- Email: [email protected]
Solutions of the nonlocal nonlinear Schrödinger hierarchy via reduction
Accepted Manuscript Solutions of the nonlocal nonlinear Schr¨odinger hierarchy via reduction
Kui Chen, Da-jun Zhang
PII: DOI: Reference:
S0893-9659(17)30184-2 http://dx.doi.org/10.1016/j.aml.2017.05.017 AML 5270
To appear in:
Applied Mathematics Letters
Received date : 27 April 2017 Accepted date : 29 May 2017 Please cite this article as: K. Chen, D. Zhang, Solutions of the nonlocal nonlinear Schr¨odinger hierarchy via reduction, Appl. Math. Lett. (2017), http://dx.doi.org/10.1016/j.aml.2017.05.017 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
Solutions of the nonlocal nonlinear Schr¨odinger hierarchy via reduction Kui Chen, Da-jun Zhang∗ Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
April 27, 2017 Abstract In this letter we propose an approach to obtain solutions for the nonlocal nonlinear Schr¨odinger hierarchy from the known ones of the Ablowitz-Kaup-Newell-Segur hierarchy by reduction. These solutions are presented in terms of double Wronskian and some of them are new. The approach is general and can be used for other systems with double Wronskian solutions which admit local and nonlocal reductions. MSC: 35Q51, 35Q55 Keywords: nonlocal nonlinear Schr¨ odinger hierarchy, AKNS hierarchy, double Wronskian, solutions, reduction
1
Introduction
Recently integrable continuous and semidiscrete nonlocal nonlinear Schr¨ odinger (NLS) equations describing wave propagation in nonlinear PT symmetric media were found [1, 2] and have drown much attention. The nonlocal NLS hierarchy [1] are derived from the Ablowitz-KaupNewell-Segur (AKNS) hierarchy through the reduction r(x, t) = ∓q ∗ (−x, t) where ∗ denotes complex conjugate. As for solutions, these nonlocal models have been solved through Inverse Scattering Transform, Darboux transformation, bilinear approach, etc. (cf. [3–12]). In this letter, we propose an approach to construct solutions for the nonlocal NLS hierarchy directly from the known solutions of the AKNS hierarchy. These solutions are presented in terms of double Wronskian. Wronskian technique [13] is an elegant way to construct and verify solutions for soliton equations. The double Wronskian solutions for the focusing NLS equation was first given in [14]. In 1990 Liu [15] showed the whole AKNS hierarchy admitted solutions in double Wronskian form. In this letter we directly apply the reduction to the solutions of the AKNS hierarchy obtained in [15] and construct solutions for the nonlocal focusing and defocusing NLS hierarchies. Some of solutions are new, which correspond to multiple poles of transmission coefficient. The letter is organized as follows. We will first in Sec.2 review known results on the AKNS hierarchy and their double Wronskian solutions. Then in Sec.3 we prove the condition under which double Wronskian solutions of the even order AKNS hierarchy are reduced to those of the nonlocal NLS hierarchies. In Sec.4 we present concrete Wronskian elements and some solutions. Finally, conclusion is given in Sec.5.
2
AKNS hierarchy and double Wronskian solutions
It is well known that the AKNS spectral problem [16] φ1 λ q φ1 = φ2 x r −λ φ2 ∗
Corresponding author. Email: [email protected]
1
(1)
yields the isospectral AKNS hierarchy qtn qtn+1 , (n ≥ 0) =L rtn rtn+1
(2)
with setting t1 = x and (qt0 , rt0 )T = (−q, r)T , where L is a recursion operator −∂x + 2q∂x−1 r 2q∂x−1 q L= , −2r∂x−1 r ∂x − 2r∂x−1 q in which ∂x =
∂ ∂x
(3)
and ∂x = ∂x−1 = ∂x−1 ∂x = 1. By introducing rational transformation g h q=2 , r=2 , f f
(4)
the hierarchy (2) can be written as the following bilinear form [17] (2Dtn+1 − Dx Dtn )g · f = 0,
(5a)
(2Dtn+1 − Dx Dtn )f · h = 0, Dx2 f
(5b)
· f = 8gh,
(5c)
for n ≥ 1, where Hirota’s bilinear operator D is defined as [18]
Dxm Dyn f (x, y) · g(x, y) = (∂x − ∂x0 )m (∂y − ∂y0 )n f (x, y)g(x0 , y 0 )|x0 =x,y0 =y .
