# Rational Solutions of a Weakly Coupled Nonlocal Nonlinear Schrödinger Equation

Authors: Huijuan Zhou; Chuanzhong Li; Yueh-Lung Lin
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

In this article, we investigate an integrable weakly coupled nonlocal nonlinear Schrödinger (WCNNLS) equation including its Lax pair. Afterwards, Darboux transformation (DT) of the weakly coupled nonlocal NLS equation is constructed, and then the degenerated Darboux transformation can be got from Darboux transformation. Applying the degenerated Darboux transformation, the new solutions (q[1], r[1]) and self-potential function (V[1]) are created from the known solutions (q, r). The (q[1], r[1]) satisfy the parity-time (PT) symmetry condition, and they are rational solutions with two free phase parameters of the weakly coupled nonlocal nonlinear Schrödinger equation. From the plots of solutions, the compression effects of the real refractive index profile and the gain-or-loss distribution are produced.


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## Body

## 1. Introduction

A nonlocal nonlinear Schrödinger (NLS) equation was recently found and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the local case, the nonlinearly induced potential is PT-symmetric [1]; thus the nonlocal NLS equation is also PT-symmetric. According to the work of Bender and Boettcher [2], the PT-symmetry plays a vital role in the spectrum of the Hamiltonian. They proved that a broad class of non-Hermitian Hamiltons [3] with PT-symmetry have real and positive spectrum; the work drew attention for several researchers who study PT-symmetry in quantum mechanics [4–7]. For a non-Hermitian Hamiltonian H=d2/dx2+V(x), it is PT-symmetric when V(x)=V(-x)∗. Under this condition, the Schrödinger equation iΨt=HΨ is PT-symmetric. It has been shown that optics can provide a good condition for testing the theory of PT-symmetry or observing the phenomenon when the PT-symmetry is broken [8–11].As is well known, the nonlinear Schrödinger equation(1)qtx,t-iqxxx,t-2iqx,t2qx,t=0,which is PT-symmetric, received an extensive study since the work by Shabat and Zakharov [12]. The NLS equation has many important physical applications, such as water wave [13], nonlinear optics [14], plasma physics [15], and so on.Recently, Musslimani and Ablowitz [16] considered the solution of the nonlocal NLS equation,(2)qtx,t-iqxxx,t±2iVx,tqx,t=0,Vx,t=qx,tq∗-x,t.We first recall that the nonlocal focusing NLS equation(3)qtx,t-iqxxx,t+2iVx,tqx,t=0,from zero background by using the IST method, which implied the PT-symmetry, in other words, x→-x, t→-t and the complex conjugation of the field envelope. 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 an optical potential, respectively [7, 8], and the asterisk denotes the complex conjugation. In optics, the real part of V(x,t) defined by VR(x,t) indicates the refractive index profile, and the imaginary part of V(x,t) represented by VI(x,t) denotes the gain or loss [17] distribution. Based on VR(x,t)=VR(-x,t) and VI(x,t)=-VI(-x,t), a PT-symmetric system can be designed.Nevertheless, the nonlocal property of the soliton equations is not new, for example, nonlocal symmetry [18]. In this purpose, we consider another new nonsymmetric coupled nonlocal NLS equation called weakly coupled nonlocal NLS equation which is as follows:(4)qtx,t-iqxxx,t+2iVx,tqx,t=0,rtx,t-irxxx,t+2iqx,t2r∗-x,t+4iVx,trx,t=0,Vx,t=qx,tq∗-x,t.The concept of PT-symmetry, based on the non-Hermitian Hamiltonians [2, 19–22], has recently attracted much attention [23], in particular in the fields of optics and photonics [9, 24, 25]. This concept offers a fertile ground for PT-related notions and experiments. Furthermore, applying such idea for the design of photonic devices has opened up many new possibilities, which allow a controlled interplay between gain and loss. The loss is abundant in physical systems but is typically considered as a problem. The gain, however, as afforded by lasers, is valuable in optoelectronics because it provides means to overcome loss [26]. Several studies have shown that the PT-symmetric optical structures can exhibit particular properties that are otherwise unattainable in traditional Hermitian structures, for example, the possibility for breaking this symmetry through an abrupt phase transition [11, 27], the unidirectional invisibility [28], and so on. For a few overview papers in the area of linear and nonlinear PT-symmetric systems, see [29–31].If a Hamiltonian is PT-symmetric, there are two possibilities: Either the eigenvalues are entirely real, in which case the Hamiltonian is said to be in an unbroken PT-symmetric phase, or else the eigenvalues are partly complex and partly real, in which case the Hamiltonian is said to be in a broken PT-symmetric phase. Moreover, there is a PT-symmetry-breaking threshold, where the transition is between broken and unbroken symmetry. At this point, the behavior of Hamiltonians becomes even more interesting. In this paper, the solutions of (4) at the threshold value of eigenvalues are derived by degenerated DT. It is found that the solutions possess a similar structure with the second-order soliton, but both of them are different because the asymptotic amplitudes at two orbits for our rational solutions depend on each other and there does not exist any phase shift after the interaction. Besides, we consider the analyticity related to the phase parameters and classify the solution according to its asymptotic amplitudes on the phase parameters. We have provided the rational solutions (q[1],r[1]) from the weakly nonlocal NLS equation, and the analytic properties of (q[1],r[1]) are proved in detail.This paper is arranged as follows: In Section2, we construct a weakly coupled nonlocal NLS equation and its Lax representation. In Section 3, we calculate the Darboux transformation of the weakly coupled nonlocal NLS equation. In Section 4, we consider the rational solutions of the weakly coupled nonlocal NLS equation. Please see Section 5 for the conclusion.


