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Darboux transformation and soliton solutions in the parity-time-symmetric nonlocal vector nonlinear Schrödinger equation
Accepted Manuscript Darboux transformation and soliton solutions in the parity-time-symmetric nonlocal vector nonlinear Schr¨odinger equation
Hai-Qiang Zhang, Meng-Yue Zhang, Rui Hu
PII: DOI: Reference:
S0893-9659(17)30280-X http://dx.doi.org/10.1016/j.aml.2017.09.002 AML 5329
To appear in:
Applied Mathematics Letters
Received date : 10 July 2017 Revised date : 5 September 2017 Accepted date : 5 September 2017 Please cite this article as: H.-Q. Zhang, M. Zhang, R. Hu, Darboux transformation and soliton solutions in the parity-time-symmetric nonlocal vector nonlinear Schr¨odinger equation, Appl. Math. Lett. (2017), http://dx.doi.org/10.1016/j.aml.2017.09.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Darboux transformation and soliton solutions in the parity-time-symmetric nonlocal vector nonlinear Schr¨odinger equation Hai-Qiang Zhang ∗, Meng-Yue Zhang and Rui Hu College of Science, P. O. Box 253, University of Shanghai for Science and Technology, Shanghai 200093, China Abstract In this Letter, we study a nonlocal vector nonlinear Schr¨odinger (NVNLS) equation with self-induced parity-time-symmetric potential. We construct the N -fold Darboux transformation in terms of compact determinant forms. Starting from the nonvanishing background, we give the general solution of spectral problem, which allows us to derive many different types of exact analytical solutions of the NVNLS equation, like the breathers, dark and anti-dark solitons. With three-component case as an example, we display three types of two-soliton elastic collision behaviors: breather and dark soliton, breather and anti-dark soliton, dark soliton and anti-dark soliton. Keywords: Darboux transformation, Solitons, Nonlocal vector nonlinear Schr¨odinger equation
∗
Corresponding author, with e-mail address as [email protected]
1
1. Introduction Recently, a new nonlocal nonlinear Schr¨odinger (NLS) equation was proposed by Ablowitz and Musslimani [1] iqz (x, z) = qxx (x, z) + 2σq(x, z)q ∗ (−x, z)q(x, z),
(1)
where q(x, z) is a complex valued function of the real variables x and z, the star stands for the complex conjugation, and σ = ±1 correspond to the self-focusing case σ = 1 and defocusing case σ = −1, respectively. In Eq. (1), the nonlinear term has a self-induced nonlinear potential V (x, z) = q(x, z)q ∗ (−x, z) satisfying the parity-time-symmetric condition V (x, z) = V ∗ (−x, z). It has been shown that Eq. (1) is completely integrable in the sense that it can be solved by the inverse scattering transformation (IST) [1, 2]. The various aspects related to the integrability of Eq. (1) have been studied extensively, and a large number of exact solutions have been found, such as the dark and antidark soliton solutions [3, 4], periodic solutions [5] and rational soliton solutions [6–8]. The nonlocal NLS Eq. (1) can be easily extended to a multi-component case, the so-called vector nonlocal NLS equation [2, 9] iQz = Qxx + 2σQQP Q,
(2)
