RATIONAL SOLUTION OF THE NONLOCAL NONLINEAR SCHRODINGEREQUATION AND ITS APPLICATION IN OPTICS

YONGSHUAI ZHANG1, DEQIN QIU2, YI CHENG1, JINGSONG HE3,∗

1School of Mathematical Sciences, USTC, Hefei, Anhui 230026, P. R. China2College of Mathematics and Statistics, Jishou University, Hunan 416000, P. R. China

3Department of Mathematics, Ningbo University, Zhejiang 315211, P. R. ChinaE-mail∗: hejingsong@nbu.edu.cn, jshe@ustc.edu.cn (corresponding author)

Received December 31, 2016

Abstract. We derive a kind of rational solution with two free phase parametersof the nonlocal nonlinear Schrodinger equation, which satisfies the parity–time (PT)symmetry condition. In addition, the analyticity of the rational solution about the phaseparameters and the classification of this solution according to its asymptotic amplitudeare studied in detail. Furthermore, the gain/loss profile related to an optical systemis remarkably similar to the profile of the imaginary part of a PT-symmetric potentialgenerated by the above-mentioned rational solution.

Key words: Parity-time symmetry, nonlocal nonlinear Schrodinger equation,gain and loss.

1. INTRODUCTION

In 1895, Korteweg and de Vries [1] derived an equation, which was now knownas the Korteweg-de Vries (KdV) equation

ut(x,t)−6u(x,t)ux(x,t) +uxxx(x,t) = 0. (1)

The KdV equation can be used to describe the one-dimensional, long-time asymp-totic behavior of small, but finite amplitude shallow water waves [2]. Besides, thisequation had one kind of permanent wave solution, which was named “solitons” byZabusky and Kruskal [3], in 1965, because they found by numerical studies that thecollision of two such solutions was elastic and that there existed phase shifts on evo-lutionary trajectories after collision. Shortly after this discovery, Gardner, Greene,Kruskal, and Miura [4] introduced a new method of mathematical physics, whichwas able to generate the analytic expression of soliton solutions for the KdV equa-tion. Zakharov and Shabat [5] extended this method to another famous nonlinearintegrable equation, namely, the nonlinear Schrodinger equation (NLS)

qt(x,t)− iqxx(x,t)−2i|q(x,t)|2q(x,t) = 0, (2)

which was related to the KdV equation [6, 7]. Several nonlinear partial differen-tial equations solved by this method were found in the published literatures [8–10].

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Article no. 108 Yongshuai Zhang et al. 2

Owing to the frequent use of direct and inverse scattering data, this highly powerfulmethod was referred to as the celebrated “inverse scattering transformation (IST)”.

Except the IST method, there are many other methods used to generate solu-tions for diverse nonlinear integrable equations, such as the Darboux transformation(DT) [11]-[18], Hirota direct method [19, 20] and so on [21, 22]. Based on theseapproaches, a series of interesting solutions for the nonlinear integrable equationsare derived, such as breather solution (BS) [23–25] and its limit case, namely, roguewave solution (RWS) [26]. The above two solutions are not only known as supple-ments on nonzero background to soliton solution (generated from vacuum), but alsoadmit new interesting properties. For instance, BS evolves in a periodic way, RWSis a quasi-rational solution, whose modulus is a rational expression, and higher orderRWSs have many interesting structures [15], [27]-[35], not just displaying the simplecollisions of lower order RWSs (for other relevant works on different types of RWSsin a series of physical settings, see Refs. [36]-[46]).

Recently, Ablowitz and Musslimani [47, 48] considered the solution of thenonlocal NLS equation

qt(x,t)− iqxx(x,t)±2iV (x,t)q(x,t) = 0, V (x,t) = q(x,t)q∗(−x,t), (3)

from zero background by using the IST method, which implied the parity–time (PT)symmetry, namely, x→−x, t→−t and the complex conjugation of the field enve-lope. Here q(x,t) and V (x,t) (called the PT-symmetric potential) imply the electric–field envelope of the optical beam and complex refractive–index distribution or anoptical potential, respectively [49, 50], and the asterisk denotes the complex conju-gation. In optics, the real part of V (x,t), defined by VR(x,t), denotes the refractive–index profile, and the imaginary part of V (x,t) denotes the gain/loss distribution,defined by VI(x,t). Based on VR(x,t) = VR(−x,t) and VI(x,t) = −VI(−x,t), aPT–symmetric system can be designed.

The concept of PT symmetry, based on the non–Hermitian Hamiltonians [51]-[62], has recently attracted much attention [63]-[67], in particular in the fields ofoptics and photonics [49, 50, 68–72].