## 2. Weakly Coupled Nonlocal NLS Equation and Its Lax Representation

The weakly coupled nonlocal NLS equation (4) can be expressed by the compatibility condition of the following Lax pair:(5)Φxx,t;λ=Mx,t;λΦx,t;λ,Φtx,t;λ=Nx,t;λΦx,t;λ,where Φ(x,t;λ) is a matrix valued function, λ∈C is the spectral parameter, and(6)Mx,t;λ=-iλEqx,tE+rx,tΓq∗-x,tE+r∗-x,tΓiλE,Nx,t;λ=N1N2N3N4,where(7)N1=-2iλ2-iqx,tq∗-x,tE-iqx,tr∗-x,t+q∗-x,trx,tΓ,N2=2λqx,t+iqxx,tE+2λrx,t+irxx,tΓ,N3=2λq∗-x,t-iqx∗-x,tE+2λr∗-x,t-irx∗-x,tΓ,N4=2iλ2+iqx,tq∗-x,tE+iqx,tr∗-x,t+q∗-x,trx,tΓ,(8)E=1001,Γ=0010.That is,Mt-Nx+[M,N]=0 leads to a weakly coupled nonlocal NLS equation (4),(9)qtx,t-iqxxx,t+2iVx,tqx,t=0,rtx,t-irxxx,t+2iqx,t2r∗-x,t+4iVx,trx,t=0,Vx,t=qx,tq∗-x,t.


