTHOGONALLY POLARIZED AND
EQUAL-AMPLITUDE VECTOR SOLITONS
The coupled NLS equations under study are
iA
t
A
xx
A
2
B
2
A
0,
2.1
iB
t
B
xx
B
2
A
2
B
0,
2.2
where A and B are complex amplitudes of wave envelopes in
two orthogonal polarizations or two WDM channels 3,29 .
The parameter
is the XPM coefcient that is always posi-
tive in optics applications. The system 2.1 and 2.2 is
phase, position, and Galilean invariant. Solitary waves in this
system are of the form
A x,t
r
1
x
v<i>t
x
0
e
(1/2)i
v<i>x
i[
1
2
(
v
2
/4