Interaction of pulses in the nonlinear Schrodinger model

Abstract

The interaction of two rectangular pulses in the nonlinear Schrödinger model is studied by solving the appropriate Zakharov-Shabat system. It is shown that two real pulses may result in an appearance of moving solitons. Different limiting cases, such as a single pulse with a phase jump, a single chirped pulse, in-phase and out-of-phase pulses, and pulses with frequency separation, are analyzed. The thresholds of creation of new solitons and multisoliton states are found.

Full text

arXiv:nlin/0303061v1 [nlin.PS] 26 Mar 2003
Interaction of pulses in nonlinear Schr¨odinger model
E. N. Tsoya∗and F. Kh. Abdullaeva,b
aPhysical-Technical Institute of the Uzbek Academy of Sciences,
2-B, Mavlyanov street, Tashkent, 700084, Uzbekistan
bInstituto de Fisica Teorica, UNESP, Sao Paulo, Brasil
(June 20, 2018)
The interaction of two rectangular pulses in nonlinear Schr¨odinger model is studied by solving the
appropriate Zakharov-Shabat system. It is shown that two real pulses may result in appearance of
moving solitons. Different limiting cases, such as a single pulse with a phase jump, a single chirped
pulse, in-phase and out-of-phase pulses, and pulses with frequency separation, are analyzed. The
thresholds of creation of new solitons and multi-soliton states are found.
(Submitted to Phys. Rev. E, 2003)
42.65.Tg, 42.65.Jx
I. INTRODUCTION
The nonlinear Schr¨odinger (NLS) equation is an important model of the theory of modulational waves. It describes
the propagation of pulses in optical fibers1,2, the dynamics of laser beams in a Kerr media, or the nonlinear difraction3,
waves in plasma4, and the evolution of a Bose-Einstein condensate wave function5. The NLS equation is written in
dimensionless form as the following:
iuz+uxx/2 + |u|2u= 0,(1)
where u(x, z) is a slowly varying wave envelope, zis the evolutional variable, and xis associated with the spatial
variable.
An exact solution of the NLS equation has a form of a soliton:
u(x, z) = 2ηsech[2η(x+ 2ξz −x0)] exp[−2iξx −2i(ξ2−η2)z+iφ0].(2)
where 2ηand 2ξare amplitude, or the inverse width, and the velocity of the soliton, x0and φ0are the initial position
and phase, respectively. The soliton represents a basic mode and plays a fundamental role in nonlinear processes.
The dynamics of NLS solitons and single pulses even in the presence of various perturbations is well understood
(see e.g. Refs. 1,2and reference therein). However, the evolution of several pulses is not studied in details. In
recent works6(see also book2) mostly an interaction of solitons and near-soliton pulses was considered. A study
of near-soliton pulses, especially a use of the effective particle approach, often results in small variation of soliton
parameters, including the soliton velocities and as a consequence weak repulsion or attraction of solitons. Such a
study do not involve a possibility of appearance of additional solitons. However, for many applications it is necessary
to consider the interaction of pulses with arbitrary amplitudes or pulses with different parameters. For example,
in optical communication systems with the wavelength division multiplexing (WDM), the initial signal consist of
several solitons with different frequencies. An estimation of the critical separation between pulses is important for
determination of the repetition rate of a particular transmission scheme.
In present work the interaction of two pulses in the NLS model is studied both theoretically and numerically. We
show the presence of different scenarios of the behaviour, depending on the initial parameters of the pulses, such as
the pulse areas, the relative phase shift, the spatial and frequency separations. One of our main observation is a fact
that a pure real initial condition of the NLS equation can result in additional moving solitons. As a consequence the
number of solitons, emerging from two pulses separated by some distance, can be larger than the sum of the numbers
of solitons, emerging from each pulse. Such properties were also found for the Manakov system7, which is a vector
generalization of the NLS equation. The scalar NLS equation was studied in7as a particular case. Later similar
results and approximation formulas for the soliton parameters were obtained in papers8,9(see also10 ). In works7–10
∗Corresponding author: etsoy@physic.uzsci.net
1