where Q = (q1 (x, z), q2 (x, z), . . . , qn (x, z)) and QP = (q1∗ (−x, z), q2∗ (−x, z), . . . , qn∗ (−x, z))T (the superscript T represents the vector transpose). Recently, Ref. [9] has studied the integrable properties for the two-component vector nonlocal NLS equation, and obtained the soliton solution by the IST method. In this Letter, we will first construct the N -fold Darboux transformation for Eq. (2) in terms of compact determinants. With non-vanishing background, we will give the general solution of spectral problem, and obtain many different types of exact analytical solutions of Eq. (2). With three-component case as an example, we will display three types of two-soliton elastic collision behaviors from the simplest case of the Darboux transformation. 2. Lax pair The Lax pair for Eq. (2) can be represented in the form ) ( λ Q Ψ, (3a) Ψx = −σ QP −λ In×n ) ( −2iλ2 − iσQQP −2iλQ − iQx Ψ, (3b) Ψz = 2iλ2 In×n + iσQP Q 2iσλ QP − iσQP x where Ψ = (ψ1 , ψ2 , · · · , ψn+1 )T is the vector eigenfunction, λ is the spectral parameter, In×n is the n × n identity matrix, and the compatibility condition Ψxz = Ψzx is exactly equivalent to Eq. (2). 2
It is well known that for a nonlinear partial differential equation the Lax pair assures the integrability and permits many applications of integrable properties [10], such as the conservation laws, the Darboux transformation and the B¨acklund transformation. On the basis of the Lax pair (3a) and (3b), one can further derive an infinite number of conservation laws of Eq. (2). Here, the first few conserved quantities associated with Eq. (2) are given by ∫ +∞ C0 = QQP dx, (4) −∞ ∫ +∞ P C1 = QQP (5) x + Qx Q dx, −∞ ∫ +∞ ( ) P 2 C2 = Qx QP dx. (6) x − σ QQ −∞
In the context of parity-time symmetric classical optics, the quantity C0 is referred to as the “quasipower”, and C2 is the Hamiltonian quantity. 3. Darboux transformation The Darboux transformation method is a very effective tool in soliton theory to construct wide classes of exact solutions of integrable nonlinear equations [11–15]. In this section, we will construct the Darboux transformation for Eq. (2) with σ = −1. We assume the N -fold eigenfunction transformation on the Lax pair (3a) and (3b) to be of the form N∑ −1 N∑ −1 N∑ −1 j j j aj λ b1,j λ · · · bn,j λ j=0 j=0 j=0 N∑ N∑ −1 N∑ −1 −1 (1) (1) j j j c λ d λ · · · d λ 1,j 1,j n,j , (7) j=0 j=0 j=0 Ψ[N ] = D[N ]Ψ, D[N ] = λN In×n − .. .. .. .. . . . . N∑ N∑ −1 −1 N∑ −1 (n) j (n) j j dn,j λ cn,j λ d1,j λ · · · j=0
j=0
j=0
where Ψ[N ] is the N -fold eigenfunction that satisfies the same linear Eqs. (3a) and (3b), and (m) the functions aj (x, z), bk,j (x, z), ck,j (x, z), dk,j (x, z) (1 6 k, m 6 n; 0 6 j 6 N − 1) can be uniquely determined by requiring that D[N ]|λ=λj Ψj = 0, (1)
(m)
D[N ]|λ=λ∗j Φj
= 0,
(j, m = 1, 2, . . . , n)
(8)
(n)
where Ψj = (fj (x, z), gj (x, z), · · · gj (x, z))T is the solution for Lax pair (3a) and (3b) with (m) Φj
m−1 n−m z }| { z }| { (m)∗ (−gj (−x, z), 0, · · · , 0, fj∗ (−x, z), 0, · · · , 0)
= λ = λj , and are all orthogonal to Ψj . Moreover, from the invariance of Lax pair (3a) and (3b), we can further derive the N -fold potential transformation: qj [N ](x, z) = qj (x, z) + 2bj,N −1 = qj (x, z) + 2(−1)jN −1 3
τj , τ
(j = 1, 2, . . . , n)
(9)
with (1) F GN N (1)∗ FN∗ −GN .. .. . . τ = (j)∗ −GN 0 . .. .. . (n)∗ −GN 0
(j)
(n)
· · · GN ··· 0 .. .. . . · · · FN∗ .. .. . .
· · · GN ··· 0 .. .. . . ··· 0 .. .. . .