This concept provides a fertile ground for PT-related notions and experiments.Furthermore, applying such concept for the design of photonic devices has openedup many new possibilities, which allow a controlled interplay between gain and loss.Loss is abundant in physical systems, but is typically considered as a problem. Gain,however, as afforded by lasers, is valuable in optoelectronics because it providesmeans to overcome loss [70, 72, 73]. As it has been shown in several studies, thePT-symmetric optical structures can exhibit peculiar properties that are otherwiseunattainable in traditional Hermitian structures, such as the possibility for breakingthis symmetry through an abrupt phase transition [74–76], the band merging effects[49, 68, 69], the unidirectional invisibility [77, 78] and so on; for a few overview

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3 Rational solution of the nonlocal NLS eq. and its application in optics Article no. 108

papers in the area of linear and nonlinear PT-symmetric systems see Refs. [79–81].If a Hamiltonian is PT symmetric, there are two possibilities: Either the eigen-

values are entirely real, in which case the Hamiltonian is said to be in an unbrokenPT-symmetric phase, or else the eigenvalues are partly real and partly complex, inwhich case the Hamiltonian is said to be in a broken PT-symmetric phase. More-over, there is a PT-symmetry-breaking threshold, where the transition is betweenunbroken and broken symmetry. At this point the behavior of Hamiltonians becomeseven more interesting. In this paper, the solution of Eq. (3) at the threshold valueof eigenvalues is derived by virtue of DT. It is found that the solution possesses asimilar structure with the second-order soliton, but both of them are totally differ-ent, because the asymptote amplitudes at two orbits for our rational solution dependon each other and there does not exist any phase shifts after interaction. Besides,we consider the analyticity related to the phase parameters and classify the solutionaccording to its asymptote amplitudes on the phase parameters. Compared to thesolution that has been given by Xu and Li [82], we prove the analyticity of solutionrelated to the phase parameters in detail. The trajectories of the peaks (or valleys)and the parameters in our paper are more general, which include the case studied in[82]. Furthermore, the gain/loss effect related to the solution is considered in thispaper.

This paper is organized as follows: In Sec. 2, we construct a rational solutionof Eq. (3) under PT symmetry with two phase parameters and consider its analyticityrelated to these two parameters. In Sec. 3, we classify in detail the solution accordingto the amplitude. In Sec. 4, we consider the gain/loss effect related to imaginarypart of V (x,t) Eq. (3) generated by the above-mentioned rational solution. Theconclusion and discussion are given in the final Section.

2. THE RATIONAL SOLUTION OF THE NONLOCAL NLS EQUATION

2.1. THE LAX PAIR AND REDUCTION CONDITION

The nonlocal NLS equation (3) can be expressed by the compatibility conditionof the following Lax pair:

Φx(x,t;λ) =M(x,t;λ)Φ(x,t;λ) = [−iλσ+Q(x,t)]Φ(x,t;λ), (4)

Φt(x,t;λ) =N(x,t;λ)Φ(x,t;λ) = [−2iλ2σ+ Q(x,t)]Φ(x,t;λ). (5)

where Φ(x,t;λ) is a 2× 1 matrix valued function, λ ∈ C is the spectral parameter,and

σ =

(1 00 −1

), Q(x,t) =

(0 q(x,t)

r(x,t) 0

),

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Article no. 108 Yongshuai Zhang et al. 4

Q(x,t) = 2Q(x,t)λ− iσQ2(x,t) + iσQx(x,t), r(x,t) = q∗(−x,t).

That is, Mt−Nx + [M, N ] = 0 leads to the nonlocal NLS equation (3). Under thecondition r(x,t) = q∗(−x,t), the eigenfunctions possess the following property:Lemma 2.1 Under the reduction condition

q(x,t) = r∗(−x,t), (6)

the eigenfunctions possess the following property:(φ11(x,t;λ1)φ12(x,t;λ1)

)=

(φ∗12(−x,t;−λ∗1)−φ∗11(−x,t;−λ∗1),

)(7)

where Φ1(x,t;λ1) =

(φ11(x,t;λ1)φ12(x,t;λ1)

)is an eigenfunction associated with λ= λ1.

Form the x-part of the Lax pair, one has(φ11(x,t;λ1)φ12(x,t;λ1)

)x

=

(−iλ1 q(x,t)

q∗(−x,t) iλ1

)(φ11(x,t;λ1)φ12(x,t;λ1)

).

Let x=−x and taking the complex conjugate to both sides, then(φ∗11(−x,t;λ1)φ∗12(−x,t;λ1)

)x

=

(−iλ∗1 −q∗(−x,t)−q(x,t) iλ∗1

)(φ∗11(−x,t;λ1)φ∗12(−x,t;λ1)

),

or (φ∗12(−x,t;λ1)−φ∗11(−x,t;λ1)

)x

=

(iλ∗1 q(x,t)

q∗(−x,t) −iλ∗1

)(φ∗12(−x,t;λ1)−φ∗11(−x,t;λ1)

).