mostly the interaction of real pulses was analyzed, while here we consider pulses with non-zero relative phase shift
and frequency separation. A preliminary version of this study was presented in work11.
The paper is organized as the following. The linear scattering problem associated with the NLS equation is
considered in Section II. We also present the general solution of the problem for the case of two rectangular pulses.
In Section III we study different particular cases, such as two in-phase pulses, two out-of-phase pulses, a single pulse
with a phase jump, a single chirped pulse, and two pulses with the frequency separation. The results and conclusions
are summarized in Section IV.
II. DIRECT SCATTERING PROBLEM
In this paper we are interested only in an asymptotic state of the pulse interaction. In order to simplify the problem
and to obtain exact results we consider the interaction of two rectangular pulses (“boxes”). Therefore we take the
following initial conditions for Eq. (1):
u(x, 0) ≡U(x) = 


Q1exp[2iν1x],for x1< x < x2,
Q2exp[2iν2x],for x3< x < x4,
0,otherwise ,
(3)
where Q1and Q2are the complex constant amplitudes, w1≡x2−x1and w2≡x4−x3are the pulse widths, and 2ν1
and 2ν2are the detunings.
It is known that NLS equation is integrable by the inverse scattering transform method3. As follows from this fact,
initial conditions, which decreases sufficiently fast at x=±∞, results in a set of solitons and linear waves (so called,
radiation). The number Nand parameters of solitons emerging from an initial condition are found from the solution
of the Zakharov-Shabat scattering problem3:
i∂ψ1
∂x −iU (x)ψ2=λψ1,
−i∂ψ2
∂x −iU ∗(x)ψ1=λψ2(4)
with the following boundary conditions
Ψx→−∞ =1
0e−iλx,Ψx→∞ =a(λ)e−iλx
b(λ)eiλx .(5)
Here Ψ(x) is an eigenvector, λis an eigenvalue, a(λ) and b(λ) are the scattering coefficients, and a star means a
complex conjugate. The number Nis equal to the number of poles λn≡ξn+iηn, where n= 1,...,N, and ηn>0, of
the transmission coefficient 1/a(λ). Each λnis invariant on z. If all ξnare different then u(x, z) at z→ ∞ represents
a set of solitons, each in the form of Eq. (2) with η=ηnand ξ=ξn. If real parts of several λnare equal then a
formation of a neutrally stable bound state of solitons is possible.
The solution of the Zakharov-Shabat problem (4) with the potential (3) is written as
a(λ) = ei(λ+ν1)w1ei(λ+ν2)w2
cos(k1w1)−i(λ+ν1)
k1
sin(k1w1)×
cos(k2w2)−i(λ+ν2)
k2
sin(k2w2)−
Q∗
1Q2
k1k2
e−2i(λ+ν1)x2e2i(λ+ν2)x3sin(k1w1) sin(k2w2),(6)
b(λ) = ei(λ+ν1)w1e−i(λ+ν2)w2
−Q∗
1
k1
e−2i(λ+ν1)x2sin(k1w1)
cos(k2w2) + i(λ+ν2)
k2
sin(k2w2)−
2

