1. INTRODUCTION
The field of nonlinear optics has developed in recent years as
nonlinear materials have become available, and widespread applications
have become apparent. This is particularly true for optical solitons and
other types of nonlinear pulse transmission in optical fibers and laser
plasma interaction (Osman et al., 2000,
2004a, 2004b; Hora, 2004).
This form of light propagation can be utilized in the future for very high
capacity dispersion-free communications. The purpose of this paper is to
describe the use of a very powerful tool to solve the generalized
nonlinear Schrödinger equation (NLSE) that has stable solutions
called optical solitons (Drazin & Johnson,
1990). The solitary wave (or soliton) is a wave that consists of a
single symmetrical hump that propagates at uniform velocity without
changing its form. The physical origin of solitons is the Kerr effect,
which relies on a nonlinear dielectric constant that can balance the group
dispersion in the optical propagation medium. The resulting effect of this
balance is the propagation of solitons, which has the form of a hyperbolic
secant (Zakharov & Shabat, 1972).
2. NONLINEAR SCHRÖDINGER EQUATION
The nonlinear Schrödinger equation (NLSE) used in this paper is
generalized as:
where n is the order of dissipation of the Schrödinger
equation being used, and δn is an arbitrary
constant.
Akhmediev and Ankiewicz (1997) covered the
third and fourth orders adequately and Karpman (1998) and Karpman and Shagalow (1999) dealt with the solution of the fifth order. The
purpose of this paper is to build upon these excellent results, explaining
the three-dimensional (3D) form of these solitons and extending this to
cover orders six and seven and beyond. We will be using a simplified
version of the Zakharov and Shabat (1972)
solution to this equation as our basis for solving it numerically using
Mathematica©.
The above research has indicated the presence of radiations emanating
from soliton waves. This phenomenon is the subject of this research,
particularly into determining which solitons, or solitons from which
parenting NLSE, are emanating radiations, and ultimately how this
radiation may be controlled or eliminated completely.
Practical research into soliton emissions has indicated that these
emissions are largely restricted to solitons derived from the cubic NLSE,
namely a higher order equation with an odd numbered power.
3. INITIAL CONDITIONS AND PROGRAMMING FOR
HIGHER ORDERS
In this problem, an analogous role is played by the particular
solutions of the nonlinear Schrödinger equation [8]:
Where η,ξ,φ,t,z0 are
scaling parameters traditionally used in nonlinear Schrödinger
equation [8]. This form of the solution can also be known as a
soliton that has a stable formation. For the purpose of this paper, we
will use the simplified solution:
z is used here as the soliton solution in a complex variable.
In the Mathematica representation, the soliton is represented as a complex
function in ξ,τ space. Customary care should be taken to view
these symbols within their relative context.
Using this programming method, with its wide range of available
iteration models available from Mathematica; we get a 3D representation of
the soliton. For each graphic representation of a soliton, we plotted its
contour graph. This is most helpful because the direction of movement of
soliton is much clearer. It is common for the viewer to assume that the
wave moves in the direction normal to its axis, in this case the ξ
axis. This is not so for a soliton which moves in the direction described
by the wave ridge. The contour graph also gives a clearer picture of the
radiations, which are crucial to this exercise. The third plot drawn here
shows the two-dimensional (2D) cross-section of the wave at the end of the
time axis, in these cases mostly τ = 2. In these graphs, the wave at
the end of it's time axis is shown in green. The red line represents
the wave at τ = 0. The fourth plot given here represents the maximum
possible error at time τ. The value of this information is realized
when we note that this gives an accurate indication of where the
radiations/wave breaking phenomenon begins. Figure
1a shows the basic soliton wave where the coefficient of dispersion
is zero; there being no dispersion term.
(A) The 3D plot of the basic soliton wave. (B) The
contour plot of the basic soliton wave.
(C) The cross-section plot for Figure 1a. The red line is the
starting point, time 0, the green line is the cut-off point at time 2.
