The nth-Darboux Transformation and Explicit Solutions of the PT-Symmetry Second-Type Derivative Nonlinear Schrödinger Equation

Non-local problems have become one of the research hotspots in recent years since Ablowitz–Musslimani constructed an integrable non-local nonlinear Schro¨dinger\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Schr}\ddot{\mathrm{o}}\hbox {dinger}$$\end{document} (NLS) equation in 2013. In this paper, we first derive the PT-symmetry second derivative nonlinear Schro¨dinger\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Schr}\ddot{\mathrm{o}}\hbox {dinger}$$\end{document} (PT-DNLSII) equation. Then we present the nth-Darboux transformation (DT) of the PT-DNLSII equation. As applications, starting from the zero seed solution and non-zero periodic seed solution, the explicit expressions of the multi-soliton solutions, bright soliton solution, dark soliton solution and breather soliton solution of the PT-DNLSII equation are worked out.


Introduction
Usually, many physical problems will occur in two or more places, and are related to each other, it is called multi-point problems. In order to work on such multipoint problems, the Alice-Bob system [1] is proposed. The Alice-Bob system describes that if A(x,t) is the state of Alice and B(x,t) is the state of Bob, then there will be an operator f that can describe the state of Alice and Bob. The relationship between Equivalently, the operator is required to satisfy the condition f 2 = 1 . Generally, (x, t) is far away from (x 0 , t 0 ) , so the two-point system, or Alice-Bob system, is a nonlocal system. Some special non-local models have been proposed. For example, in 2013, Ablowitz and Musslimani proposed a nonlinear Schrödinger equation (NLS) with f = PC transformation [2], where P is the spatial inversion transformation and C is the charge conjugate transformation. After the pioneering work of Ablowitz-Musslimani, various non-local integrable models have been proposed. For example, the non-local AKNS model obtained by the Ablowitz-Kaup-Newell-Segur (AKNS) system [3], local KdV model [4], non-local NLS model obtained from discrete nonlocal NLS model [5,6], coupled non-local NLS model [7,8], non-local MKdV model [9] and so on, are still in the rapid development stage. These existed results inspire us to study the non-local equation further.
In 1979, the Chen-Lee-Liu (CLL) equation, also called the second derivative nonlinear Schrödinger equation (DNLSII), was proposed and proved to be integrable. In 2007, the physical meaning of the DNLSII equation was experimentally proved in optics aspect: this equation controls the propagation of the optical pulse which only has the self-steepening effect but without the self-phase-modulation. Also, this equation has a wide range of applications in the fields of weakly nonlinear dispersive water waves, nonlinear optics, etc. Therefore, it is interesting to study the PT-symmetry DNLSII equation.
In this paper, based on the coupled CLL equation, we construct the PT-symmetry DNLSII equation in Sect. 2. Then, the nth-DT of the PT-symmetry DNLSII equation are constructed in Sect. 3. In Sect. 5, the multi-soliton solutions, periodic solution, bright soliton, dark soliton and breather solution are obtained. A conclusion is given in the last section.

PT-Symmetry DNLSII Equation
After Chen-Lee-Liu equated the linearisation equation of the nonlinear Hamilton system with the Lax pair time part [10] and started to find the Lax pair of the given equations, many equations have been obtained. One of them is an integrable coupled derivative NLS equation Let r = q * , the DNLSII equation can be obtained First, we construct an integrable coupled CLL system [11], as shown below If p = q , s = r , coupled CLL system can be obtained from the coupled system (4).
Then let t = it , x = −ix and r = r 8 , we can get The integrability of the coupled CLL system (4) can be guaranteed by the following Lax pairs:

3
For the CLL system (4), let p = q , s = r , r = r 8 , and then take the reduced condition r = q(−x + x 0 , −t + t 0 ) , we get the DNLSII equation of the non-local system with PT-symmetry condition