## 3. Darboux Transformation for the Weakly Coupled Nonlocal NLS Equation

It was pointed out that the Darboux transformation is a method proposed by J. G. Darboux more than 100 years ago and it is an efficient method to generate soliton solutions of integrable equations. In this section, we would like to introduce the Darboux transformation [32, 33] and degenerated Darboux transformation for the weakly coupled nonlocal NLS equation. First, we will introduce a simple gauge transformation of spectral problems (5) as follows:(10)Φ1=TΦ.It can transform linear problems (5) into a new one(11)Φx1=M1Φ1,Φt1=N1Φ1,where M[1], N[1] have the same forms with M, N except that the q(x,t), r(x,t), q∗(-x,t), r∗(-x,t) in the matrices M, N are replaced with q[1](x,t), r[1](x,t), q[1]∗(-x,t), r[1]∗(-x,t) in the matrices M[1], N[1]. It is easy to obtain the equations(12)M1T=Tx+TM,N1T=Tt+TN,which lead to(13)M1=TxT-1+TMT-1,N1=TzT-1+TNT-1.In general, if the transformationT is a polynomial of the parameter λ, we can start from(14)T=T1λ+T0,where(15)T1=a10E+a11Γb10E+b11Γc10E+c11Γd10E+d11Γ,T0=a0E+a1Γb0E+b1Γc0E+c1Γd0E+d1Γ.For (13), we can get (16)Tx=T1xλ+T0x=a10xE+a11xΓb10xE+b11xΓc10xE+c11xΓd10xE+d11xΓλ+a0xE+a1xΓb0xE+b1xΓc0xE+c1xΓd0xE+d1xΓ=0-2ib10E-2ib11Γ2ic10E+2ic11Γ0λ2+H1H2H3H4λ+K1K2K3K4,where (17)H1=q1x,tc10-b10q∗-x,tE+r1x,tc10+q1x,tc11-b10r∗-x,t-b11q∗-x,tΓ,H2=-2ib0+q1x,td10-a10qx,tE+-2ib1+q1x,td11+r1x,td10-a10rx,t-a11qx,tΓ,H3=2ic0+q1∗-x,ta10-d10q∗-x,tE+2ic1+q1∗-x,ta11+r1∗-x,ta10-d10r∗-x,t-d11q∗-x,tΓ,H4=q1∗-x,tb10-c10qx,tE+q1∗-x,tb11+r1∗-x,tb10-c10rx,t-c11qx,tΓ,K1=q1c0-q∗-x,tb0E+r1x,tc0+q1x,tc1-r∗-x,tb0-b1q∗-x,tΓ,K2=q1d0-a0qx,tE+q1x,td1+r1x,td0-a0rx,t-a1qx,tΓ,K3=q1∗-x,ta0-d0q∗-x,tE+q1∗-x,ta1+r1∗-x,ta0-r∗-x,td0-d1q∗-x,tΓ,K4=q1∗-x,tb0-c0qx,tE+q1∗-x,tb1+r1∗b0-c0rx,t-c1qx,tΓ.By comparing the coefficient in terms ofλ, we can see(18)a10x=a11x=0,b10=b11=0,d10x=d11x=0,c10=c11=0,q1x,td10-2ib0-qx,ta10=0,r1x,td10+q1x,td11-2ib1-qx,ta11-rx,ta10=0.So functionsa10, a11, d10, d11 are independent of x. For Tt+TN=N[1]T, we can get that functions a10, a11, d10, d11 should be independent of t using the same method. This suggests that a10, a11, d10, and d11 are constants. Then the form of Darboux transformation operator is as follows:(19)T=T1λ=a10E+a11Γ00d10E+d11Γλ+a0E+a1Γb0E+b1Γc0E+c1Γd0E+d1Γ.Now we will try to determine the specific expression of the matrixT. In order to solve this problem, we must use eigenfunction which contains λ to determine their expressions through the parameterized method. Next, we will introduce the properties of eigenfunction in spectral problem, with lemma as follows.Lemma 1.