System (5) admit double Wronskian solutions [15, 19] f = W n+1,m+1 (ϕ, ψ) = |ϕ b(n) ; ψb(m) |, g = W n+2,m (ϕ, ψ) = |ϕ b(n+1) ; ψb(m−1) |, h=W
n,m+2
(n−1)
(ϕ, ψ) = |ϕ b
; ψb(m+1) |.
(6a) (6b) (6c)
Here, ϕ and ψ are respectively (n + m + 2)-th order column vectors
ϕ = (ϕ1 , ϕ2 , · · · , ϕn+m+2 )T , ψ = (ψ1 , ψ2 , · · · , ψn+m+2 )T
(7)
defined by (with t1 = x) ϕ = exp
∞ X j=1
∞ X A tj C , ψ = exp − Aj tj C − , j
+
(8)
j=1
where A = (kij )(n+m+2)×(n+m+2) is a (n + m + 2) × (n + m + 2) complex constant matrix and ± ± T C ± are (n + m + 2)-th order complex constant column vectors C ± = (c± 1 , c2 , · · · , cn+m+2 ) ; ϕ b(n) and ψb(m) respectively denote (n + m + 2) × (n + 1) and (n + m + 2) × (m + 1) Wronski matrices ϕ b(n) = ϕ, ∂x ϕ, ∂x2 ϕ, · · · , ∂xn ϕ , ψb(m) = ψ, ∂x ψ, ∂x2 ψ, · · · , ∂xm ψ . (9)
Return to the AKNS hierarchy (2). For a concrete system in this hierarchy, for example, the second order AKNS system qt2 = −qxx + 2q 2 r, rt2 = rxx − 2r2 q,
(10)
ϕ = exp (Ax + A2 t2 )C + , ψ = exp (−Ax − A2 t2 )C − .
(11)
one can treat tj (j > 2) as dummy variables which are then absorbed into C ± and present their solutions through (4) and (6) where ϕ and ψ are defined as With the same idea, for the even order AKNS hierarchy qtn −q n = Kn = L , n = 2, 4, 6, · · · , rtn r
(12)
their solutions can be described through (4) and (6) where ∞ ∞ X X + 2j ϕ = exp Ax + A t2j C , ψ = exp −Ax − A2j t2j C − .
(13)
j=1
j=1
2
3
Reduction to the nonlocal NLS hierarchies
The nonlocal nonlinear Schr¨odinger hierarchies [1] iq(x)τ2k = Kk± [q(x), q(−x)], k = 1, 2, 3, . . .
(14)
can be reduced from the even order AKNS hierarchy (12) with the reductions r(x) = ±q ∗ (−x), t2k = iτ2k , (x, τ2k ∈ R),
(15)
where i2 = −1 and the ± in Kk± corresponds to the sign “±” in (15). For the case of k = 1, the nonlocal focusing NLS equation reads iq(x)τ2 = K1− = q(x)xx + 2q(x)2 q ∗ (−x),
(16)
and the nonlocal defocusing NLS equation reads iq(x)τ2 = K1+ = q(x)xx − 2q(x)2 q ∗ (−x).
(17)
In the following, we derive solutions for the nonlocal hierarchies (14) by imposing the reduction relations (15) on the solutions of the even order AKNS hierarchy (12). Theorem 1. The nonlocal isospectral NLS hierarchies (14) admit the following solutions q(x) = 2
|ϕ b(n+1) , ψb(n−1) | , |ϕ b(n) , ψb(n) |
(18)
where ϕ and ψ are (2n + 2)-th order column vectors (i.e. m = n in (7)), defined by (13) with t2k = iτ2k and satisfy ψ(x) = T ϕ∗ (−x), −
C = TC
+∗
(19a)
,
(19b)
in which T is a constant matrix determined through AT = T A∗ ,
(20a)
∗
T T = δI, δ = ∓1.
(20b)
Proof. We are going to prove that with the assumptions (19) and (20), q and r defined by (4) with double Wronskians (6) satisfy the reduction (15). To achieve that, first, under assumptions of (19b) and (20a), it is easy to verify that P ∗ −1 )2j τ )T C +∗ ψ(x) = exp (−(T A∗ T −1 )x − i ∞ 2j j=1 (T A T P∞ 2j + ∗ = T (exp (−Ax + i j=1 A τ2j )C ) = T ϕ∗ (−x). Next, to examine relation between Wronskians, based on (9), we introduce notation 2 n ϕ b(n) (ax)[bx] = ϕ(ax), ∂bx ϕ(ax), ∂bx ϕ(ax), · · · , ∂bx ϕ(ax)
and similar one for ψb(m) (ax)bx , where a, b = ±1. With such notations, setting n = m and noticing the constraint (19a), f, g, h in (6) are expressed as f (x) = |ϕ b(n) (x)[x] ; ψb(n) (x)[x] | = |ϕ b(n) (x)[x] ; T ϕ b∗(n) (−x)[x] |, g(x) = |ϕ b(n+1) (x)[x] ; ψb(n−1) (x)[x] | = |ϕ b(n+1) (x)[x] ; T ϕ b∗(n−1) (−x)[x] |, (n−1)
h(x) = |ϕ b
b(n+1)
(x)[x] ; ψ
(n−1)
(x)[x] | = |ϕ b 3
∗(n+1)
(x)[x] ; T ϕ b
(−x)[x] |.