So if λ1 =−λ∗1, then(φ∗12(−x,t;−λ∗1)−φ∗11(−x,t;−λ∗1)

)x

=

(−iλ1 q(x,t)

q∗(−x,t) iλ1

)(φ∗12(−x,t;−λ∗1)−φ∗11(−x,t;−λ∗1)

).

Taking a similar procedure, the symmetry property also holds for the t-part of the

Lax pair, that is, if λ1 = −λ∗1, the eigenfunction(φ∗12(−x,t;−λ∗1)−φ∗11(−x,t;−λ∗1)

)is also the

eigenfunction corresponding to λ1, because it satisfies the same Lax equations.Q.E.D.

To preserve the reduction condition, we let λ2 =−λ∗1 in the following, unless other-wise specified.

2.2. THE RATIONAL SOLUTION OF THE NONLOCAL NLS

Starting from the seed solution

q(x,t) = r∗(−x,t) = cexp(−2ic2t), (8)

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5 Rational solution of the nonlocal NLS eq. and its application in optics Article no. 108

with c denoting the height of the background, and taking the following transforma-tion:

Φ(x,t;λ) = ψ(x,t;λ)exp(−ic2tσ), (9)then

ψx(x,t;λ) = Mψ(x,t;λ) =

(−iλ cc iλ

)ψ(x,t;λ), (10)

ψt(x,t;λ) = Nψ(x,t;λ) =

(−2iλ2 2cλ2cλ 2iλ2

)ψ(x,t;λ). (11)

Thus, solving the Lax pair equations (4) and (5) is equivalent to solving equations(10) and (11). Solving the equations (10) and (11), it leads to

ψ(x,t;λ) = J

(1

s(λ)+iλc

)exp[s(λ)x+ 2λs(λ)t] (12)

+K

(s(λ)+iλ

c−1

)exp[−s(λ)x−2λs(λ)t]

withs(λ) =

√c2−λ2, J = J1 + iJ2, K =K1 + iK2.

Here the Lemma 2.1 and the superposition principle are adopted to construct the non-trivial eigenfunction. In [82], the authors set J to be dependent on K (J ×K =−1)and |J | = |K| = 1, and then just describe the analyticity of rational solution of Eq.(3) without proof. In the following, we will consider the more general case where Jand K are free, and prove in detail the analyticity of rational solution of Eq. (3).

Without loss of generality, let K1 = 1 and K2 = 0. In addition, the eigenval-ues ±s(λ) of matrix M are real when |c| > |λ| , in which case Eq. (3) displaysthe unbroken PT symmetry. Otherwise, the eigenvalues ±s(λ) are pure imaginarywhen |c| < |λ|, in which case Eq. (3) displays the broken PT symmetry. Both theeigenvalues and eigenvectors of matrix M are the same when λ= c, showing all thecharacteristics of an exceptional point singularity [59, 83]. Interestingly, based onthe eigenfunctions and applying the degenerated DT and limit technique [15], we getone kind of rational solution of the nonlocal NLS equation (3) at the threshold λ= c

q1 =−F1

G1exp(−2it), r1 =

F1

G1exp(2it) (13)

where

F1 =(8J1

2 + 8J22−16J2 + 8

)t2 +

(8iJ1

2 + 8iJ22−16iJ2 + 8i + 8J1

)t

−(2J1

2 + 2J22−4J2 + 2

)x2 +

(2iJ1

2 + 2iJ22−2i

)x

+ 4iJ1−J12−J22 + 4J2−1,

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Article no. 108 Yongshuai Zhang et al. 6

F1 =(−8J1

2−8J22 + 16J2−8

)t2 +

(8iJ1

2 + 8iJ22−16iJ2 + 8i−8J1

)t

+(2J1

2 + 2J22−4J2 + 2

)x2−

(2iJ1

2 + 2iJ22−2i

)x

+ 4iJ1 +J12 +J2

2−4J2 + 1,

G1 =(8J1

2 + 8J22−16J2 + 8

)t2 + 8 tJ1−

(2J1

2 + 2J22−4J2 + 2

)x2

+(2iJ1

2 + 2iJ22−2i

)x+J1

2 +J22 + 1.

Here, we have set c = 1 for convenience, namely, the height of background is 1. Itcan be verified that the solution (13) satisfies the PT symmetry and the reductioncondition (6). Furthermore, this solution is analytic under certain condition.Theorem 2.1 If J2

1 +J22 6= 1, then |q1| and |r1| are analytic. Otherwise, |q1| and |r1|

have singular points.First, we display that |q1| and |r1| are analytic when J2

1 +J22 6= 1. In Eq. (13) |q1|

and |r1| are rational solutions, so the singular points just come from zero points ofthe denominator |G1|, that is, it is enough to prove |G1| 6= 0 when J2

1 +J22 6= 1.

Note that

|G1|= 0 ⇔ ReG1 = 0 & ImG1 = 0,

where ReG1 and ImG1 denote the real part and imaginary part of G1 respectively.If ReG1 = 0 and J2

1 +J22 6= 1, then

ReG1 = 2x(J1

2 +J22−1

)= 0 ⇔ x= 0.

