Nonlocal-derivative NLS equations and group-invariant soliton solutions

*Correspondence: yyqinw@126.com 1Department of Applied Mathematics, China Agricultural University, Beijing, China Full list of author information is available at the end of the article Abstract A coupled Chen–Lee–Liu (CLL) system is proposed and its linear Lax pair is given. Many kinds of nonlocal-derivative NLS (DNLS) equations arise from the group symmetry reductions of the coupled CLL system. P̂T̂ Ĉ-symmetry invariant one-soliton solution and periodic two-soliton solution of a two-place DNLS (TDNLS) system are obtained. A group symmetry invariant two-soliton solution of a four-place DNLS (FDNLS) system is worked out. New characteristics of the two-soliton interactions for the TDNLS system and FDNLS system are analyzed.

With the advent of the nonlocal systems, the methods mentioned above have been developed to construct the soliton solutions of the nonlocal systems [49][50][51][52][53][54][55][56][57]. Meanwhile, there is few work about the soliton solutions of the nonlocal four-place systems. In this paper, new nonlocal two-place DNLS (TDNLS) and four-place DNLS (FDNLS) systems are derived based on the coupled Chen-Lee-Liu (CLL) system and thePTĈ-symmetry group. A linear Lax pair is given which guarantees the integrability of the nonlocal TDNLS system and FDNLS system. In order to construct the group-invariant soliton solutions, we first rewrite the solutions of the DNLS equation [58] in the form expressed by hyperbolic and triangular functions. Then thePTĈ-symmetry invariant one-soliton solution and periodic two-soliton solution of a new TDNLS system are obtained. Further, we also work out the group-invariant two-soliton solution of a FDNLS system. There is some interesting dynamics appearing in the TDNLS system and FDNLS system, different from the dynamics of the local DNLS equation.
The paper is organized as follows. In Sect. 2, we construct the coupled CLL system and its Lax pair is given. Some new nonlocal TDNLS system and FDNLS system arise from the group symmetry reductions of the coupled CLL system. In Sect. 3, the expressions of group-invariant soliton solutions for the nonlocal DNLS system are presented and the multi-soliton solutions of the TDNLS system and the FDNLS system are worked out. A conclusion is given in the last section.

Nonlocal multi-place derivative NLS system
The derivative NLS (DNLS) equation can be reduced from the Chen-Lee-Liu (CLL) system [59] q t = q xx + 2qrq x , by setting r = q * and replacing t by it and x by -ix. From the coupled system (4), some different kinds of nonlocal integrable DNLS equations can be obtained by using thePTĈsymmetry reductions. In the following, we first find some kinds of integrable coupled CLL systems. Here is the first non-trivial coupled CLL system q t = q xx + 2(p + q) 2 (r + s)(prqs) + 2q(r + s)(p x + q x ) + 2 (p + q)(qspr) x , It is obvious that the coupled CLL system (5) can be reduced to the standard CLL system if we take p = q and r = s. The integrability of the coupled CLL system (5) can be guaranteed by the following Lax pair: with n 11 n 12 0 0 n 21 -n 11 0 0 n 31 n 32 n 11 n 12 n 41 -n 31 n 21 -n 11 where The fullPTĈ-symmetry group Θ possesses the form [10] Θ = {1,P,TĈ,PTĈ} ∪Ĉ{1,P,TĈ,PTĈ} = Θ 1 ∪ Θ C 1 .