The eigenfunctions possess the following property:(20)ϕ110x,t;λ1E+ϕ111x,t;λ1Γϕ120x,t;λ1E+ϕ121x,t;λ1Γ=ϕ120∗-x,t;-λ1∗E+ϕ121∗-x,t;-λ1∗Γ-ϕ110∗-x,t;-λ1∗E-ϕ111∗-x,t;-λ1∗Γ, where(21)Φ1x,t;λ1=ϕ110x,t;λ1E+ϕ111x,t;λ1Γϕ120x,t;λ1E+ϕ121x,t;λ1Γ is an eigenfunction associated with λ=λ1.Proof.
IfΦ1(x,t;λ1) is the solution of (5), (22)ϕ110xx,t;λ1=-iλ1ϕ110x,t;λ1+qx,tϕ120x,t;λ1,ϕ111xx,t;λ1=-iλ1ϕ111x,t;λ1+qx,tϕ121x,t;λ1+rx,tϕ120x,t;λ1,ϕ120xx,t;λ1=iλ1ϕ120x,t;λ1+q∗-x,tϕ110x,t;λ1,ϕ121xx,t;λ1=iλ1ϕ121x,t;λ1+q∗-x,tϕ111x,t;λ1+r∗-x,tϕ110x,t;λ1.
Take the conjugate for the above equation on both sides at the same time, and letx=-x:(23)ϕ110x∗-x,t;λ1=iλ1∗ϕ110∗-x,t;λ1+q-x,t∗ϕ120∗-x,t;λ1,ϕ111x∗-x,t;λ1=iλ1∗ϕ111∗-x,t;λ1+q-x,t∗ϕ121∗-x,t;λ1+r-x,t∗ϕ120∗-x,t;λ1,ϕ120x∗-x,t;λ1=-iλ1∗ϕ120∗-x,t;λ1+qx,tϕ110∗-x,t;λ1,ϕ121x∗-x,t;λ1=-iλ1∗ϕ121∗-x,t;λ1+qx,tϕ111∗-x,t;λ1+rx,tϕ110∗-x,t;λ1,so if λ1=-λ1∗, then(24)ϕ110x∗-x,t;-λ1∗=-iλ1ϕ110∗-x,t;-λ1∗+q-x,t∗ϕ120∗-x,t;-λ1∗,ϕ111x∗-x,t;-λ1∗=-iλ1ϕ111∗-x,t;-λ1∗+q-x,t∗ϕ121∗-x,t;-λ1∗+r-x,t∗ϕ120∗-x,t;-λ1∗,ϕ120x∗-x,t;-λ1∗=iλ1ϕ120∗-x,t;-λ1∗+qx,tϕ110∗-x,t;-λ1∗,ϕ121x∗-x,t;-λ1∗=iλ1ϕ121∗-x,t;-λ1∗+qx,tϕ111∗-x,t;-λ1∗+rx,tϕ110∗-x,t;-λ1∗.Taking a similar procedure, the symmetry property also holds for the t-part of the Lax pair; that is, if λ1=-λ1∗, the eigenfunction(25)ϕ120∗-x,t;-λ1∗∗E+ϕ121∗-x,t;-λ1∗Γ-ϕ110∗-x,t;-λ1∗E-ϕ111∗-x,t;-λ1∗Γ is also the eigenfunction corresponding to λ1, because it satisfies the same Lax equations. We can let(26)ϕ210x,t;λ1=ϕ120∗-x,t;-λ1∗,ϕ211x,t;λ1=ϕ121∗-x,t;-λ1∗,ϕ220x,t;λ1=-ϕ110∗-x,t;-λ1∗,ϕ221x,t;λ1=-ϕ111∗-x,t;-λ1∗,and λ2=-λ1∗ in the following.According to Lemma1, by choosing a10=a11=1, d10=d11=1, we can get the onefold Darboux transformation for the weakly coupled NLS equation as follows:(27)T=T1λ=λ+a0E+λ+a1Γb0E+b1Γc0E+c1Γλ+d0E+λ+d1Γ=E+Γ00E+Γλ+a0E+a1Γb0E+b1Γc0E+c1Γd0E+d1Γ,and then the new solution q[1], r[1] can be obtained by the Darboux transformation(28)q1x,t=qx,t+2ib0,r1x,t=rx,t+2ib1-b0.The Darboux transformation matrixT1 must satisfy the following equation:(29)T1λ,λjΦj=0,j=1,2.From the above equation, we can get the expressions of functionsb0, b1 easily by the eigenfunctions as follows:(30)b0=λ1-λ2ϕ110ϕ210ϕ110ϕ220-ϕ120ϕ210,b1=λ1-λ2ϕ1102ϕ211ϕ220-ϕ210ϕ221+ϕ2102ϕ110ϕ121-ϕ111ϕ120ϕ110ϕ220-ϕ120ϕ2102.Combining with (28), the following theorem can be derived.Theorem 2.
The Darboux transformation of the weakly coupled NLS equation can be written:(31)q1=q+2iλ1-λ2ϕ110ϕ210ϕ110ϕ220-ϕ120ϕ210,r1=r+2iλ1-λ2ϕ1102ϕ220ϕ211-ϕ210-ϕ210ϕ221+ϕ2102ϕ110ϕ120+ϕ121-ϕ111ϕ120ϕ110ϕ220-ϕ120ϕ2102,Then, degenerated Darboux transformation of the weakly coupled NLS equation is given explicitly by a series expansion of theϕijk(λ=λ1+ϵ), i,j=1,2, k=0,1. Degenerated Darboux transformation of the weakly coupled nonlocal NLS equation will help us to find the rational solution in the next section.