Substituting x= 0 into ImG1 = 0 yields

ImG1 =(8J1

2 + 8J22−16J2 + 8

)t2 + 8 tJ1 + 1 +J1

2 +J22 = 0,

which implies

t=−2J1±

√∆

4(J1

2 +J22−2J2 + 1

) ,with

∆≡ −2J14−4J1

2J22−2J2

4 + 4J12J2 + 4J2

3−4J22 + 4J2−2.

Through simple calculation, we find ∆ arrives at its maximum value 0 when J1 = 0and J2 = −1. Since J2

1 + J22 6= 1, ∆ is always negative, which implies there is

no t ∈ R making ImG1 = 0. Therefore, the rational solution (13) is analytic whenJ21 +J2

2 6= 1.Second, let J2

1 +J22 = 1 in Eq. (13), then it is clear that ReG1 = 0. Thus, it

is enough to prove that there always exist x ∈ R and t ∈ R making ImG1 be zero.For convenience, we consider the polar coordinates, and define J1 = cos(θ) andJ2 = sin(θ). Then it yields that

ImG1 = (8 sin(θ)−8) t2 + (−2 sin(θ) + 2)x2−4 cos(θ) t−1. (14)

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7 Rational solution of the nonlocal NLS eq. and its application in optics Article no. 108

Now we need to prove the discriminant of ImG1 about x

∆ = 8 (sin(θ)−1)(8 t2 sin(θ)−4 cos(θ) t−8 t2−1

), 8 (sin(θ)−1)f(θ)

is positive. Here sin(θ) 6= 1 otherwise it leads the coefficient of x2 to be zero. Owingto sin(θ)−1< 0, we need f(θ)< 0. Let f ′(θ0) = 0, then θ0 =−arctan(2t), takingθ0 into f(θ), leading to

f(θ0) =−4 t√

4 t2 + 1−8 t2−1.

Obviously, there exists t∈R making f(θ0)< 0 so that the discriminant ∆> 0. Thus,q and r have singular points when J2

1 +J22 = 1.

Q.E.D.Note that q1 is analytic when (J1, J2) = (0, 1), but it is trivial, owing to q1 = 1

in this case. The dynamical structure of |q1|2 under the condition J21 + J2

2 6= 1 isshown in Fig. 1. In these figures, we find that |q1|2 has four types of patterns. Thefirst case, shown in Fig. 1(a), displays two wavefronts, a bright one and a dark one,moving from t = −∞ to t =∞ on two straight lines and intersecting each other inthe neighborhood of the origin. At the intersection point, amplitudes of the bright andthe dark wavelet decrease rapidly, and then their amplitudes simultaneously increaserapidly to their original amplitudes before the intersection. The second case, shownin Fig. 1(b), is similar to the first case except that the bright and dark wavefrontexchange their locations. The third and fourth cases, shown in Fig. 1(c) and (d),display the evolution of two bright wavefronts, and the amplitudes of these brightwavefronts have similar properties as the wavefronts in the first and second case. Weterm these four cases dark-bright (DB), bright-dark (BD) and bright-bright (BB–I andBB–II) structures, respectively. The corresponding region on J1 and J2 are displayedin the following Section.

3. THE EVOLUTION OF SOLUTION

Note that |q1| runs forward in two straight orbits and preserves almost the sameamplitude in each orbit except for the neighborhood of the intersection of the twoorbits. In order to find the location of the peak (or the valley), we try to find itfrom G1 ≈ 0 when t >> 1, which implies two lines: x = ±2t from the real part of

G1. Thus, let L : x = ±2t+ k (k ∈ R) and t go to infinity in∂|q1|2

∂x

∣∣∣∣L

=0, then

k = ± J1J1

2+J22−2J2+1

through a simple calculation. That is, two asymptotic orbitsfor q1 are obtained as

L1 : x=−2t−|k|, L2 : x= 2t+ |k|. (15)

Figure 2 shows the two lines L1 and L2 are good approximations of the lo-cations of peaks (or valleys). Note that there exist no phase shift for |q1|, which is

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Article no. 108 Yongshuai Zhang et al. 8

(a) |q1|2 (b) |q1|2

(c) |q1|2 (d) |q1|2

Fig. 1 – (Color online). |q1|2 displays four structures with the height of background being 1. (a) Thedark-bright (DB) structure: J1 = 2 and J2 = 3. (b) The bright-dark (BD) structure: J1 = 0.2 andJ2 = 0.2. (c) The bright–bright–I (BB–I) structure: J1 = 0.2 and J2 =−0.1. (d) The bright-bright–II(BB–II) structure: J1 = 3 and J2 =−1.

different from the action of two solitons [3]. Let H1 and H2 denote the amplitudesof |q1| on the line L1 and L2, respectively, then we have

H1 = |1 +m|, H2 = |3 +m|, (16)

where

m=4(1−J2)J21 +J2

2 −1. (17)

Theorem 3.1 The amplitude of |q1| has the following properties when t goes to in-finity:

• H1 is always unequal to H2 except the trivial case q1 = 1.