···
···
0
FN∗
where
(1) F GN N +1 (1)∗ −GN +1 FN∗ .. .. . . , τj = (j)∗ −GN +1 0 .. .. . . (n)∗ −GN +1 0
(j)
(n)
· · · GN −1 · · · GN ··· 0 ··· 0 .. .. .. .. . . . . ∗ · · · FN −1 · · · 0 .. . .. .. .. . . . ··· 0 · · · FN∗
,
[ ] [ ∗ m−1 ∗ ] ∗ FN = λm−1 f (x, z) , F = (λ ) f (−x, z) , j N j j j 16j,m6N 16j,m6N ] [ ] [ (k) (k) (k)∗ ∗(k) GN = λm−1 gj (x, z) 16k6n , GN = (λ∗j )m−1 gj (−x, z) 16k6n . j 16j,m6N
16j,m6N
Therefore, the N -fold Darboux transformation for Eq. (2) is comprised of the eigenfunction transformation (7) and potential transformation (9). Then, the wide classes of exact solutions of Eq. (2) can be derived by virtue of the potential transformation (9) from different backgrounds. 3. Soliton solutions from non-vanishing background It is easy to find that Eq. (2) has the following plane wave solution 2i
qj (x, z) = ρj e
n ∑
ρ2l z
l=1
,
(j = 1, 2, . . . , n)
(10)
where ρj ’s are all real parameters. Substituting the above seed solution into the Lax pair (3a) and (3b) with λ = λj , we obtain the general solution ϕj
fj = βj e
φj
+ γj e
(k) , gj
−2iz
=e
n ∑
l=1
where θj = −λj x + (n)
αj
=−
n−1 ∑ k=1
2iωj2 z,
ρ2l
(
) (k) (k) (k) αj eθj + βj eϕj + γj eφj , (j, k = 1, 2, . . . , n)
ϕj = ωj x + iz
n ∑ k=1
(k) ρk
αj
ρn
,
(k)
βj
=
βj ρk , λj + ωj
ρ2k
− 2iλj ωj z, φj = −ωj x + iz (k)
γj
=
γj ρk , λj − ωj
n ∑
(11)
ρ2k + 2iλj ωj z,
k=1
v u n ∑ u 2 t ωj = λj + ρ2k . k=1
With the substitution of solution (11), the expression (9) for N = 1 can describe three different types of two-soliton collision behaviors: breather and dark soliton, breather and anti-dark soliton, dark soliton and anti-dark soliton. Through the analysis of the expres(k) sion (9), it is found that when the parameters αj = 0 (1 6 j 6 n, 1 6 k 6 n − 1), the expression (9) only can describe the elastic collisions between dark and anti-dark solitons, 4
(k)
while αj ̸= 0, the expression (9) can exhibit the collisions between breathers and dark or anti-dark solitons. With three-component case as an example, Figs. 1-3 display the above three types of elastic collision behaviors. In each component, they (breathers, dark and antidark solitons) undergo the standard of elastic interactions and their respective amplitudes and velocities are the same as those before collision.
(a)
(b)
(c)
Fig. 1. Elastic collision between a breather and an anti-dark soliton via solution (9) with N = 1. The parameters are α1 = α2 = γ1 = 1, β1 = 2i, λ1 = i, ρ1 = 0.4, ρ2 = 0.9 and ρ3 = 0.6.
(a)
(b)
(c)
Fig. 2. Elastic collision between a breather and a dark soliton via solution (9) with N = 1. The parameters are α1 = α2 = 1, γ1 = −2, β1 = 2i, λ1 = i, ρ1 = 0.4, ρ2 = 1 and ρ3 = 0.8.
(a)
(b)
(c)
Fig. 3. Elastic collision between a dark soliton and an anti-dark soliton via solution (9) with N = 1. The parameters are α1 = α2 = 0, γ1 = 1, β1 = 2i, λ1 = i, ρ1 = 0.4, ρ2 = 1 and ρ3 = 0.6.
4. Concluding remarks In this Letter, we have constructed the Darboux transformation for a nonlocal vector nonlinear Schr¨odinger equation with self-induced parity-time-symmetric potential. We have presented the explicit determinant representation for N -fold iterative transformation. With three-component case as an example, we have displayed three types of two-soliton elastic 5
collision behaviors: breather and dark soliton, breather and anti-dark soliton, dark soliton and anti-dark soliton. Finally, it is also interesting to explore the rational soliton solutions of Eq. (2) by the generalized Darboux transformation, which will be published elsewhere. Acknowledgments This work is supported by the Shanghai Leading Academic Discipline Project under Grant No. XTKX2012, by the Technology Research and Development Program of University of Shanghai for Science and Technology under Grant No. 2017KJFZ122 and by Hujiang Foundation of China under Grant No. B14005. H. Q. Zhang also thanks the support by Young Scholars’ Visiting Program sponsored by Shanghai Municipal Education Commission.