Q∗
2
k2
e−2i(λ+ν2)x3sin(k2w2)
cos(k1w1)−i(λ+ν1)
k1
sin(k1w1) (7)
where kj= [(λ+νj)2+|Qj|2]1/2.
Since the linear operator in (4) is not Hermitian, complex eigenvalues are possible even for real u(x, 0) (e. g.
see Section III A). Though this is an obvious fact, “an interesting “folklore” property seems to have arisen in the
literature over the last 25 years, namely, that only pure imaginary EVs (eigenvalues) can occur for symmetric real
valued potentials”8. As demonstrated below, the statement [Theorem (III) in §2] of paper12 , that claims this result, is
incorrect. An existence of eigenvalues with non-zero real parts for Zakharov-Shabat problem with pure real potential
was first shown in paper7.
Equations (6,7) represent a general solution of the scattering problem (4,5) with initial condition (3). Applications
of these equations to particular cases of the pulse interaction are considered in the next section.
III. RESULTS
A. Interaction of in-phase pulses with equal amplitudes
1. Properties of eigenvalues
Here we analyze a simple case of two real pulses, separated by a distance L≡x3−x2, with zero detuning, i.e.
Q1=Q2=Q0,w1=w2≡w, and ν1=ν2= 0, where Q0is real. Then using Eq. (6), the equation for discrete
spectrum is written as
F(λ, Q0, w)±Q0
keiλL sin(kw) = 0 ,(8)
where F(λ, Q, w)≡cos(k w)−iλ sin(k w)/k, and k= (λ2+Q2)1/2. Note that F(λ, Q0, w) = 0 determines the discrete
spectrum for a single box with zero detuning13 . Therefore the second term in Eq. (8) can be associated with the
result of nonlinear interference. Recall also that for a single box with amplitude Q0and width w, the number NSB
of emerging solitons is determined as3NS B = int(Q0w/π + 1/2), where int() means an integer part. Results for the
two boxes are reduced to those for a single box in limiting cases L= 0 and L=∞.
As shown by Klaus and Shaw8, the Zakharov-Shabat problem with a “single-hump” real initial condition admits
pure imaginary eigenvalues only, i.e. solitons with zero velocity. We show that the case of two pulses provides much
richer dynamics.
Let us now compare the properties of eigenvalues at different S≡Q0w(Fig. 1). In Fig. 1, as well as in subsequent
figures of the paper, all variables are dimensionless. In the first two cases, S= 1.8 and S= 2.0, there is one soliton
at L= 0 and there are two solitons at L=∞, while in the case S= 2.5 there are two solitons in both limits. The
dependence of eigenvalues on Lat S= 2.5 is obvious, while that at S= 1.8 and 2.0 looks unexpected. Firstly, the
number of solitons at intermediate Lis larger then that in the limits L= 0 and L=∞. Secondly, the two real boxes
lead to eigenvalues with non-zero real part. Third, for S= 2.0 there is a “fork” bifurcation at L=LF≈4.1, when
two eigenvalues coincide. At larger Lthree pure imaginary eigenvalues constitute a three-soliton state, so that the
limiting two-soliton case at L→ ∞ is realized as a limit of a three-soliton solution with an amplitude of the third
soliton tending to zero.
Results of numerical simulations of the NLS equation (1) agree with analysis of Eq. (8). For example, as shown
in Fig. 2, in accordance with Fig. 1b there are one fixed and two moving solitons at S= 2.0 and L= 2, and there
are a three-soliton state and two moving solitons at S= 2.0 and L= 5. Note that an appearance of moving solitons
and multi-soliton states is not related to the rectangular form of initial pulses. For example, an initial condition
u(x, 0) = 0.7[sech(x+ 2.5) + sech(x−2.5)] also results in moving solitons.
Below we discuss in details the behaviour of the eigenvalues, namely we find a threshold of appearance of new roots,
estimate a number of emerging solitons, and calculate a threshold for the “fork” bifurcation. It should be mentioned
that eigenvalues with non-zero real part do not exist only at S= [3π/4,3.3] and S= [7π/4,5.51] (see Section III A 2),
so that the dependence at S= 2.5 is rather an exception than a general rule. This results allows to understand why
moving solitons are not observed in interaction of near-soliton pulses with area S≈π.
3