• If (J1,J2) is inside the unit circle on (J1, J2)-plane then H1 > H2, otherwiseH1 <H2.

1. Let H1 =H2, then

|1 +m|= |3 +m| ⇔ m=−2

yields4(1−J2)J21 +J2

2 −1=−2 ⇔ (J2−1)2 +J2

1 = 0.

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9 Rational solution of the nonlocal NLS eq. and its application in optics Article no. 108

(a) |q1|2 (b) |q1|2

(c) |q1|2 (d) |q1|2

Fig. 2 – (Color online). Density plots of |q1|2 on (x,t)-plane associated with Fig. 1. (a) DB structurewith (J1, J2) = (2, 3), (b) BD structure with (J1, J2) = (0.2, 0.2), (c) BB–I structure with J1 = 0.2and J2 =−0.1, (d) BB–II structure with (J1, J2) = (3,−1).

Note that (J1, J2) = (0, 1) leads to a trivial case q1 = 1, thus H1 6=H2.

2. Let H1 >H2, then

|1 +m|> |3 +m| ⇔ 4m+ 8< 0 ⇔ m<−2.

Substituting m<−2 into (17), we have

4(1−J2)J21 +J2

2 −1<−2.

For J21 +J2

2 −1> 0,

2(1−J2)<−(J21 +J2

2 ) + 1 ⇔ (J2−1)2 +J21 < 0.

It gives a meaningless result.

For J21 +J2

2 −1< 0,

2(1−J2)>−(J21 +J2

2 ) + 1 ⇔ (J2−1)2 +J21 > 0.

Since (J1, J2) 6= (0, 1), the above expression is always preserved. Thus, whenJ21 +J2

2 < 1, H1 >H2 always holds.

3. Similarly, we find the fact that J21 +J2

2 > 1 leads to m>−2 and H1 <H2.

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Fig. 3 – (Color online). Inside the gray region H1 >H2 and outside the gray region H1 <H2. On theboundary line (the unit circle), |q1| is not analytic.

Q.E.D.From the Theorem 2.1, we find that the unit circle on the (J1, J2)-plane is a

crucial cut line, yielding two areas, which determine H1 >H2 and H1 <H2, respec-tively. An explicit description about this property is given in Fig. 3.Theorem 3.2 |q1| has a dark structure on the line L1 when J2 > 1 and a brightstructure J2 < 1. |q1| has a dark structure on the line L2 when J2

1 + (J2− 12)2 < 1

4and a bright structure when J2

1 + (J2− 12)2 > 1

4 .

• Let H1 < 1, then

|m+ 1|< 1 ⇔ −2<m< 0.

From m> −2, we have J21 +J2

2 > 1 from Theorem 3.1, and then J2 > 1 can bederived from m< 0. Hence, |q1| on the line L1 is dark when J2 > 1. Similarly,we get H1 > 1 and |q1| is bright on the line L1 when J2 < 1.

• Let H2 < 1, then

|m+ 3|< 1 ⇔ −4<m<−2.

Fromm<−2, we have J21 +J2

2 < 1 from Theorem 3.1. Substituting J21 +J2

2 < 1into

m=4(1−J2)J21 +J2

2 −1>−4,

then

1−J2 <−J21 −J2

2 + 1, ⇔ J21 +J2

2 < J2,

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11 Rational solution of the nonlocal NLS eq. and its application in optics Article no. 108

which yields

J21 + (J2−

1

2)2 <

1

4.

Similarly, let H2 > 1, and we have J21 + (J2− 1

2)2 > 14 .

Q.E.D.

Fig. 4 – (Color online). |q1| has a DB structure in the blue area, a BD structure in the red area, and twoBB (BB–I and BB–II) structures in the complementary area without including the boundary: J2 = 1,J21 +(J2− 1

2 )2 = 1

4 and J21 +J2

2 = 1.

The region in which |q1| displays the DB, BD, BB–I, and BB–II structures isclear, and is shown explicitly in Fig. 4 and Table 1. The four panels in Fig. 1 exactlyverify the results of Theorem 3.2. From Table 1, it is shown that on the boundaryJ21 +(J2− 1

2)2 = 14 and J2 = 1, H1 = 1 and H2 = 1, respectively, i.e., q1 just has one

peak on the boundary J21 +(J2− 1

2)2 = 14 or J2 = 1. It is important for us to consider

the evolution of q1 in the neighborhood of the origin on two asymptotic lines L1 andL2.

