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 degenerated Darboux transformation can be got from Darboux transformation. Applying the degenerated Darboux transformation, the new solutions (, ) and self-potential function are created from the known solutions (, ). The (, ) 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.
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 ; thus the nonlocal NLS equation is also PT-symmetric. According to the work of Bender and Boettcher , the PT-symmetry plays a vital role in the spectrum of the Hamiltonian. They proved that a broad class of non-Hermitian Hamiltons  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 , it is PT-symmetric when . Under this condition, the Schrödinger equation 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 equationwhich is PT-symmetric, received an extensive study since the work by Shabat and Zakharov . The NLS equation has many important physical applications, such as water wave , nonlinear optics , plasma physics , and so on.
Recently, Musslimani and Ablowitz  considered the solution of the nonlocal NLS equation,We first recall that the nonlocal focusing NLS equationfrom zero background by using the IST method, which implied the PT-symmetry, in other words, , and the complex conjugation of the field envelope. 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 indicates the refractive index profile, and the imaginary part of represented by denotes the gain or loss  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 . 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, 19–22], has recently attracted much attention , 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 . 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 , 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 from the weakly nonlocal NLS equation, and the analytic properties of 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.
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:where is a matrix valued function, is the spectral parameter, andwhere
That is, leads to a weakly coupled nonlocal NLS equation (4),
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:It can transform linear problems (5) into a new onewhere , have the same forms with , except that the , , , in the matrices , are replaced with , , , in the matrices , . It is easy to obtain the equationswhich lead to
In general, if the transformation is a polynomial of the parameter , we can start fromwhere
For (13), we can get where
By comparing the coefficient in terms of , we can see
So functions , , , are independent of . For , we can get that functions , , , should be independent of using the same method. This suggests that , , , and 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.
Lemma 1. The eigenfunctions possess the following property: where is an eigenfunction associated with .
Proof. If is the solution of (5), Take the conjugate for the above equation on both sides at the same time, and let :so if , thenTaking a similar procedure, the symmetry property also holds for the t-part of the Lax pair; that is, if , the eigenfunction is also the eigenfunction corresponding to , because it satisfies the same Lax equations. We can letand in the following.
According to Lemma 1, by choosing , , we can get the onefold Darboux transformation for the weakly coupled NLS equation as follows:and then the new solution , can be obtained by the Darboux transformation
The Darboux transformation matrix must satisfy the following equation:
From the above equation, we can get the expressions of functions , easily by the eigenfunctions as follows:Combining with (28), the following theorem can be derived.
Theorem 2. The Darboux transformation of the weakly coupled NLS equation can be written:
Then, degenerated Darboux transformation of the weakly coupled NLS equation is given explicitly by a series expansion of the , , . 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 , , being arbitrary real constants. Substituting seed solution into (4), through a proper simplification, we can obtain a constraintTaking the transformationthenSo solving the Lax pair equation (34) is equivalent to solving (5).
By definingthe following basic solutions , , , are obtained in terms of from the the spectral problem of (5) as
Here Lemma 1 and the superposition principle are adopted to construct the nontrivial eigenfunctions , , , :where , , , , , are arbitrary real constants. It can be proved that , , , and are also the solutions of the Lax equation with . Now applying the degenerated Darboux transformation, we can get one kind of rational solution of the nonlocal NLS equation at the threshold :where Here, for convenience, we have set . It can be verified that the solutions (38) satisfy the PT-symmetry. Furthermore, the solutions are analytic under certain condition.
Theorem 3. If , then and are analytic. Otherwise, and have singular points.
Proof. First, we display that and are analytic when . In (38) and are rational solutions, so the singular points just come from zero points of the denominator ; that is, it is enough to prove when .
We know that is equivalent to and , where and denote the real part and imaginary part of , respectively.
If , we can get in the case of . Substituting into , thenwhich impliesThrough simple calculation, we find that arrives at its maximum value 0 when and . Since is always negative, this implies there is no making . Therefore, the rational solutions (38) are analytic when .
Second, let in (38). For convenience, we define and . Then we can getwhere
When , are analytic but trivial owing to in this case. So when we need to prove that there always exist and making be zero.
Now we need to prove the discriminant of about is positive. Owing to , let ; we need to prove . Let ; then , taking into , leading to Obviously, there exists making so that the discriminant . Thus, and have singular points when .
The dynamical structures of under the condition were shown in Figures 1 and 3. When , , we can know , with =1. In Figure 1, we find that 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 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 , the PT-symmetric potential = +i=,