(21a)
(21b) (21c)
By calculation we find f ∗ (−x) = |ϕ b(n) (−x)[−x] ; T ϕ b∗(n) (x)[−x] |∗
= δ n+1 |T ∗ ||T ϕ b∗(n) (−x)[−x] ; ϕ b(n) (x)[−x] | 2
= δ n+1 (−1)(n+1) |T ∗ ||ϕ b(n) (x)[x] ; T ϕ b∗(n) (−x)[x] | 2
= δ n+1 (−1)(n+1) |T ∗ |f (x),
where we have made use of ∂−x = −∂x and assumed T T ∗ = δI where δ is a constant and I is the unit matrix. In a similar way we have g ∗ (−x) = (−1)n(n+2) |T ∗ |δ n+2 h(x). With these relations, by making a comparison of q ∗ (−x) = 2g ∗ (−x)/f ∗ (−x) and r(x) = 2h(x)/f (x) one can find that r(x) = ±q ∗ (−x) when δ = ∓1. Thus we finish the proof. We note that to reduce (12) to solutions for the local NLS hierarchies, we need same reduction condition as in Theorem 1 except (20a) replaced with AT + T A∗ = 0.
4
Solutions of the nonlocal NLS hierarchies
In this section, we list Wronskian elements of solutions for the nonlocal NLS hierarchies (14).
4.1 4.1.1
Soliton solutions Focusing case
For the nonlocal focusing NLS hierarchy iq(x)τ2k = Kk− [q(x), q(−x)], k = 1, 2, 3, . . . , according to Theorem 1, we take δ = 1 and 0 I A1 0 , C + = (c1 , c2 , · · · , c2n+2 )T , , T = A= I 0 0 A∗1
(22)
(23a)
where A1 = Diag(k1 , k2 , · · · , kn+1 ), kj ∈ C and kj are distinct. The Wronskian elements are taken as Φ1 ϕ(x) = , ψ(x) = T ϕ∗ (−x), Φ2
(23b)
(24a)
where Φj = (ϕj,1 , ϕj,2 , · · · , ϕj,n+1 )T , (j = 1, 2), ξl (x,kl∗ )
ϕ1,l = eξl (x,kl ) , ϕ2,l = e
and ξl (x, kl ) = kl x + i
∞ X
,
kl2j τ2j .
(24b) (24c)
(24d)
j=1
The simplest solution to the nonlocal focusing NLS equation (16) is obtained by taking n = 0 in double Wronskian and the solution is given as 2
−4ic1 c2 Im(k1 )e2k1 x+i2k1 τ2 q1ss (x) = , |c1 |2 ei4Im(k1 )x e−8Re(k1 )Im(k1 )τ2 − |c2 |2
(25)
which is the same as the one obtained in [6] and [11], up to some scalar transformations. (25) is a nonsingular periodic solution of the case of Re(k1 ) = 0, Im(k1 ) 6= 0 and |c1 | = 6 |c2 |. 4
4.1.2
Defocusing case
For the nonlocal defocusing NLS hierarchy iq(x)τ2k = Kk+ [q(x), q(−x)], k = 1, 2, 3, . . . ,
(26)
from Theorem 1, we take A and C + as in Sec.4.1.1 but δ = −1 and 0 −I T = , I 0
(27)
where ϕ and ψ follow the structure (24) but with T defined as in (27). Solution to the nonlocal defocusing NLS equation (17) is (n = 0) 2
−4ic1 c2 Im(k1 )e2k1 x+i2k1 τ2 , q1ss (x) = |c1 |2 ei4Im(k1 )x e−8Re(k1 )Im(k1 )τ2 + |c2 |2
(28)
which is a nonsingular solution of the case of Re(k1 ) = 0, Im(k1 ) 6= 0 and |c1 | = 6 |c2 |.