On the one hand, substituting x= 2t+k, J2 = 1 into |q1|2, then

|q1|2x=2t+k = 9− 1

2(t+ 1

2J1

)2+ 1

8

, g1(t). (18)

On the other hand, substituting x = −2t− k, J2 = 1 into |q1|2, leads to |q1|2 = 1,the same as the case for t→∞, which implies that not only when t =∞ but alsowhen t <∞ does |q1|2 equal to the background amplitude 1 at the line L1 for J2 = 1.That is, there exists only one orbit when J2 = 1. Moreover, g1(t) in (18) indicatesthe variation of amplitude |q1|2 with t on the line L2. Similarly, the variation of

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Article no. 108 Yongshuai Zhang et al. 12

Table 1

The region of J1, J2 related to the classification of q1.

Region Values of H1 and H2 Structures

J2 > 1 H1 < 1, H2 > 1, H1 <H2 DB

J21 + (J2− 1

2)2 < 14 H1 > 1, H2 < 1, H1 >H2 BD

J21 + (J2− 1

2)2 > 14 J2

1 +J22 < 1 H1 > 1, H2 > 1, H1 >H2 BB–I

J21 +J2

2 > 1 J2 < 1 H1 > 1, H2 > 1, H1 <H2 BB–II

J2 = 1 H1 = 1, H2 > 0 —

J21 + (J2− 1

2)2 = 14 H1 > 0, H2 = 1 —

amplitude |q1|2 with t on the line L1 is

|q1|2x=−2t−k = 9− 1

2(t− cos(θ)

2(sin(θ)−1)

)2+ 1

8

, g2(t). (19)

Here, we have set J1 = 12 cos(θ), J2 = 1

2 sin(θ) + 12 . From the expression of g1(t)

and g2(t), it is found that the rate of decay of amplitude is proportional to O(t2).

(a) V1I (b) V1I

Fig. 5 – (Color online). The imaginary part of the PT-symmetric potential generated by the rationalsolution. (a) The profile of V1I with J1 =−5, J2 = 2. (b) Four trajectories for V1I , which show theirvalidity.

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13 Rational solution of the nonlocal NLS eq. and its application in optics Article no. 108

(a) (b)

(c) (d)

Fig. 6 – (Color online). The profiles of V1I along x at different moments, which are very similar to thegain/loss distribution of an optical system. (a) t= 0. (b) t= 1. (c) t= 2. (d) t= 3.

4. PT-SYMMETRIC POTENTIAL

In this Section, we consider a PT-symmetric potential V1(x,t) = V1R+ iV1I =q1(x,t)q

∗1(−x,t), which is generated from the rational solution of the nonlocal NLS

equation. The PT symmetry related to V1(x,t) can be directly verified, which impliesV1R is an even function and V1I is an odd function. Taking the same method to V1Ias we operated to q1 in Sec. 3, we find that V1I evolves on four lines:

l1 : x= 2t+

√3(J2

1 +J22 −1) + 6J1

6[J21 + (J2−1)2]

, l2 : x= 2t+−√

3(J21 +J2

2 −1) + 6J16[J2

1 + (J2−1)2],

l3 : x=−2t+

√3(J2

1 +J22 −1)−6J1

6[J21 + (J2−1)2]

, l4 : x=−2t+−√

3(J21 +J2

2 −1)−6J16[J2

1 + (J2−1)2].

When t goes to infinity, the amplitudes of the bright part and the dark part of V1Iare 3

√3(J2

1+J22−2J2+1)2

2(J21+J

22−1)2

and −3√3(J2

1+J22−2J2+1)2

2(J21+J

22−1)2

, respectively. The profile of V1Iand these four lines are displayed in Fig. 5. It is enough to consider the structureof V1I when x > 0, because of it being a odd function. When x > 0, V1I displaysa dark state on the lines l1 and l4, and a bright state on the lines l2 and l3, whichmay impliy that, in the PT-symmetric waveguide system, when a part of the systeminvolves loss, an equal amount of gain is observed near this part, which may be

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Article no. 108 Yongshuai Zhang et al. 14

applied to overcome the loss effects in the system. Moreover, the experimental resultsof El–Ganainy et al. [57] may support our result. Figure 1 in [57] displayed a PT-symmetric waveguide system that involved refractive index and gain/loss profiles,and the gain/loss line is almost totally similar to our results about V1I(x,0), V1I(x,1),V1I(x,2), and V1I(x,3), which are shown in Fig. 6. Thus, our result may guidethe interplay between gain and loss, and the manufacture of a new generation ofmultifunction optical devices and networks.