References [1] M. J. Ablowitz, Z. H. Musslimani, Phys. Rev. Lett. 110 (2013) 064105. [2] M. J. Ablowitz, Z. H. Musslimani, Stud. Appl. Math. (2016) DOI: 10.1111/sapm.12153. [3] A. K. Sarma, M.-A. Miri, Z. H. Musslimani, D. N. Christodoulides, Phys. Rev. E 89 (2014) 052918. [4] M. Li, T. Xu, Phys. Rev. E 91 (2015) 033202. [5] A. Khare, A. Saxena, J. Math. Phys. 56 (2015) 032104. [6] M. Li, T. Xu, D. X. Meng, J. Phys. Soc. Jpn. 85 (2016) 124001. [7] X. Y. Wen, Z. Y. Yan, Y. Yang, Chaos 26 (2016) 063123. [8] G. Q. Zhang, Z. Y. Yan, Y. Chen, App. Math. Lett. 69 (2017) 113. [9] D. Sinha, P. K. Ghosh, Phys. Lett. A 381 (2017) 124. [10] M. J. Ablowitz, B. Prinari, A. D. Trubatch, Discrete and Continuous Nonlinear Schr¨odinger Systems, Cambridge Univ. Press, Cambridge, 2004. [11] V. B. Matveev, M. A. Salle, Darboux Transformations and Solitons, Springer Press, Berlin, 1991. [12] H. Q. Zhang, B. Tian, T. Xu, H. Li, C. Zhang, H. Zhang, J. Phys. A: Math. Theor. 41 (2008) 355210. [13] T. Xu, B. Tian, J. Math. Phys. 51 (2010) 033504. [14] H. Q. Zhang, R. Hu, M. Y. Zhang, Applied Math. Lett. 69 (2017) 101. [15] H. Q. Hao, R. Guo, J. W. Zhang, Nonlinear Dyn. 88 (2017) 1615; X. J. Zhao, R. Guo, H. Q. Hao, Applied Math. Lett. 75 (2018) 114; R. Guo, H. Q. Hao, Commun. Nonlinear Sci. Numer. Simulat. 18 (2013) 2426. 6
Hai-Qiang Zhang, Meng-Yue Zhang, Rui Hu
PII: DOI: Reference:
S0893-9659(17)30280-X http://dx.doi.org/10.1016/j.aml.2017.09.002 AML 5329
To appear in:
Applied Mathematics Letters
Received date : 10 July 2017 Revised date : 5 September 2017 Accepted date : 5 September 2017 Please cite this article as: H.-Q. Zhang, M. Zhang, R. Hu, Darboux transformation and soliton solutions in the parity-time-symmetric nonlocal vector nonlinear Schr¨odinger equation, Appl. Math. Lett. (2017), http://dx.doi.org/10.1016/j.aml.2017.09.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Darboux transformation and soliton solutions in the parity-time-symmetric nonlocal vector nonlinear Schr¨odinger equation Hai-Qiang Zhang ∗, Meng-Yue Zhang and Rui Hu College of Science, P. O. Box 253, University of Shanghai for Science and Technology, Shanghai 200093, China Abstract In this Letter, we study a nonlocal vector nonlinear Schr¨odinger (NVNLS) equation with self-induced parity-time-symmetric potential. We construct the N -fold Darboux transformation in terms of compact determinant forms. Starting from the nonvanishing background, we give the general solution of spectral problem, which allows us to derive many different types of exact analytical solutions of the NVNLS equation, like the breathers, dark and anti-dark solitons. With three-component case as an example, we display three types of two-soliton elastic collision behaviors: breather and dark soliton, breather and anti-dark soliton, dark soliton and anti-dark soliton. Keywords: Darboux transformation, Solitons, Nonlocal vector nonlinear Schr¨odinger equation
∗
Corresponding author, with e-mail address as [email protected]
1
1. Introduction Recently, a new nonlocal nonlinear Schr¨odinger (NLS) equation was proposed by Ablowitz and Musslimani [1] iqz (x, z) = qxx (x, z) + 2σq(x, z)q ∗ (−x, z)q(x, z),
(1)
where q(x, z) is a complex valued function of the real variables x and z, the star stands for the complex conjugation, and σ = ±1 correspond to the self-focusing case σ = 1 and defocusing case σ = −1, respectively. In Eq. (1), the nonlinear term has a self-induced nonlinear potential V (x, z) = q(x, z)q ∗ (−x, z) satisfying the parity-time-symmetric condition V (x, z) = V ∗ (−x, z). It has been shown that Eq. (1) is completely integrable in the sense that it can be solved by the inverse scattering transformation (IST) [1, 2]. The various aspects related to the integrability of Eq. (1) have been studied extensively, and a large number of exact solutions have been found, such as the dark and antidark soliton solutions [3, 4], periodic solutions [5] and rational soliton solutions [6–8]. The nonlocal NLS Eq. (1) can be easily extended to a multi-component case, the so-called vector nonlocal NLS equation [2, 9] iQz = Qxx + 2σQQP Q,