4.2
Jordan block solutions
Jordan block solutions mean the solutions corresponding to the matrix A taking the form as in (23a) where A1 is a (n + 1) × (n + 1) Jordan block A1 = k1 I + σ, I = (δj,l )(n+1)×(n+1) , σ = (δj,l+1 )(n+1)×(n+1) . 4.2.1
(29)
Focusing case
For the nonlocal focusing NLS hierarchy (22), to get their solutions, we take δ = 1, T to be defined as in (23a), and C + = (C1 , C1 )T , C1 = (1, 1, · · · , 1)n+1 . The Wronskian elements take the form (24a) in which n h X 1 l ξ1 (x,k1 ) l i T ξ1 (x,k1 ) Φ1 (x, k1 ) = e + (∂ e )σ C1 , l! k1
h ∗ Φ2 (x, k1∗ ) = eξ1 (x,k1 ) +
l=1 n X l=1
with ξ1 (x, k1 ) = k1 x + i
(30a)
1 l ξ1 (x,k1∗ ) l i T (∂ ∗ e )σ C1 , l! k1
∞ X
(30b)
k12j τ2j .
(31)
j=1
For the nonlocal NLS equation (16), when n = 1, from (31) we have ξ1 (x, k1 ) = k1 x + ik12 τ2 , and the corresponding solution reads ∗
q(x) =
4(k1∗ − k1 )(e2ξ1 (x,k1 ) a(k1 , k1∗ ) − e2ξ1 (x,k1 ) a(k1∗ , k1 )) , ∗ ∗ e2ξ1 (x,k1 )−2ξ1 (x,k1 ) + e2ξ1 (x,k1 )−2ξ1 (x,k1 ) − b
(32)
with a(k1 , k1∗ ) = (k1∗ − k1 )x + 2ik1∗ (k1∗ − k1 )τ2 + 1, b = 4(k1 −
k1∗ )2 x2
+ 8i(k1 −
k1∗ )2 (k1
5
+
k1∗ )xτ2
(33a) −
16k1 k1∗ (k1
−
k1∗ )2 τ22
+ 2.
(33b)
4.2.2
Defocusing case
For the nonlocal defocusing NLS hierarchy (26), in Theorem 1 we take A, C, Φj same as in Sec.4.2.1, but δ = −1 and 0 −I T = . (34) I 0 For the nonlocal defocusing NLS equation (17), the simplest solution of Jordan block case
is
∗
4(k1∗ − k1 )(e2ξ1 (x,k1 ) a(k1 , k1∗ ) + e2ξ1 (x,k1 ) a(k1∗ , k1 )) , q(x) = ∗ ∗ e2ξ1 (x,k1 )−2ξ1 (x,k1 ) + e2ξ1 (x,k1 )−2ξ1 (x,k1 ) + b where a(k1 , k1∗ ) and b are denoted by (33).
5
(35)
Conclusions
In this letter by direct reduction we have derived doubled Wronskian solutions of the nonlocal NLS hierarchies from the known ones of the even-order AKNS hierarchy. Compared with the double Wronskian solution obtained in [7] via Darboux transformation where all {kj } are pure imaginary, here in our solutions all {kj } are complex, which admits more freedom. Besides, we also obtain Jordan block solutions which correspond to multiple-pole case of the transmission coefficient. Since equation (20) admits block diagonal extension, it is easy to extend our results to case where A is a block-diagonal composed by diagonal matrix and Jordan blocks. The reduction procedure developed in this letter is general and can apply to other systems which have double Wronskian solutions and admit local and nonlocal reductions.
Acknowledgments This work was supported by the NSF of China [grant numbers 11371241, 11631007].