## 4. The Rational Solutions of the Weakly Coupled Nonlocal NLS Equation

From the previous section, degenerated Darboux transformation of the weakly coupled nonlocal NLS equation has been discussed. In this section, we will be focused on a kind of solutions which starts from seed solutions by the degenerated Darboux transformation. The solutions are called rational solutions. The seed solutions can be assumed as(q(x,t),r(x,t))=(ceibt,iceibt), b, c being arbitrary real constants. Substituting seed solution into (4), through a proper simplification, we can obtain a constraint(32)b=-2c2.Taking the transformation(33)Φxx,t;λ=ψxx,t;λeibtσ,σ=100-1,then(34)ψxx,t;λ=M^ψx,t;λ=-iλEcEcEiλE,ψtx,t;λ=N^ψx,t;λ=-2iλ2E2cλE2cλE2iλ2E.So solving the Lax pair equation (34) is equivalent to solving (5).By defining(35)R=c2-λ2,the following basic solutions ψ110, ψ111, ψ120, ψ121 are obtained in terms of R from the the spectral problem of (5) as(36)ψ110R=e2tRλ+xR,ψ111R=e2tRλ+xR,ψ120R=R+iλce2tRλ+xR,ψ121R=R+iλce2tRλ+xR.Here Lemma1 and the superposition principle are adopted to construct the nontrivial eigenfunctions ϕ110, ϕ120, ϕ111, ϕ121:(37)ϕ110=Jψ110Rx,t,λ+Kψ120∗R-x,t,-λ=Je2tRλ+xR+KR+iλce-2tRλ-xR,ϕ111=Jψ111Rx,t,λ+Kψ121∗R-x,t,-λ=Je2tRλ+xR+KR+iλce-2tRλ-xR,ϕ120=Jψ120Rx,t,λ-Kψ110∗R-x,t,-λ=JR+iλce2tRλ+xR-Ke-2tRλ-xR,ϕ121=Jψ121Rx,t,λ-Kψ111∗R-x,t,-λ=JR+iλce2tRλ+xR-Ke-2tRλ-xR,where J=J1+iJ2, K=K1+iK2, J1, J2, K1, K2 are arbitrary real constants. It can be proved that ϕ110, ϕ111, ϕ120, and ϕ121 are also the solutions of the Lax equation with λ1=c+ϵ. Now applying the degenerated Darboux transformation, we can get one kind of rational solution of the nonlocal NLS equation at the threshold λ=c:(38)q1=F1G1e-2it,r1=F2+F3+F4G12e-2it,where (39)F1=-8J12+J22-2J2+1t2-8iJ12+iJ22-2iJ2+i+J1t-22J2-J12-J22-1x2-2iJ22+iJ12-ix-4iJ1+J12+J22-4J2+1,G1=8J12+J22-2J2+1t2+8J1t+22J2-J12-J22-1x2+2iJ12+J22-1x+J12+J22+1,F2=2+i64J12+J222-256J2J22+J12+1+J22+128iJ12+646+it4+64iJ22+J122+4J2-12J1-iJ2+2J1iJ1+12-J11+2J2+2J12-iJ2+2iJ22J1-1+it3+16i+1J12+J222+1+32J123i1+J1-iJ2+5+32J223iJ1-iJ2+i-1-32iJ2+96iJ11-2J2t2+82iJ2J2-1-J12-J22+J16iJ1+2i+1+J1221+i-iJ1+iJ12+J222+J14+1t,F3=42+iJ12+J222-4J22-6J2+8J2+2J121-2J2+1x4+2i-18-8J12+J222+J2J12+16J22-1x3+J1281+iJ2-iJ1-12J12-8iJ2224-12J22+8iJ2-12+J12+J222+J11-J22x2+4i-1J12+J222-1+8J1J2iJ1-J2+1-J13-iJ2+iJ23x,F4=J12642+i2J2-1-24J22+2+i128J2-32J12+J222+J23-192J22-32t2x2+322i-1J12+J222+J121+2J21-2i-2J21-2i1-J22-2it2x+32J222iJ2-2i-J12+i-32iJ1-J22-16iJ12+J222-16i-32J13i+2+32J14J2-i-2+64iJ2J12+1tx2+32J23+J2J12-1-16J12+J222+2i-1J1J12+J22-1tx+i+2iJ122J2+1+2iJ221+2J1+iJ12+J222+4iJ1+J13+4J2+4J23. Here, for convenience, we have set c=1. It can be verified that the solutions (38) satisfy the PT-symmetry. Furthermore, the solutions are analytic under certain condition.Theorem 3.
IfJ12+J22≠1, then q1 and r1 are analytic. Otherwise, q1 and r1 have singular points.Proof.
First, we display thatq1 and r1 are analytic when J12+J22≠1. In (38) q1 and r1 are rational solutions, so the singular points just come from zero points of the denominator G1; that is, it is enough to prove G1≠0 when J12+J22≠1.
We know thatG1=0 is equivalent to G1R=0 and G1I=0, where G1R and G1I denote the real part and imaginary part of G1, respectively.
IfG1I=0, we can get x=0 in the case of J12+J22≠1. Substituting x=0 into G1R=0, then(40)G1R=8J12+8J22-16J2+8t2+8J1t+J12+J22+1=0,
which implies(41)t=-2J1±Δ14J12+J22-2J2+1,Δ1=-2J12+J22-J22-2J2-12.
Through simple calculation, we find thatΔ1 arrives at its maximum value 0 when J1=0 and J1=1. Since J12+J22≠1 is always negative, this implies there is no t∈R making G1R=0. Therefore, the rational solutions (38) are analytic when J12+J22≠1.
Second, letJ12+J22=1 in (38). For convenience, we define J1=cos⁡(θ) and J2=sin⁡(θ). Then we can get(42)q1=-8itsin⁡θ-8t2sin⁡θ+2x2sin⁡θ+2icos⁡θ+8it+4tcos⁡θ+8t2-2x2+2sin⁡θ-1Q,r1=isin⁡θ-16it2+4ix2+8t2-2x2+2i+8t+cos⁡θ8it-2-4t-8t2+2x2+16it2-4ix2-8t-1Q,where Q=2-2sin⁡θx2+8sin⁡θt2-4tcos⁡(θ)-8t2-1.