(2)
where Q = (q1 (x, z), q2 (x, z), . . . , qn (x, z)) and QP = (q1∗ (−x, z), q2∗ (−x, z), . . . , qn∗ (−x, z))T (the superscript T represents the vector transpose). Recently, Ref. [9] has studied the integrable properties for the two-component vector nonlocal NLS equation, and obtained the soliton solution by the IST method. In this Letter, we will first construct the N -fold Darboux transformation for Eq. (2) in terms of compact determinants. With non-vanishing background, we will give the general solution of spectral problem, and obtain many different types of exact analytical solutions of Eq. (2). With three-component case as an example, we will display three types of two-soliton elastic collision behaviors from the simplest case of the Darboux transformation. 2. Lax pair The Lax pair for Eq. (2) can be represented in the form ) ( λ Q Ψ, (3a) Ψx = −σ QP −λ In×n ) ( −2iλ2 − iσQQP −2iλQ − iQx Ψ, (3b) Ψz = 2iλ2 In×n + iσQP Q 2iσλ QP − iσQP x where Ψ = (ψ1 , ψ2 , · · · , ψn+1 )T is the vector eigenfunction, λ is the spectral parameter, In×n is the n × n identity matrix, and the compatibility condition Ψxz = Ψzx is exactly equivalent to Eq. (2). 2
It is well known that for a nonlinear partial differential equation the Lax pair assures the integrability and permits many applications of integrable properties [10], such as the conservation laws, the Darboux transformation and the B¨acklund transformation. On the basis of the Lax pair (3a) and (3b), one can further derive an infinite number of conservation laws of Eq. (2). Here, the first few conserved quantities associated with Eq. (2) are given by ∫ +∞ C0 = QQP dx, (4) −∞ ∫ +∞ P C1 = QQP (5) x + Qx Q dx, −∞ ∫ +∞ ( ) P 2 C2 = Qx QP dx. (6) x − σ QQ −∞
In the context of parity-time symmetric classical optics, the quantity C0 is referred to as the “quasipower”, and C2 is the Hamiltonian quantity. 3. Darboux transformation The Darboux transformation method is a very effective tool in soliton theory to construct wide classes of exact solutions of integrable nonlinear equations [11–15]. In this section, we will construct the Darboux transformation for Eq. (2) with σ = −1. We assume the N -fold eigenfunction transformation on the Lax pair (3a) and (3b) to be of the form N∑ −1 N∑ −1 N∑ −1 j j j aj λ b1,j λ · · · bn,j λ j=0 j=0 j=0 N∑ N∑ −1 N∑ −1 −1 (1) (1) j j j c λ d λ · · · d λ 1,j 1,j n,j , (7) j=0 j=0 j=0 Ψ[N ] = D[N ]Ψ, D[N ] = λN In×n − .. .. .. .. . . . . N∑ N∑ −1 −1 N∑ −1 (n) j (n) j j dn,j λ cn,j λ d1,j λ · · · j=0
j=0
j=0
where Ψ[N ] is the N -fold eigenfunction that satisfies the same linear Eqs. (3a) and (3b), and (m) the functions aj (x, z), bk,j (x, z), ck,j (x, z), dk,j (x, z) (1 6 k, m 6 n; 0 6 j 6 N − 1) can be uniquely determined by requiring that D[N ]|λ=λj Ψj = 0, (1)
(m)
D[N ]|λ=λ∗j Φj
= 0,
(j, m = 1, 2, . . . , n)
(8)
(n)
where Ψj = (fj (x, z), gj (x, z), · · · gj (x, z))T is the solution for Lax pair (3a) and (3b) with (m) Φj
m−1 n−m z }| { z }| { (m)∗ (−gj (−x, z), 0, · · · , 0, fj∗ (−x, z), 0, · · · , 0)
= λ = λj , and are all orthogonal to Ψj . Moreover, from the invariance of Lax pair (3a) and (3b), we can further derive the N -fold potential transformation: qj [N ](x, z) = qj (x, z) + 2bj,N −1 = qj (x, z) + 2(−1)jN −1 3
τj , τ
(j = 1, 2, . . . , n)
(9)
with (1) F GN N (1)∗ FN∗ −GN .. .. . . τ = (j)∗ −GN 0 . .. .. . (n)∗ −GN 0
(j)
(n)
· · · GN ··· 0 .. .. . . · · · FN∗ .. .. . .