References [1] M.J. Ablowitz, Z.H. Musslimani, Phys. Rev. Lett., 110 (2013) No.064105 (5pp). [2] M.J. Ablowitz, Z.H. Musslimani, Phys. Rev. E, 90 (2014) No.032912 (5pp). [3] M.J. Ablowitz, Z.H. Musslimani, Nonlinearity, 29 (2016) 915-946. [4] A.K. Sarma, M.A. Miri, Z.H. Musslimani, Phys. Rev. E., 89 (2014) No.052918 (7pp). [5] L.Y. Ma, Z.N. Zhu, Appl. Math. Lett., 59 (2016) 115-121. [6] L.Y. Ma, Z.N. Zhu, J. Math. Phys., 57 (2016) No.083507 (20pp). [7] M. Li, T. Xu, Phys. Rev. E, 91 (2015) No.033202 (8pp). [8] D. Sinha, P.K. Ghosh, Phys. Rev. E, 91 (2015) No.042908 (14pp). [9] A. Khare, A. Saxena, J. Math. Phys., 56 (2015) No.032104 (27pp). [10] Z. Yan, Z. Wen, C. Hang, Phys. Rev. E, 92 (2015) No.022913 (10pp). [11] X. Huang, L. Ling, Euro. Phys. J. Plus, 131 (2016) No.16148 (11pp). [12] Z.X. Zhou, arXiv:1612.04892. [13] N.C.Freeman, J.J.C. Nimmo, Phys. Lett. A, 95 (1983) 1-3. [14] J.J.C. Nimmo, Phys. Lett. A, 99 (1983) 279-280. [15] Q.M. Liu, J. Phys. Soc. Jpn., 59 (1990) 3520-3527. [16] M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Phys. Rev. Lett., 31 (1973) 125-127. [17] A.C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphin, 1985. [18] R. Hirota, Prog. Theor. Phys., 52 (1974) 1498-1512. [19] F.M. Yin, Y.P. Sun, F.Q. Cai, Comm. Theor. Phys., 49 (2008) 401-408.
6
Kui Chen, Da-jun Zhang
PII: DOI: Reference:
S0893-9659(17)30184-2 http://dx.doi.org/10.1016/j.aml.2017.05.017 AML 5270
To appear in:
Applied Mathematics Letters
Received date : 27 April 2017 Accepted date : 29 May 2017 Please cite this article as: K. Chen, D. Zhang, Solutions of the nonlocal nonlinear Schr¨odinger hierarchy via reduction, Appl. Math. Lett. (2017), http://dx.doi.org/10.1016/j.aml.2017.05.017 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
Solutions of the nonlocal nonlinear Schr¨odinger hierarchy via reduction Kui Chen, Da-jun Zhang∗ Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
April 27, 2017 Abstract In this letter we propose an approach to obtain solutions for the nonlocal nonlinear Schr¨odinger hierarchy from the known ones of the Ablowitz-Kaup-Newell-Segur hierarchy by reduction. These solutions are presented in terms of double Wronskian and some of them are new. The approach is general and can be used for other systems with double Wronskian solutions which admit local and nonlocal reductions. MSC: 35Q51, 35Q55 Keywords: nonlocal nonlinear Schr¨ odinger hierarchy, AKNS hierarchy, double Wronskian, solutions, reduction
1
Introduction
Recently integrable continuous and semidiscrete nonlocal nonlinear Schr¨ odinger (NLS) equations describing wave propagation in nonlinear PT symmetric media were found [1, 2] and have drown much attention. The nonlocal NLS hierarchy [1] are derived from the Ablowitz-KaupNewell-Segur (AKNS) hierarchy through the reduction r(x, t) = ∓q ∗ (−x, t) where ∗ denotes complex conjugate. As for solutions, these nonlocal models have been solved through Inverse Scattering Transform, Darboux transformation, bilinear approach, etc. (cf. [3–12]). In this letter, we propose an approach to construct solutions for the nonlocal NLS hierarchy directly from the known solutions of the AKNS hierarchy. These solutions are presented in terms of double Wronskian. Wronskian technique [13] is an elegant way to construct and verify solutions for soliton equations. The double Wronskian solutions for the focusing NLS equation was first given in [14]. In 1990 Liu [15] showed the whole AKNS hierarchy admitted solutions in double Wronskian form. In this letter we directly apply the reduction to the solutions of the AKNS hierarchy obtained in [15] and construct solutions for the nonlocal focusing and defocusing NLS hierarchies. Some of solutions are new, which correspond to multiple poles of transmission coefficient. The letter is organized as follows. We will first in Sec.2 review known results on the AKNS hierarchy and their double Wronskian solutions. Then in Sec.3 we prove the condition under which double Wronskian solutions of the even order AKNS hierarchy are reduced to those of the nonlocal NLS hierarchies. In Sec.4 we present concrete Wronskian elements and some solutions. Finally, conclusion is given in Sec.5.