3PTĈ-invariant multi-soliton solutions of the DNLS type multi-place system
In Ref. [58], the bilinear form of the generalized DNLS equation is worked out and by taking the variable transformation q = g f , r = h s and making use of some identities. As a case of reduction, taking r = q * , i.e. s = f * , h = g * and replacing t by it and x by -ix, the generalized DNLS equation (16) reduces to the DNLS equation (3) and the bilinear equation (17) reduces to iD t + D 2 x g · f = 0, Equation (18) just is the bilinear equation of the the DNLS equation (3). Its N-soliton solutions can be uniformly written as where with arbitrary complex constants ξ (0) j , j = 1, 2, . . . , n. The summations A 1 (μ) and A 2 (μ) are taken over all possible combinations of μ j = 0, 1 (j = 1, 2, . . . , 2n) and satisfy the following conditions: It is clear that the solution (19) is notPTĈ-invariant for arbitrary ξ (0) j . So it is not the solution of the DNLS type multi-place system. In order to findPTĈ-invariant solutions from (19), we rewrite ξ j as We can prove that the solution (19) with (20) can be written as hyperbolic and triangular functions, which guarantees thePTĈ-invariance when an appropriate constant is chosen. However, the general expression in terms of the hyperbolic and triangular functions is very complicated. So we only write down two examples for n = 1 and n = 2.

3.1PTĈ-invariant solutions of a nonlocal two-place DNLS equation
In this section, we will give thePTĈ-invariant multi-soliton solutions of a new nonlocal two-place DNLS equation. Replacing t by it, x by -ix and q by 1 2 q in Eq. (12), we obtain the following nonlocal two-place DNLS equation: Setting q = q * (-x, -t), Eq. (23) can be reduced to the DNLS equation (3), so q| in Eq. (21) can solve the nonlocal two-place DNLS equation (23) withf =PT. Thus Eq. (23) has the following one-soliton solution: with This is a traveling wave at the speed of 2k 1R k 1I with an initial phase 1 2 arctant k 1I k 1R . Figure 1 shows the shape and motion of the one-soliton case for t = 0 and t = 1.
When take k j = ik jI (j = 1, 2) in Eq. (22), we can obtain the following two-soliton solutions of Eq. (23): Figure 1 Motion of the one-soliton for the nonlocal two-place DNLS equation (23). The green one is the density distributions |q| 2 for t = 0 and the red one is for t = 1 with k 1R = 1, k 1I = 2 (2020) 2020:126 Page 9 of 13

3.2PTĈ-invariant multi-soliton solutions of a nonlocal four-place DNLS equation
Replacing t by it, x by -ix and q by 1 2 q in the nonlocal four-place DNLS equation (12), it becomes In this section, we want to construct the soliton solutions of the nonlocal four-place DNLS equation (26). First, setting q = q * (x, -t) in Eq. (26), it reduces to Then setting q(-x, t) = q, Eq. (27) becomes the DNLS case (3). Thus, the two-soliton solution of the nonlocal four-place DNLS equation (26) can be obtained when we take k 2 = -k 1 in Eq. (22), with The interaction of the two-soliton solution (28) is illustrated in Fig. 3. The first column show the density plots of the two-soliton solution with k 1R = 1, k 1I = 1, k 1R = 3, k 1I = 1 and k 1R = 1, k 1I = 3, respectively. Pictures (a2)-(a4), (b2)-(b4) and (c2)-(c4) reveal the interaction process of the corresponding two-soliton for different choice of the real part and imparity part. Figure 3 reveals that the real part k 1R and imparity k 1I influence the interaction of the two-soliton solution.

Conclusions
Multi-place systems are important in both mathematical and physical fields. In this paper, we first construct the coupled CLL system and address its Lax pair which guarantees the integrability of the coupled CLL system. Then some kinds of nonlocal TDNLS equations and FDNLS equations are proposed by using thePTĈ-symmetry.
PTĈ-symmetry can be used not only to establish multi-place systems but also to solve the multi-place systems. With the help of thePTĈ-symmetry, we not only obtain the onesoliton solution and periodic two-soliton solution of a nonlocal TDNLS equation but also work out the two-soliton solution of a nonlocal FDNLS equation for the first time. It is interesting to find that the arbitrary constant in the real part of η j can influence the interaction process of the two-soliton for the TDNLS equation and new dynamical behaviors are analyzed in Fig. 2. For the FDNLS equation, it is interesting to find that the real part k jR and the imparity k jI of the parameter k j influence the interaction process of the two-soliton and the dynamics as demonstrated in Fig. 3.
From the results of this paper, we find that there are some new interesting phenomena in the nonlocal multi-place systems. So it is significant to study the nonlocal multi-place systems.