5. CONCLUSION AND DISCUSSION

We have derived a rational solution of the nonlocal NLS equation under the PT-symmetry condition and we have found that it is analytic if the phase parameters areoff the unit circle. Based on the asymptotic amplitudes H1 and H2, we have dividedthe obtained solutions into four categories: BB–I, BB–II, BD, and DB solutions andwe have graphically illustrated these four types of soliton solutions. Finally, thesimilarity between the profiles of the gain/loss profile of a PT-symmetric waveguidesystem and the imaginary part V1I of the PT-symmetric potential generated by therational solution of the nonlocal NLS equation is discussed. This observation maybe useful for synthesizing new artificial optical structures and materials by mixingtogether the refractive index distribution and the gain/loss profile. In the future, wemay consider the PT symmetry for another kinds of nonlinear evolution equations[84]. The higher-order rational solution of nonlocal equations, such as the nonlocalNLS and nonlocal derivative NLS equations [84], can also be obtained by means ofthe determinant representation of the degenerate DT [15] as we have done for theNLS equation.

Acknowledgements. This work is supported by the NSF of China under Grant Nos. 11271210and 11671219, and the K. C. Wong Magna Fund in Ningbo University.

REFERENCES

1. D. Korteweg and G. de Vries, Phil. Mag. 39, 422–443 (1895).2. J. J. Stoker, Water waves: The mathematical theory with applications, John Wiley & Sons, 2011.3. N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15, 240–243 (1965).4. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Phys. Rev. Lett. 19, 1095–1097

(1967).5. V. E. Zakharov and A. Shabat, Sov. Phys. JETP 34, 62–69 (1972).6. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Phys. Rev. Lett. 31, 125–127 (1973).7. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Stud. Appl. Math. 53, 249–315 (1974).8. M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, SIAM, 1981.9. S. Novikov, S. Manakov, L. Pitaevskii, and V. Zakharov, Theory of solitons: The inverse scattering

method, New York and London, Consultants Bureau, 1984.

(c) RJP 62(Nos. 3-4), id:108-1 (2017) v.2.0*2017.4.21#f9b0ec75

15 Rational solution of the nonlocal NLS eq. and its application in optics Article no. 108

10. F. Calogero and A. Degasperis, Spectral transform and solitons, Elsevier, 2011.11. V. B. Matveev and M. A. Salle, Darboux transformations and solitons, Springer, 1991.12. E. Fan, J. Phys. A: Math. Gen. 33, 6925–6933 (2000).13. B. Guo, L. Ling, and Q. Liu, Phys. Rev. E 85, 026607 (2012).14. B. Guo, L. Ling, and Q. Liu, Stud. Appl. Math. 130, 317–344 (2013).15. J. S. He, H. R. Zhang, L. H. Wang, K. Porsezian, and A. S. Fokas, Phys. Rev. E 87, 052914 (2013).16. L. Ling, B.-F. Feng, and Z. Zhu, Physica D 327, 13–29 (2016).17. B.-F. Feng, L. Ling, and Z. Zhu, Phys. Rev. E 93, 052227 (2016).18. J.-L. Ji and Z.-N. Zhu, Commun. Nonl. Sci. Num. Simul. 42, 699–708 (2017).19. R. Hirota, Phys. Rev. Lett. 27, 1192–1194 (1971).20. R. Hirota, Direct methods in soliton theory, pp. 157–176, Springer, 1980.21. Z. N. Zhu, Phys. Lett. A 180, 409–412 (1993).22. Z.-N. Zhu and H.-W. Tam, J. Phys. A: Math Gen. 37, 3175 (2004).23. E. Kuznetsov, Sov. Phys. Dokl. 22, 507–508 (1977).24. Y.-C. Ma, Stud. Appl. Math. 60, 43–58 (1979).25. N. Akhmediev, V. Eleonsky, and N. Kulagin, Sov. Phys. JETP 62, 894–899 (1985).26. D. Peregrine, J. Austral. Math. Soc. Ser. B 25, 16–43 (1983).27. P. Dubard, P. Gaillard, C. Klein, and V. Matveev, Eur. Phys. J. Special Topics 185, 247–258 (2010).28. P. Dubard and V. Matveev, Nat. Hazard. Earth Sys. 11, 667–672 (2011).29. P. Gaillard, J. Phys. A: Math. Theor. 44, 435204 (2011).30. P. Gaillard, J. Math. Phys. 54, 013504 (2013).31. A. Ankiewicz, Y. Wang, S. Wabnitz, and N. Akhmediev, Phys. Rev. E 89, 012907 (2014).32. J. He, L. Guo, Y. Zhang, and A. Chabchoub, Proc. Roy. Soc. London A: Mathematical, Physical

and Engineering Sciences 470, 20140318 (2014).33. Y. Zhang, L. Guo, J. He, and Z. Zhou, Lett. Math. Phys. 105, 853–891 (2015).34. D. Qiu, J. He, Y. Zhang, and K. Porsezian, Proc. R. Soc. A 471 (2180), 20150236 (2015).35. D. Qiu, Y. Zhang, and J. He, Commun. Nonl. Sci. Num. Simul. 30, 307–315 (2016).36. S. Chen, Phys. Lett. A 378, 2851–2856 (2014).37. S. Chen, J. M. Soto-Crespo, and Ph. Grelu, Phys. Rev. E 90, 033203 (2014).38. S. Chen and D. Mihalache, J. Phys. A: Math. Theor. 48, 215202 (2015).39. D. Mihalache, Proc. Romanian Acad. A 16, 62–69 (2015).40. D. Mihalache, Rom. Rep. Phys. 67, 1383–1400 (2015).41. S. W. Xu, K. Porsezian, J. S. He, and Y. Cheng, Rom. Rep. Phys. 68, 316–340 (2016).42. S. Chen, J. M. Soto-Crespo, F. Baronio, Ph. Grelu, and D. Mihalache, Opt. Express 24, 15251–