(D) The error graph for the basic soliton wave.
The excellent graphics obtained from the use of this software can be
used to further enhance the understanding of the motion, as well as the
physical form of the soliton wave. The untrained eye does not immediately
comprehend the way in which the soliton is moving here. A lifetime of
watching the movement of water waves has trained the senses to
automatically assume that the wave is moving along the spatial, or ξ
axis.
It is also clear from this graph that there exists a small amount of
radiation being emitted from the front, or bow of the basic soliton wave.
This should be carefully noted here as being distinct from other types of
radiation which will come into play later as we progress to higher order
terms of the NLSE. By way of an explanation of this phenomenon, let us say
that any object moving through its propagating medium will, in respect of
its motion, produce an effect on the stability of the medium. This type of
miniscule radiation is separate and distinct from that which will appear
later.
Let us now observe the contour plot as shown in Figure 1b. From the outward radiations and the
pointed leading edge of the wave, it is immediately clear that it is
moving along the temporal, or τ axis. Please take care to note that in
Figure 1a the graph is plotted from left to
right, as opposed to right to left in the other graphs. Moving on to Figure 1c, we show the 2D cross-section at the
outward extremity of the time axis. The green line is the value at τ =
2, the red line is the basic wave form. Finally, we show in Figure 1d, the maximum error plot for this (zero)
order.
A plotted range of 0 ≤ τ ≤ 2 and −10 ≤ ξ ≤
10 is sufficient to give a clear picture of what is happening, while
showing the point where the soliton starts to break up into increasingly
evident radiations. Following on from the basic wave graphs, and before we
progress to the results obtained, we thought it sufficiently important
that we include here a wave plot showing the effect of lengthening the
axes. To this end, we show Figure 2, where the
time axis is extended to τ = 5 and the spatial axis to −20 ≤
ξ ≤ 20. This is deemed sufficient to illustrate the expanding wave
form. We therefore have limited all our other plots to the range shown for
Figure 1.
This shows the same soliton as Figure 1,
with extended axes. Here the viewer can see that the wave break-up, and
the radiations resulting from it, increase as we progress along the time
axis. It is deemed unnecessary, for further plotting, to venture beyond
τ = 2 in order that the perspective of this study may be
maintained.
Figure 3 illustrate the effect on soliton of
the zero-order term, iγu. Here we showed the effect
on soliton in Figure 1 of making γ = 1. The
effect is one of damping. In this plot we can clearly see that the soliton
and its radiations were very effectively damped by adding a unitary
coefficient to the zero order term −iγu in Eq.
(1).
The soliton in Figure 1, with the zero
order term at coefficient 1. In this plot we can clearly see that the
soliton and it's radiations have been very effectively damped by
adding a unitary coefficient to the zero order term
−iγu in (1).
4. RESULTS
We have found by the above method that for the dispersion orders from
3 upward in odd numbers there is a point along the evolution of the
coefficient for each order of dispersion where radiation becomes visibly
evident.
The first wave to be shown next, in Figure
4b, is the 3D wave form for the third order soliton with dispersion
coefficient 0.5. To supplement this we follow up with the Contour plot,
Figure 4b, the cross-section plot, 4c and the
error plot 4d. We then finish off our representation of the third order by
showing the 3D soliton graph for coefficient 3.5 in Figure 5a, followed up by its contour plot in Figure 5b, its cross-section plot in Figure 5c and its error plot in Figure 5d.
(A) This is the wave form of the third order at dispersion
coefficient 0.5. (B) This shows the contour plot for the wave in
Figure 4a.
(C) The cross-section plot for Figure 4A. (D) This
is the error plot for the soliton in Figure 4A.
(A) This shows the soliton at third order coefficient 3.5.
(B) This is the contour plot for Figure 5A.
(C) Here we show the cross-section plot for Figure 5A.
(D) Here we represent the error plot for Figure 5A.