Whensin⁡(θ)=1, (q[1],r[1]) are analytic but trivial owing to q[1]=-1,r[1]=2+I in this case. So when sin⁡(θ)≠1 we need to prove that there always exist x∈R and t∈R making Q be zero.
Now we need to prove the discriminant ofQ about x(43)Δ2=8sin⁡θ-18sin⁡θt2-4cos⁡θt-8t2-1is positive. Owing to sin⁡θ-1<0, let fθ=8t2sin⁡θ-1-4cos⁡θt-1; we need to prove f(θ)<0. Let f(θ0)′=0; then θ0=-arctan⁡(2t), taking θ0 into f(θ), leading to f(θ0)=-4t4t2+1-8t2-1. Obviously, there exists t∈R making f(θ0)<0 so that the discriminant Δ2>0. Thus, q[1] and r[1] have singular points when J12+J22=1.The dynamical structures of(q12,r12) under the condition J12+J22≠1 were shown in Figures 1 and 3. When x→∞, t→∞, we can know q12→1, r12→5 with c=1. In Figure 1, we find that (q12) has four types of patterns. The first and second cases, shown in Figures 1(a) and 1(b), display two wavefronts, a bright one and a dark one, moving from t=-∞ to t=∞ on two straight lines and intersecting with each other in the neighborhood of the origin. At the intersection point, amplitudes of the bright and the dark wavelet decrease rapidly, and then their amplitudes simultaneously increase rapidly to their original amplitudes before the intersection. The only difference between Figures 1(a) and 1(b) is that the bright and dark wavefronts exchange their locations. The third and fourth cases, shown in Figures 1(c) and 1(d), display the evolution of two bright wavefronts, and the amplitudes of these bright wavefronts have similar properties as the wavefronts in the first and second cases. In Figure 3, we find that (|r1|2) has two types of patterns. The first case, shown in Figure 3(a), is similar to Figure 1(b). The second case, shown in Figure 3(b), is similar to Figure 1(a). Figure 2 and Figure 4 are density plots of Figure 1 and Figure 3, respectively.Figure 1
q 1 2 displays four structures with the height of background being 1. (a) The dark-bright structure: (J1,J2)=(2,3). (b) The bright-dark structure: (J1,J2)=(0.2,0.2). (c) The bright-bright 1 structure: (J1,J2)=(0.2,-0.1). (d) The bright-bright 2 structure: (J1,J2)=(3,-1).
(a)
( q 1 2 ) (b)
( q 1 2 ) (c)
( q 1 2 ) (d)
( q 1 2 )Figure 2
Density plots of|q[1]|2 on (x,t)-plane associated with Figure 1. (a) The dark-bright structure: (J1,J2)=(2,3). (b) The bright-dark structure: (J1,J2)=(0.2,0.2). (c) The bright-bright 1 structure: (J1,J2)=(0.2,-0.1). (d) The bright-bright 2 structure: (J1,J2)=(3,-1).
(a)
( q 1 2 ) (b)
( q 1 2 ) (c)
( q 1 2 ) (d)
( q 1 2 )Figure 3
| r [ 1 ] | 2 displays two structures with the height of background being 5. (a) The bright-bright 1 structure: (J1,J2)=(0.2,-0.1). (b) The bright-bright 2 structure: (J1,J2)=(3,-1).
(a)
( | r 1 | 2 ) (b)
( | r 1 | 2 )Figure 4
Density plots of|r[1]|2 on (x,t)-plane associated with Figure 1. (a) The bright-bright 1 structure: (J1,J2)=(0.2,-0.1). (b) The bright-bright 2 structure: (J1,J2)=(3,-1).
(a)
( | r [ 1 ] | 2) (b)
(|r[1]|2)In the end of the section, we study the self-induced potential functionV[1](x,t), the PT-symmetric potential V[1](x,t) =VR[1] +iVI[1]=q[1](x,t)q[1](-x,t),(44)V1x,t=-F5F6,(45)F5=8itJ12+2ixJ12+8it-2ix+8t2J12+8t2J22-2x2J12-2x2J22-16itJ2-16t2J2+4x2J2+4iJ1+8itJ22+2ixJ22+8t2+8tJ1-2x2-J12-J22+4J2-11-8t2-4J2+8itJ12+8it-2ixJ12+2ix+J12+J22-16itJ2-8t2J12-8t2J22+16t2J2+2x2J12+2x2J22-4x2J2-8tJ1+2x2+4iJ1+8itJ22-2ixJ22,F6=8t2J12+8t2J22-16t2J2+8t2-2x2J12-2x2J22+2ixJ12+2ixJ22+4x2J2+8tJ1-2x2-2ix+J12+J22+12.Figures5 and 6 imply that in the PT-symmetric waveguide system that involved refractive index and gain or loss profiles, when a part of the system involves loss, an equal amount of gain is observed near this part, which may be applied to overcome the loss effects in the system. Thus, our results may guide the interplay between gain and loss and the manufacture of a new generation of multifunction optical devices and networks.Figure 5
The real part of the PT-symmetric potential generated by the rational solution. The profile ofV1I with (J1,J2)=(2,3).Figure 6
The imaginary part of the PT-symmetric potential generated by the rational solution. The profile ofV1I with (J1,J2)=(2,3).