2. Appearance of new eigenvalues
Solving numerically Eq. (8), one can conclude that new eigenvalues penetrate to the upper half-plane of λin pairs
by crossing the real axis. Therefore, the bifurcation parameter can be found from Eq. (8), assuming that λ=β,
where βis real:
cotan y=±p2S2−y2
y,(9)
β=±Q0sin(βL).(10)
Here y=κw,κ= (β2+Q2
0)1/2, and the signs are taken such that tan(y) tan(βL)<0 is satisfied. As follows from
the definition of yand Eq. (9), one has S≤y < 2S.
Analysis of Eqs. (9,10) results in the following conclusions:
(i) As follows from Eq. (9), the number NP P of the penetration points depends only on Sand is determined from:
NP P = 4(m−n+ 1) −2θS−n+1
4π−
2θS−n+3
4π−4θ(Sm−S) for S≥3π/4,(11)
where m= int(√2S/π), n= int(S/π), θ(x) is the Heavyside function, and Smis a root of
tan(p2S2
m−1) = p2S2
m−1,(12)
which satisfies mπ ≤(2S2
m−1)1/2<(m+ 1)π. It is easy to find that NP P = 0 for S < π/4 and NP P = 2 for
π/4< S < 3π/4. Equation (12) defines such values of S=Sm, when the right hand side of Eq. (9) with plus sign
touches cotanycurve. All penetration points βj, where j= 1,...,NPP , are symmetrically situated with respect to
β= 0.
(ii) All roots |βj| ≤ Q0, which follows from 2S2−y2≥0.
(iii) For every βj, Eq. (10) defines the separation distance L=LC, when eigenvalues cross the real axis.
(iv) As follows from Eq. (10) there is an infinite number of thresholds LCfor a given βj. However, the total number
of eigenvalues in the upper half-plane of λis, most probably, finite, because for some LCeigenvalues pass to the upper
half-plane, and for other LCeigenvalues go to the lower half-plane. The direction of eigenvalue motion is defined by
the derivative dλ/dL at λ=βj.
The positions of penetration points, βjas a function of Sis shown in Fig. 3a, where only positive βjare presented.
As follows from Eq. (9) the number NP P decreases by two, when Spasses (2l+ 1)π/4, where l= 1,2..., and NP P
increases by four, when Sexceeds Sm[see Eq. (12)]. Therefore one can obtain that Eq. (9) has no roots only at
S= [3π/4, S2] and at S= [7π/4, S3], where S2≈3.26 and S3≈5.51 are found from Eq. (12). This property is clearly
seen in Fig. 3. The dependence of LCon Sis presented in Fig. 3b. Only the thresholds, such that βjLC= [0,2π],
are shown for each βj.
3. Thresholds of “fork” bifurcation
Here we analyze a bifurcation, when a pair of complex eigenvalues becomes pure imaginary, e.g. LF≈4.1 in Fig. 1b.
The equation which determine pure imaginary eigenvalues can be obtained from Eq. (6) with Re[λ] = 0, i.e. λ=iγ:
cotan y=−pS2−y2±Sexp[−pS2−y2L/w]
y.(13)
Here y=κ w ,κ= (−γ2+Q2
0)1/2. It is easy to show that κ2should be positive (there is no real solution for κ2<0).
As a consequence, all pure imaginary eigenvalues satisfy γj≤Q0.
The value of L=LF, when new pure imaginary root of Eq. (13) appears, corresponds to the “fork” bifurcation.
The bifurcation threshold LFcan be found from the condition that the functions, corresponding to the right-hand
side of Eq. (13), touch the cotan ycurve.
Low bound of the number of the pure imaginary eigenvalues is found as NLB = int(2S/π + 1/2). This number is
an actual number of the pure imaginary eigenvalues for all S, except of the regions where
4

