Rational Solutions of a Weakly Coupled Nonlocal Nonlinear Schrödinger Equation

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 degeneratedDarboux transformation can be got fromDarboux transformation. Applying the degeneratedDarboux 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.


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 PTsymmetry have real and positive spectrum; the work drew attention for several researchers who study PT-symmetry in quantum mechanics [4][5][6][7].For a non-Hermitian Hamiltonian H =  2 / 2 + (), it is PT-symmetric when () = (−) * .Under this condition, the Schrödinger equation Ψ  = 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 PTsymmetry is broken [8][9][10][11].
Here (, ) and (, ) (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 (, ) defined by   (, ) Advances in Mathematical Physics indicates the refractive index profile, and the imaginary part of (, ) represented by   (, ) denotes the gain or loss [17] distribution.Based on   (, )=  (−, ) and   (, ) = −  (−, ), 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: The concept of PT-symmetry, based on the non-Hermitian Hamiltonians [2,[19][20][21][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 PTsymmetric systems, see [29][30][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 PTsymmetric 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 ( [1] ,  [1] ) from the weakly nonlocal NLS equation, and the analytic properties of ( [1] ,  [1] ) are proved in detail.
This paper is arranged as follows: In Section 2, 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.

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: where
So functions  10 ,  11 ,  10 ,  11 are independent of .For   +  =  [1] , we can get that functions  10 ,  11 ,  10 ,  11 should be independent of  using the same method.This suggests that  10 ,  11 ,  10 , and  11 are constants.Then the form of Darboux transformation operator is as follows: Now we will try to determine the specific expression of the matrix .In order to solve this problem, we must use eigenfunction which contains  to determine their expressions through the parameterized method.Next, we will introduce the properties of eigenfunction in spectral problem, with lemma as follows.
Taking a similar procedure, the symmetry property also holds for the t-part of the Lax pair; that is, if  1 = − * 1 , the eigenfunction and  2 = − * 1 in the following.
The Darboux transformation matrix  1 must satisfy the following equation: From the above equation, we can get the expressions of functions  0 ,  1 easily by the eigenfunctions as follows: Combining with (28), the following theorem can be derived.

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 ((, ), (, )) = (  ,   ), ,  being arbitrary real constants.Substituting seed solution into (4), through a proper simplification, we can obtain a constraint Taking the transformation (34) So solving the Lax pair equation ( 34) is equivalent to solving (5).By defining where where Here, for convenience, we have set  = 1.It can be verified that the solutions (38) satisfy the PT-symmetry.Furthermore, the solutions are analytic under certain condition. [1]| and | [1] | are analytic.Otherwise, | [1] | and | [1] | have singular points.
The dynamical structures of (| 1 | 2 , | 1 | 2 ) under the condition  2 1 +  2 2 ̸ = 1 were shown in Figures 1 and 3. When  → ∞,  → ∞, we can know | [1] | 2 → 1, | [1] | 2 → 5 with =1.In Figure 1, we find that (| 1 | 2 ) 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  = −∞ to  = ∞ 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 (| 1 | 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.
In the end of the section, we study the self-induced potential function  [1] (, ), the PT-symmetric potential  [1] (, ) = [1]   +i [1]   = [1] (, ) [1] (−, ),  [1] (, ) = −  5  6 , (44) (45) Figures 5 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.

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 ( [1] ,  [1] ) generated from known solutions (, ).By choosing some special eigenvalues  (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 ( [1]   ), and the gain-or-loss distribution ( [1]   ) of the weakly coupled nonlocal NLS equation under the PT-symmetry condition and we have proved that it is analytic if  2 1 +  2 2 ̸ = 1 [17].We have divided the obtained solutions  [1] ,  [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.