Darboux Transformation of the PT-Symmetry DNLSII Equation
In this section, we will construct the DT of the PT-symmetry DNLSII equation. We first briefly review the DT of the coupled DNLSII equation which has the following Lax pair: with Then, we suppose is the first-order DT of the coupled DNLSII equation, which transform U and V into U [1] and V [1] with q and r replaced by the new potentials q [1] and r [1] . Taking the gauge transformation [12] the Eq. (9) turns into To determine a 0 , b 0 , c 0 , d 0 , a 1 , b 1 , c 1 , d 1 , the functions of x and t,we compare the coefficients of j , (j = 0, 1, 2, 3) of Eq. (12). In order to obtain non-trival solutions, we take a 0 = d 0 = b 1 = c 1 = 0 . After that we get the relationship between the new potentials q [1] , r [1] and the old ones , we obtain the following constraints: In the same way, from correlation relationship q [n] = r [n] (−x + x 0 , −t + t 0 ) , we can get the following relationships: where (12) Φ [1] x = (T [1] x + T [1] U)(T [1] ) −1 Φ [1] = U [1] Φ [1] ,

Remark 1
In some cases, H can be solved. For example, when q = r = 0 , H is a constant. When r( Thus, the matrix of T 1 can be written as follows From the explicit form of T 1 above, the recursive form of q 1 starting from q can be expressed as Remark 2 Compared with the classical DT for the Kaup-newell spectral problem [13], the Darboux matrix in our paper is more general and the determination of the the pending coefficients in the Darboux matrix depends on Lemma 1 rather than the symmetry relations.
Same as above, we can obtain the explicit form of T n , where According to the article [12], the recursive solution of q n can be get from the matrix form of T n above.
When n = 2k + 1, (24) (31) In the same way, the following relationship holds for order k , j (x, t) = 2i 2 j (x + 2 2 j t), j = 1, 2. Then for n = 3 , it is still a single-soliton solution. The structure of |q 2 | 2 is shown in Fig. 1. The first column is a standard left propagating single-soliton. The second column shows a stationary soliton whose amplitude changes with time. The third column shows a rightward propagating soliton-like solution with time-varying amplitude and the fourth column reveal a kink-shape solution. Figure 2 shows that, the two-soliton solutions are obtained from zero seed when n = 4 . The first two solitons will meet and then separate over time. And there is no change in the waveform before and after the encounter, which is in line with the characteristics of solitons. Taking special values of j , two kinds of periodic solutions are presented, as shown in Fig. 3.

Dark and Bright-Soliton Solutions and Breather Solution from Periodic Seed
Here, we will discuss the solutions generated from a periodic seed solution. Set a periodic seed According to Remark 1, H = e − 1 4 ic 2 (4aht+c 2 h 2 t−2hx) , h = e i(ax 0 +bt 0 ) . Solving the Lax pair (9) with this seed solution and separating variables, we get a solution corresponding to an eigenfunction as with In the condition of q = r(−x + x 0 , −t + t 0 ) , it is obviously that is also a solution of the Lax pair (9). To satisfy the Eq.
(38) we need to construct a new solution of the Lax pair. Let Obviously, this is a solution of the Lax pair (9). Next, we will use this eigenfunction to construct the new solutions of PT-DNLSII equation. Let n = 1 , = i , the first-order solution is obtained by Eq. (21). with , j = 1, 2.

Conclusion
The study of analytical solutions to nonlinear equations is one of the important issues studied by mathematical physicists in recent years. This article introduces the PT symmetric DNLSII equation studied in this article on the basis of the research significance and background. Firstly, the recursive form of the n-th order solution of the general DNLSII equation is derived with the help of the Darboux transformation matrix. Then, according to the constraints of PT symmetry, the conditions that the parameters of the solution need to meet are deduced, and the solution of the PTsymmetry DNLSII equation is obtained. Finally, starting from zero and non-zero solutions, the expressions of explicit solutions are given. It is found that the PT-symmetry DNLSII equation possesses abundant solution structures. Starting from the zero solution, the standard multi-soliton solution, the soliton-like solution, kink-shape solution and periodic solutions are obtained. Among them, the two-soliton solution have no molecular structure [14] because it will be reduced to a single soliton when the velocities of two solitons are the same. Starting from the periodic seed solution, bright soliton, dark soliton and breather solution are obtained. Based on the breather solution, the rogue wave [15] can be obtained by series expansion and taking the limit [16]. We will study this problem in detail in another paper.