π
2+πl < S < 3π
4+πl, l = 0,1,.... (14)
If Ssatisfies Eq. (14) then the number of pure imaginary eigenvalues can be either NLB or NLB + 2, depending on
whether L < LF(S) or L > LF(S), respectively. Therefore the appearance of new pure imaginary eigenvalues, in
other words, the fork bifurcation, is possible only if Ssatisfies Eq. (14). Figure 4 represents the dependence of LFon
S, where only one interval, corresponding to l= 1 in Eq. (14), is shown; the behaviour for l > 1 is similar. One can
calculate that LF(S= 1.8) = 10.4, that is why the fork bifurcation is not seen in Fig. 1a.
B. Two out-of-phase pulses with equal amplitudes
In this section we study the influence of constant phase shift on the pulse interaction, i.e. we consider Q1=
Q0exp(−i α), Q2=Q0exp(i α), where Q0and αare real, w1=w2≡w, and ν1=ν2= 0. The non-zero relative
phase shift, 2α, changes greatly the properties of the eigenvalues, so that the behaviour presented in Section III A is
hard to realize in experiments, because it is difficult to prepare two pulses exactly in phase. The phase shift breaks
the simultaneous appearance of a pair of solitons at λ=±βj, and affects to the “fork” bifurcation.
For α6= 0, the equations for eigenvalues and for penetration points can be obtained from Eq. (8) and Eqs. (9,10)
by changing λL →λL +αand βL →βL +αin the exponent and sinus functions, respectively. Therefore the number
and positions of the penetration points are the same as for the case α= 0. As for the threshold LC, it is shifted on
the value α/βj, so that LC(α) = LC(α= 0) + α/βj, where only LC≥0 should be taken into account.
The influence of the phase shift on the distribution of eigenvalues is shown in Fig. 5. As seen, now new eigenvalues
appear one by one, not in pairs, and, as a concequense, the “fork” bifurcation disappears. Further, the real parts of
the roots do not vanish at finite L, but decreases smoothly. This means that the presence of the phase shift breaks
up a multi-soliton state, which is known to be neutrally stable to perturbations.
At L= 0, the phase shift corresponds to the phase jump of a single pulse. Such a phase jump can result in an
appearance of additional solitons as shown in Fig. 5b. The threshold of the phase shift αth, when the first new soliton
appear, can be found from the condition αth =|β1|LC(α= 0), where β1is the position of the penetration point
nearest to zero.
C. Two pulses with frequency separation
In this section we analyze the initial condition (3) with the following parameters Q1=Q2=Q0,w1=w2=w,
−ν1=ν2=ν, where Q0is a real constant. This case models the wavelength division multiplexing in optical fibers,
the case when an input signal consists of two or more pulses with different frequencies. Actually, since Q0can be taken
sufficiently large we consider the interaction of multi-soliton states. The detailed analysis of the interaction of sech-
pulses at different frequencies is presented in papers14 and in review15 . In particular, the authors of papers14,15 consider
the evolution of a superposition of Nsolitons with the same position of the centers, but with different frequencies. As
shown in these works there is a critical frequency separation, above which Nsolitons with almost equal amplitudes
emerge. Below this critical value the number of emerging solitons can be not equal Nand their amplitudes can
appreciable differ from each other. It was also demonstrated that an introduction of a time shift between pulses
results in decrease of the frequency separation threshold. The geometry described by Eq. (3) corresponds to the
combination of WDM and time-division multiplexing schemes, therefore our study can give some insight to such a
behaviour of the threshold. Moreover, the authors of works14,15 mostly used the perturbation technique and numerical
simulations, while in the present paper we deal with an exact solution of the Zakharov-Shabat problem.
First let us consider the case L= 0 that correspond to the case of a single chirped pulse of width 2w. The
dependence of the eigenvalues on ν, which plays here the role of a chirp parameter, is shown in Fig. 6. At small νthe
interaction of the pulse components is strong, so that there is one pure imaginary eigenvalue, or a single soliton with
zero velocity. At larger νthe frequency difference of the pulse components results in a repulsion of the components,
or a pulse splitting. At sufficiently large νthe velocities of emerging solitons tends, as expected, to ±2ν.
There is also a narrow region of ν, e.g. ν= [0.98,0.99] on Fig. 6, where three solitons, one fixed and two moving
solitons, exist. This region separate two different types of the evolution of a chirped pulse. The left boundary, which
corresponds to the appearance of new eigenvalues, of the region is found from the condition similar to that considered
in Section III A 2. The right boundary is found from a(λ= 0) = 0, which defines the values of νas a function of the
other parameters, when the pure imaginary root disappears.
The dependence of the spectrum on Lis presented in Fig. 7. At small ν(Fig. 7a) we see again an appearance of
additional solitons similar to the case ν= 0 (Fig. 1). At larger ν(Fig. 7b) the repulsion is so strong that it suppresses
5