· · · GN ··· 0 .. .. . . ··· 0 .. .. . .
···
···
0
FN∗
where
(1) F GN N +1 (1)∗ −GN +1 FN∗ .. .. . . , τj = (j)∗ −GN +1 0 .. .. . . (n)∗ −GN +1 0
(j)
(n)
· · · GN −1 · · · GN ··· 0 ··· 0 .. .. .. .. . . . . ∗ · · · FN −1 · · · 0 .. . .. .. .. . . . ··· 0 · · · FN∗
,
[ ] [ ∗ m−1 ∗ ] ∗ FN = λm−1 f (x, z) , F = (λ ) f (−x, z) , j N j j j 16j,m6N 16j,m6N ] [ ] [ (k) (k) (k)∗ ∗(k) GN = λm−1 gj (x, z) 16k6n , GN = (λ∗j )m−1 gj (−x, z) 16k6n . j 16j,m6N
16j,m6N
Therefore, the N -fold Darboux transformation for Eq. (2) is comprised of the eigenfunction transformation (7) and potential transformation (9). Then, the wide classes of exact solutions of Eq. (2) can be derived by virtue of the potential transformation (9) from different backgrounds. 3. Soliton solutions from non-vanishing background It is easy to find that Eq. (2) has the following plane wave solution 2i
qj (x, z) = ρj e
n ∑
ρ2l z
l=1
,
(j = 1, 2, . . . , n)
(10)
where ρj ’s are all real parameters. Substituting the above seed solution into the Lax pair (3a) and (3b) with λ = λj , we obtain the general solution ϕj
fj = βj e
φj
+ γj e
(k) , gj
−2iz
=e
n ∑
l=1
where θj = −λj x + (n)
αj
=−
n−1 ∑ k=1
2iωj2 z,
ρ2l
(
) (k) (k) (k) αj eθj + βj eϕj + γj eφj , (j, k = 1, 2, . . . , n)
ϕj = ωj x + iz
n ∑ k=1
(k) ρk
αj
ρn
,
(k)
βj
=
βj ρk , λj + ωj
ρ2k
− 2iλj ωj z, φj = −ωj x + iz (k)
γj
=
γj ρk , λj − ωj
n ∑
(11)
ρ2k + 2iλj ωj z,
k=1
v u n ∑ u 2 t ωj = λj + ρ2k . k=1
With the substitution of solution (11), the expression (9) for N = 1 can describe three different types of two-soliton collision behaviors: breather and dark soliton, breather and anti-dark soliton, dark soliton and anti-dark soliton. Through the analysis of the expres(k) sion (9), it is found that when the parameters αj = 0 (1 6 j 6 n, 1 6 k 6 n − 1), the expression (9) only can describe the elastic collisions between dark and anti-dark solitons, 4
(k)
while αj ̸= 0, the expression (9) can exhibit the collisions between breathers and dark or anti-dark solitons. With three-component case as an example, Figs. 1-3 display the above three types of elastic collision behaviors. In each component, they (breathers, dark and antidark solitons) undergo the standard of elastic interactions and their respective amplitudes and velocities are the same as those before collision.