2
AKNS hierarchy and double Wronskian solutions
It is well known that the AKNS spectral problem [16] φ1 λ q φ1 = φ2 x r −λ φ2 ∗
Corresponding author. Email: [email protected]
1
(1)
yields the isospectral AKNS hierarchy qtn qtn+1 , (n ≥ 0) =L rtn rtn+1
(2)
with setting t1 = x and (qt0 , rt0 )T = (−q, r)T , where L is a recursion operator −∂x + 2q∂x−1 r 2q∂x−1 q L= , −2r∂x−1 r ∂x − 2r∂x−1 q in which ∂x =
∂ ∂x
(3)
and ∂x = ∂x−1 = ∂x−1 ∂x = 1. By introducing rational transformation g h q=2 , r=2 , f f
(4)
the hierarchy (2) can be written as the following bilinear form [17] (2Dtn+1 − Dx Dtn )g · f = 0,
(5a)
(2Dtn+1 − Dx Dtn )f · h = 0, Dx2 f
(5b)
· f = 8gh,
(5c)
for n ≥ 1, where Hirota’s bilinear operator D is defined as [18]
Dxm Dyn f (x, y) · g(x, y) = (∂x − ∂x0 )m (∂y − ∂y0 )n f (x, y)g(x0 , y 0 )|x0 =x,y0 =y .
System (5) admit double Wronskian solutions [15, 19] f = W n+1,m+1 (ϕ, ψ) = |ϕ b(n) ; ψb(m) |, g = W n+2,m (ϕ, ψ) = |ϕ b(n+1) ; ψb(m−1) |, h=W
n,m+2
(n−1)
(ϕ, ψ) = |ϕ b
; ψb(m+1) |.
(6a) (6b) (6c)
Here, ϕ and ψ are respectively (n + m + 2)-th order column vectors
ϕ = (ϕ1 , ϕ2 , · · · , ϕn+m+2 )T , ψ = (ψ1 , ψ2 , · · · , ψn+m+2 )T
(7)
defined by (with t1 = x) ϕ = exp
∞ X j=1
∞ X A tj C , ψ = exp − Aj tj C − , j
+
(8)
j=1
where A = (kij )(n+m+2)×(n+m+2) is a (n + m + 2) × (n + m + 2) complex constant matrix and ± ± T C ± are (n + m + 2)-th order complex constant column vectors C ± = (c± 1 , c2 , · · · , cn+m+2 ) ; ϕ b(n) and ψb(m) respectively denote (n + m + 2) × (n + 1) and (n + m + 2) × (m + 1) Wronski matrices ϕ b(n) = ϕ, ∂x ϕ, ∂x2 ϕ, · · · , ∂xn ϕ , ψb(m) = ψ, ∂x ψ, ∂x2 ψ, · · · , ∂xm ψ . (9)
Return to the AKNS hierarchy (2). For a concrete system in this hierarchy, for example, the second order AKNS system qt2 = −qxx + 2q 2 r, rt2 = rxx − 2r2 q,
(10)
ϕ = exp (Ax + A2 t2 )C + , ψ = exp (−Ax − A2 t2 )C − .
(11)
one can treat tj (j > 2) as dummy variables which are then absorbed into C ± and present their solutions through (4) and (6) where ϕ and ψ are defined as With the same idea, for the even order AKNS hierarchy qtn −q n = Kn = L , n = 2, 4, 6, · · · , rtn r
(12)
their solutions can be described through (4) and (6) where ∞ ∞ X X + 2j ϕ = exp Ax + A t2j C , ψ = exp −Ax − A2j t2j C − .
(13)
j=1
j=1
2
3
Reduction to the nonlocal NLS hierarchies
The nonlocal nonlinear Schr¨odinger hierarchies [1] iq(x)τ2k = Kk± [q(x), q(−x)], k = 1, 2, 3, . . .
(14)
can be reduced from the even order AKNS hierarchy (12) with the reductions r(x) = ±q ∗ (−x), t2k = iτ2k , (x, τ2k ∈ R),
(15)
where i2 = −1 and the ± in Kk± corresponds to the sign “±” in (15). For the case of k = 1, the nonlocal focusing NLS equation reads iq(x)τ2 = K1− = q(x)xx + 2q(x)2 q ∗ (−x),
(16)
and the nonlocal defocusing NLS equation reads iq(x)τ2 = K1+ = q(x)xx − 2q(x)2 q ∗ (−x).
(17)
In the following, we derive solutions for the nonlocal hierarchies (14) by imposing the reduction relations (15) on the solutions of the even order AKNS hierarchy (12). Theorem 1. The nonlocal isospectral NLS hierarchies (14) admit the following solutions q(x) = 2
|ϕ b(n+1) , ψb(n−1) | , |ϕ b(n) , ψb(n) |
(18)
where ϕ and ψ are (2n + 2)-th order column vectors (i.e. m = n in (7)), defined by (13) with t2k = iτ2k and satisfy ψ(x) = T ϕ∗ (−x), −
C = TC
+∗
(19a)
,
(19b)
in which T is a constant matrix determined through AT = T A∗ ,
(20a)
∗
T T = δI, δ = ∓1.