## 5. Conclusion

In this article, we study a weakly coupled nonlocal NLS equation. We have shown its integrability by proposing its Lax pair. Then, we have derived the Darboux transformation of the weakly coupled nonlocal NLS equation which implies the representations of(q[1],r[1]) generated from known solutions (q,r). By choosing some special eigenvalues λ2=-λ1∗, and eigenfunctions using the reduction conditions φ210(x,t,λ2)=φ120∗(-x,t,-λ1∗), φ211(x,t,λ2)=φ121∗(-x,t,-λ1∗), φ220(x,t,λ2)=-φ110∗(-x,t,-λ1∗), φ221(x,t,λ2)=-φ111∗(-x,t,-λ1∗), the representations of (31) provided some different solutions of the weakly coupled nonlocal NLS equation. By using degenerate Darboux transformation, we have derived rational solutions, the real refractive index profile (VR[1]), and the gain-or-loss distribution (VI[1]) of the weakly coupled nonlocal NLS equation under the PT-symmetry condition and we have proved that it is analytic if J12+J22≠1 [17]. We have divided the obtained solutions q[1], r[1] into four categories solutions and two categories solutions which are graphically illustrated in Figures 1 and 3, respectively. This observation may be useful for synthesizing new artificial optical structures and materials by mixing together the refractive index distribution and the gain-or-loss profile.



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*Source: 9216286-2018-10-14.xml*