the appearance of small-amplitude solitons. Therefore there is a threshold of the frequency separation above which
the interaction of two pulses is negligible. This result is in agreement with the conclusions of the paper15.
IV. CONCLUSION
The interaction of two pulses in the NLS model is studied by means of the solution of the associated scattering
problem. The strong dependence of the dynamics on the parameters of the initial pulses is shown. For intermediate
separation distances Lthe existence of additional moving solitons is possible even in the case of two in-phase pulses with
the same frequencies. These additional solitons can be considered as a result of nonlinear interference of pulses. The
phase shift of two pulses removes a degeneracy in the behaviour, namely it affects to the symmetry of the parameters
of emerging solitons and results in a break-up of multi-soliton states peculiar to the in-phase case. It is also shown
that the strong frequency separation suppress the appearance of additional solitons. The results obtained in the
present paper can be useful for analysis of the transmission capacity of communication systems and for interpretation
of experiments on the interaction of two laser beams in nonlinear media. Recently, the generation of up to ten solitons
has been observed experimentally in quasi-1D Bose-Einstein condensate of 7Li with attractive interaction16. Our
study can be also helpful for interpretation of this experiment.
ACKNOWLEDGMENTS
This researh was partially supported by the Foundation for Support of Fundamental Studies, Uzbekistan (grant N
15-02) and by FAPESP (Brasil).
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Khabibullaev, Optical Solitons (Springer-Verlag, Heidelberg, 1993).
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10 After the submission of this paper we became aware with the work by M. Desaix, D. Anderson, L. Helczynski, and M.
Lisak, Phys. Rev. Lett., 90, 013901 (2003), which deals with real initial conditions. However, here we consider more general
complex initial conditions (see Eq. (3)) and study different scenarios of the pulse interaction depending on space and frequency
separations, phase shift and pulse areas.
11 E. N. Tsoy and F. Kh. Abdullaev, In Nonlinear Guided Waves and Their Applications, technical dijest of the OSA Interna-
tional Workshop (Streza, Italy, 2002), vol.80, NLTuD14.
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6

-1
-0.5
0
0.5
1
1.5
012345678
Eigenvalues
L
1
2,3 4,5
6,7
2 4 6
3 5 7
(a)
-1
-0.5
0
0.5
1
1.5
012345678
Eigenvalues
L
1
2,3
4,5 6,7
2
4 6
3 5 7
(b)
2
3
0
0.5
1
1.5
2
012345678
Eigenvalues
L
1
2
(c)
FIG. 1. In-phase pulses: The dependence of real (dashed lines) and imaginary (solid lines) parts of λnon the separation
distance, w= 1. Numbers near lines corresponds to n. (a) Q0= 1.8, (b) Q0= 2.0, (c) Q0= 2.5.
Intensity (a)
zx
0-20 -10 010 20
5
10
15
0
2
4Intensity (b)
zx
0-20-10 010 20
5
10
15
0
2
4
FIG. 2. Evolution of two rectangular pulses, Q0= 2, w = 1. (a) One fixed soliton and two moving solitons at L= 2. (b)
Three-soliton state at L= 5.
7

0
2
4
6
8
10
12
14
0 2 4 6 8 10 12 14
β
S
j
(a)
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10 12 14
L
S
c
(b)
FIG. 3. (a) The dependence of βjon S. (b) Threshold LC, when eigenvalues cross the real axis of λ-plane, as a function of
S.
0
5
10
15
20
1.6 1.8 2 2.2 2.4
L
S
F
FIG. 4. Threshold LFof the fork bifurcation as a function of S.
-1
-0.5
0
0.5
1
1.5
012345678
Eigenvalues
L
1
23
45 61
2
3
4
5
6
(a)
-1
-0.5
0
0.5
1
1.5
012345678
Eigenvalues
L
1
234
5 6
1
2
3
4
5
6
(b)
FIG. 5. Out-of-phase pulses: The dependence of real (dashed lines) and imaginary (solid lines) parts of λnon Lfor
Q0= 2, w = 1. Numbers near lines corresponds to n. (a) α=π/8, (b) α=π/4.
8

-4
-2
0
2
4
0 1 2 3 4 5
Eigenvalues
ν
1
2
3
2,3
FIG. 6. Single chirped pulse; The dependence of real (dashed lines) and imaginary (solid lines) of λnon νfor
Q0= 2, w = 1, L = 0. Numbers near lines corresponds to n.
-1
-0.5
0
0.5
1
1.5
012345678
Eigenvalues
L
1
2
3
2,3
4,5
4
5
(a)
-1.5
-1
-0.5
0
0.5
1
1.5
012345678
Eigenvalues
L
1,2
1
2
(b)
FIG. 7. Pulses with frequency separation: The dependence of real (dashed lines) and imaginary (solid lines) of λnon Lfor
Q0= 2, w = 1. Numbers near lines corresponds to n. (a) ν= 0.5; (b) ν= 1.25.
9