(20b)
Proof. We are going to prove that with the assumptions (19) and (20), q and r defined by (4) with double Wronskians (6) satisfy the reduction (15). To achieve that, first, under assumptions of (19b) and (20a), it is easy to verify that P ∗ −1 )2j τ )T C +∗ ψ(x) = exp (−(T A∗ T −1 )x − i ∞ 2j j=1 (T A T P∞ 2j + ∗ = T (exp (−Ax + i j=1 A τ2j )C ) = T ϕ∗ (−x). Next, to examine relation between Wronskians, based on (9), we introduce notation 2 n ϕ b(n) (ax)[bx] = ϕ(ax), ∂bx ϕ(ax), ∂bx ϕ(ax), · · · , ∂bx ϕ(ax)
and similar one for ψb(m) (ax)bx , where a, b = ±1. With such notations, setting n = m and noticing the constraint (19a), f, g, h in (6) are expressed as f (x) = |ϕ b(n) (x)[x] ; ψb(n) (x)[x] | = |ϕ b(n) (x)[x] ; T ϕ b∗(n) (−x)[x] |, g(x) = |ϕ b(n+1) (x)[x] ; ψb(n−1) (x)[x] | = |ϕ b(n+1) (x)[x] ; T ϕ b∗(n−1) (−x)[x] |, (n−1)
h(x) = |ϕ b
b(n+1)
(x)[x] ; ψ
(n−1)
(x)[x] | = |ϕ b 3
∗(n+1)
(x)[x] ; T ϕ b
(−x)[x] |.
(21a)
(21b) (21c)
By calculation we find f ∗ (−x) = |ϕ b(n) (−x)[−x] ; T ϕ b∗(n) (x)[−x] |∗
= δ n+1 |T ∗ ||T ϕ b∗(n) (−x)[−x] ; ϕ b(n) (x)[−x] | 2
= δ n+1 (−1)(n+1) |T ∗ ||ϕ b(n) (x)[x] ; T ϕ b∗(n) (−x)[x] | 2
= δ n+1 (−1)(n+1) |T ∗ |f (x),
where we have made use of ∂−x = −∂x and assumed T T ∗ = δI where δ is a constant and I is the unit matrix. In a similar way we have g ∗ (−x) = (−1)n(n+2) |T ∗ |δ n+2 h(x). With these relations, by making a comparison of q ∗ (−x) = 2g ∗ (−x)/f ∗ (−x) and r(x) = 2h(x)/f (x) one can find that r(x) = ±q ∗ (−x) when δ = ∓1. Thus we finish the proof. We note that to reduce (12) to solutions for the local NLS hierarchies, we need same reduction condition as in Theorem 1 except (20a) replaced with AT + T A∗ = 0.
4
Solutions of the nonlocal NLS hierarchies
In this section, we list Wronskian elements of solutions for the nonlocal NLS hierarchies (14).
4.1 4.1.1
Soliton solutions Focusing case
For the nonlocal focusing NLS hierarchy iq(x)τ2k = Kk− [q(x), q(−x)], k = 1, 2, 3, . . . , according to Theorem 1, we take δ = 1 and 0 I A1 0 , C + = (c1 , c2 , · · · , c2n+2 )T , , T = A= I 0 0 A∗1
(22)
(23a)
where A1 = Diag(k1 , k2 , · · · , kn+1 ), kj ∈ C and kj are distinct. The Wronskian elements are taken as Φ1 ϕ(x) = , ψ(x) = T ϕ∗ (−x), Φ2
(23b)
(24a)
where Φj = (ϕj,1 , ϕj,2 , · · · , ϕj,n+1 )T , (j = 1, 2), ξl (x,kl∗ )
ϕ1,l = eξl (x,kl ) , ϕ2,l = e
and ξl (x, kl ) = kl x + i
∞ X
,
kl2j τ2j .