For the presentation of our results we found it unnecessary to use a
large number of graphs for each of the higher orders, it being sufficient
to show the lowest coefficient where radiation occurs, and the last
coefficient before computation overflows or the error goes beyond the
point of usefulness. Consequently we show here only two sets of graphics
for each of these orders.
Moving on to the fourth order, we now show the wave resulting from the
fourth order term at coefficient α = 0.5, in Figure 6a, followed by its contour plot in Figure 6b, its cross-section plot in Figure 6c and its error plot in Figure 6d. The last plot for the fourth order at
coefficient 3.5 is the 3D plot in Figure 7a,
followed by its contour plot in Figure 7b, its
cross-section plot in Figure 7c and its error
plot in Figure 7d.
(A) The fourth order at coefficient 0.5. Here we note the
characteristic absence of radiation for the square, or even-numbered
order, other than that mentioned above with respect to all waves.
(B) The contour plot for Figure 6A.
(C) The cross-sectional plot for Figure 6A. (D) The
error plot for Figure 6A.
(A) This shows the wave from coefficient of dispersion α
= 35. Note here the sharp forward peak of the wave, characteristic of
self-focusing, but still the absence of radiations. (B) Here we
show the contour plot for Figure 7A.
(C) This is the cross-section plot for Figure 7A.
(D) Here we show the error plot for Figure 7A.
Moving on to the fifth order we next show the result for lowest
practical coefficient of order 5, being α = 0.01.
Karpman (1998) and Karpman and Shagalow
(1999) has defined the third and fourth order
dispersion equation and this will be used throughout this work as the
basis for programming the Schrödinger equation. This level of
dispersion is covered by the work of Karpman (1998) and Karpman and Shagalow (1999). Next we go to Figure
8a. This represents the 3D soliton plot for the fifth order at
bottom coefficient 0.01. This is followed by its contour plot in Figure 8b, its cross-section plot in Figure 8c, and its error plot in Figure 8d. After this we go to Figure 9a, where we show the 3D soliton for the sixth
order at coefficient 10−7. This is followed by its
contour plot in Figure 9b, its cross-section
plot in Figure 9c, and its error plot in Figure 9d. Last we show Figure
10a, which represents the highest practical coefficient for the
sixth order. We follow this with its contour plot in Figure 10b, its cross-section plot in Figure 10c, and lastly its error plot in Figure 10d.
(A) Here we show the graph at the lower end of the fifth
order scale. It is easily evident that radiations have begun to show.
(B) This is the contour plot for Figure 8A.
(C) This is the cross-section plot for Figure 8A.
(D) The error plot for Figure 8A.
(A) This shows the sixth order soliton at coefficient of
dispersion α = 10−7. (B) This is the contour
plot for Figure 9A.
(C) This shows the cross section plot for Figure 9A.
(D) Here we show the error plot for Figure 9A.
(A) Here we show the soliton produced from a dispersion
coefficient of 0.00007. (B) The contour plot for Figure
10A.
(C) Shows the cross-section for Figure 10A. (D). The
error plot for Figure 10A.
5. OBSERVATIONS
From the graphic results presented above it has been noted that
radiations related to the higher order dispersion factor are only found in
cubic orders, that is, where the higher order term in the NLSE is
represented by an uneven number. Where the order of magnitude of the
dispersion term is quadratic, i.e., it is expressed as an even number,
there are no radiations other than the basic waves mentioned above. This
leads us to draw far-reaching conclusions.
6. CONCLUSION
The most important conclusion to be drawn in the face of this evidence
is that solitonic radiations only occur in the presence of a cubic
dispersion term of the NLSE. In the cases where the order of dispersion is
quadratic there are no higher order radiations from the soliton. The
conclusions to be drawn from this evidence is that, if the solitons can be
produced in practice by an input that has no cubic term in its spectrum,
then the resulting solitons will be virtually free from radiations. The
consequences of this are that if such a soliton were to be employed in the
field of fibre optical telecommunications, for instance, there would be
virtually none of the factors that cause interference, “crossed
lines,” etc.