(a)
(b)
(c)
Fig. 1. Elastic collision between a breather and an anti-dark soliton via solution (9) with N = 1. The parameters are α1 = α2 = γ1 = 1, β1 = 2i, λ1 = i, ρ1 = 0.4, ρ2 = 0.9 and ρ3 = 0.6.
(a)
(b)
(c)
Fig. 2. Elastic collision between a breather and a dark soliton via solution (9) with N = 1. The parameters are α1 = α2 = 1, γ1 = −2, β1 = 2i, λ1 = i, ρ1 = 0.4, ρ2 = 1 and ρ3 = 0.8.
(a)
(b)
(c)
Fig. 3. Elastic collision between a dark soliton and an anti-dark soliton via solution (9) with N = 1. The parameters are α1 = α2 = 0, γ1 = 1, β1 = 2i, λ1 = i, ρ1 = 0.4, ρ2 = 1 and ρ3 = 0.6.
4. Concluding remarks In this Letter, we have constructed the Darboux transformation for a nonlocal vector nonlinear Schr¨odinger equation with self-induced parity-time-symmetric potential. We have presented the explicit determinant representation for N -fold iterative transformation. With three-component case as an example, we have displayed three types of two-soliton elastic 5
collision behaviors: breather and dark soliton, breather and anti-dark soliton, dark soliton and anti-dark soliton. Finally, it is also interesting to explore the rational soliton solutions of Eq. (2) by the generalized Darboux transformation, which will be published elsewhere. Acknowledgments This work is supported by the Shanghai Leading Academic Discipline Project under Grant No. XTKX2012, by the Technology Research and Development Program of University of Shanghai for Science and Technology under Grant No. 2017KJFZ122 and by Hujiang Foundation of China under Grant No. B14005. H. Q. Zhang also thanks the support by Young Scholars’ Visiting Program sponsored by Shanghai Municipal Education Commission.
References [1] M. J. Ablowitz, Z. H. Musslimani, Phys. Rev. Lett. 110 (2013) 064105. [2] M. J. Ablowitz, Z. H. Musslimani, Stud. Appl. Math. (2016) DOI: 10.1111/sapm.12153. [3] A. K. Sarma, M.-A. Miri, Z. H. Musslimani, D. N. Christodoulides, Phys. Rev. E 89 (2014) 052918. [4] M. Li, T. Xu, Phys. Rev. E 91 (2015) 033202. [5] A. Khare, A. Saxena, J. Math. Phys. 56 (2015) 032104. [6] M. Li, T. Xu, D. X. Meng, J. Phys. Soc. Jpn. 85 (2016) 124001. [7] X. Y. Wen, Z. Y. Yan, Y. Yang, Chaos 26 (2016) 063123. [8] G. Q. Zhang, Z. Y. Yan, Y. Chen, App. Math. Lett. 69 (2017) 113. [9] D. Sinha, P. K. Ghosh, Phys. Lett. A 381 (2017) 124. [10] M. J. Ablowitz, B. Prinari, A. D. Trubatch, Discrete and Continuous Nonlinear Schr¨odinger Systems, Cambridge Univ. Press, Cambridge, 2004. [11] V. B. Matveev, M. A. Salle, Darboux Transformations and Solitons, Springer Press, Berlin, 1991. [12] H. Q. Zhang, B. Tian, T. Xu, H. Li, C. Zhang, H. Zhang, J. Phys. A: Math. Theor. 41 (2008) 355210. [13] T. Xu, B. Tian, J. Math. Phys. 51 (2010) 033504. [14] H. Q. Zhang, R. Hu, M. Y. Zhang, Applied Math. Lett. 69 (2017) 101. [15] H. Q. Hao, R. Guo, J. W. Zhang, Nonlinear Dyn. 88 (2017) 1615; X. J. Zhao, R. Guo, H. Q. Hao, Applied Math. Lett. 75 (2018) 114; R. Guo, H. Q. Hao, Commun. Nonlinear Sci. Numer. Simulat. 18 (2013) 2426. 6