15260 (2016).43. S. Chen, F. Baronio, J. M. Soto-Crespo, Y. Liu, and Ph. Grelu, Phys. Rev. A 93, 062202 (2016).44. F. Yuan, J. Rao, K. Porsezian, D. Mihalache, and J. S. He, Rom. J. Phys. 61, 378–399 (2016).45. Y. Liu, A. S. Fokas, D. Mihalache, and J. S. He, Rom. Rep. Phys. 68, 1425–1446 (2016).46. S. Chen, P. Grelu, D. Mihalache, and F. Baronio, Rom. Rep. Phys. 68, 1407–1424 (2016).47. M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 110, 064105 (2013).48. M. J. Ablowitz and Z. H. Musslimani, Integrable nonlocal nonlinear equations, Stud. Appl. Math.,

DOI:10.1111/sapm.12153 (2016).49. K. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904

(2008).

(c) RJP 62(Nos. 3-4), id:108-1 (2017) v.2.0*2017.4.21#f9b0ec75

Article no. 108 Yongshuai Zhang et al. 16

50. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys.6, 192–195 (2010).

51. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243–5246 (1998).52. G. Levai and M. Znojil, J. Phys. A: Math. Gen. 33, 7165–7180 (2000).53. A. Mostafazadeh, J. Math. Phys. 43, 205–214 (2002).54. C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).55. C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. D 70, 025001 (2004).56. C. M. Bender, Rep. Progr. Phys. 70, 947–1018 (2007).57. R. El-Ganainy, K. Makris, D. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632–2634

(2007).58. C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel, Amer. J. Phys. 81, 173–179 (2013).59. D. N. Christodoulides and M.-A. Miri, Proc. SPIE 9162, 91621P–1–91621P–8 (2014).60. B. Liu, L. Li, and D. Mihalache, Rom. Rep. Phys. 67, 802–818 (2015).61. P. F. Li, B. Li, L. Li, and D. Mihalache, Rom. J. Phys. 61, 577–594 (2016).62. P. F. Li, D. Mihalache, and L. Li, Rom. J. Phys. 61, 1028–1039 (2016).63. M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. E 90, 032912 (2014).64. L.-Y. Ma and Z.-N. Zhu, J. Math. Phys. 57, 083507 (2016).65. Z. Yan, Appl. Math. Lett. 47, 61–68 (2015).66. Z. Yan, Z. Wen, and V. V. Konotop, Phys. Rev. A 92, 023821 (2015).67. Z.-C. Wen and Z. Yan, Phys. Lett. A 379, 2025–2029 (2015).68. Z. Musslimani, K. G. Makris, R. El-Ganainy, and D. Christodoulides, Phys. Rev. Lett. 100, 030402

(2008).69. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. D 81,

063807 (2010).70. M. I. Stockman, Phys. Rev. Lett. 106, 156802 (2011).71. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel,

Nature 488, 167–171 (2012).72. P. Berini and I. De Leon, Nat. Photonics 6, 16–24 (2012).73. M. Noginov et al., Nature 460, 1110–1112 (2009).74. A. Guo et al., Phys. Rev. Lett. 103, 093902 (2009).75. Y. Chong, L. Ge, and A. D. Stone, Phys. Rev. Lett. 106, 093902 (2011).76. S. Bittner, B. Dietz, U. Gunther, H. Harney, M. Miski-Oglu, A. Richter, and F. Schafer, Phys. Rev.

Lett. 108, 024101 (2012).77. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett.

106, 213901 (2011).78. L. Feng et al., Nat. Mater. 12, 108–113 (2013).79. V. V. Konotop, J. Yang, and D. A. Zezyulin, Rev. Mod. Phys. 88, 035002 (2016).80. S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, Laser and

Photonics Reviews 10, 177–213 (2016).81. Y. J. He, X. Zhu, and D. Mihalache, Rom. J. Phys. 61, 595–613 (2016).82. Min Li, Tao Xu, and Dexin Meng, J. Phys. Soc. Jpn. 85, 124001 (2016).83. W. Heiss, Phys. Rev. E 61, 929–932 (2000).84. Z. Wu and J. He, Rom. Rep. Phys. 68, 79–98 (2016).

(c) RJP 62(Nos. 3-4), id:108-1 (2017) v.2.0*2017.4.21#f9b0ec75