(24b) (24c)
(24d)
j=1
The simplest solution to the nonlocal focusing NLS equation (16) is obtained by taking n = 0 in double Wronskian and the solution is given as 2
−4ic1 c2 Im(k1 )e2k1 x+i2k1 τ2 q1ss (x) = , |c1 |2 ei4Im(k1 )x e−8Re(k1 )Im(k1 )τ2 − |c2 |2
(25)
which is the same as the one obtained in [6] and [11], up to some scalar transformations. (25) is a nonsingular periodic solution of the case of Re(k1 ) = 0, Im(k1 ) 6= 0 and |c1 | = 6 |c2 |. 4
4.1.2
Defocusing case
For the nonlocal defocusing NLS hierarchy iq(x)τ2k = Kk+ [q(x), q(−x)], k = 1, 2, 3, . . . ,
(26)
from Theorem 1, we take A and C + as in Sec.4.1.1 but δ = −1 and 0 −I T = , I 0
(27)
where ϕ and ψ follow the structure (24) but with T defined as in (27). Solution to the nonlocal defocusing NLS equation (17) is (n = 0) 2
−4ic1 c2 Im(k1 )e2k1 x+i2k1 τ2 , q1ss (x) = |c1 |2 ei4Im(k1 )x e−8Re(k1 )Im(k1 )τ2 + |c2 |2
(28)
which is a nonsingular solution of the case of Re(k1 ) = 0, Im(k1 ) 6= 0 and |c1 | = 6 |c2 |.
4.2
Jordan block solutions
Jordan block solutions mean the solutions corresponding to the matrix A taking the form as in (23a) where A1 is a (n + 1) × (n + 1) Jordan block A1 = k1 I + σ, I = (δj,l )(n+1)×(n+1) , σ = (δj,l+1 )(n+1)×(n+1) . 4.2.1
(29)
Focusing case
For the nonlocal focusing NLS hierarchy (22), to get their solutions, we take δ = 1, T to be defined as in (23a), and C + = (C1 , C1 )T , C1 = (1, 1, · · · , 1)n+1 . The Wronskian elements take the form (24a) in which n h X 1 l ξ1 (x,k1 ) l i T ξ1 (x,k1 ) Φ1 (x, k1 ) = e + (∂ e )σ C1 , l! k1
h ∗ Φ2 (x, k1∗ ) = eξ1 (x,k1 ) +
l=1 n X l=1
with ξ1 (x, k1 ) = k1 x + i
(30a)
1 l ξ1 (x,k1∗ ) l i T (∂ ∗ e )σ C1 , l! k1
∞ X
(30b)
k12j τ2j .
(31)
j=1
For the nonlocal NLS equation (16), when n = 1, from (31) we have ξ1 (x, k1 ) = k1 x + ik12 τ2 , and the corresponding solution reads ∗
q(x) =
4(k1∗ − k1 )(e2ξ1 (x,k1 ) a(k1 , k1∗ ) − e2ξ1 (x,k1 ) a(k1∗ , k1 )) , ∗ ∗ e2ξ1 (x,k1 )−2ξ1 (x,k1 ) + e2ξ1 (x,k1 )−2ξ1 (x,k1 ) − b
(32)
with a(k1 , k1∗ ) = (k1∗ − k1 )x + 2ik1∗ (k1∗ − k1 )τ2 + 1, b = 4(k1 −
k1∗ )2 x2
+ 8i(k1 −
k1∗ )2 (k1
5
+
k1∗ )xτ2
(33a) −
16k1 k1∗ (k1
−
k1∗ )2 τ22
+ 2.
(33b)
4.2.2
Defocusing case
For the nonlocal defocusing NLS hierarchy (26), in Theorem 1 we take A, C, Φj same as in Sec.4.2.1, but δ = −1 and 0 −I T = . (34) I 0 For the nonlocal defocusing NLS equation (17), the simplest solution of Jordan block case
is
∗
4(k1∗ − k1 )(e2ξ1 (x,k1 ) a(k1 , k1∗ ) + e2ξ1 (x,k1 ) a(k1∗ , k1 )) , q(x) = ∗ ∗ e2ξ1 (x,k1 )−2ξ1 (x,k1 ) + e2ξ1 (x,k1 )−2ξ1 (x,k1 ) + b where a(k1 , k1∗ ) and b are denoted by (33).
5
(35)
Conclusions
In this letter by direct reduction we have derived doubled Wronskian solutions of the nonlocal NLS hierarchies from the known ones of the even-order AKNS hierarchy. Compared with the double Wronskian solution obtained in [7] via Darboux transformation where all {kj } are pure imaginary, here in our solutions all {kj } are complex, which admits more freedom. Besides, we also obtain Jordan block solutions which correspond to multiple-pole case of the transmission coefficient. Since equation (20) admits block diagonal extension, it is easy to extend our results to case where A is a block-diagonal composed by diagonal matrix and Jordan blocks. The reduction procedure developed in this letter is general and can apply to other systems which have double Wronskian solutions and admit local and nonlocal reductions.
Acknowledgments This work was supported by the NSF of China [grant numbers 11371241, 11631007].
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