Determinant Representation of Binary Darboux Transformation for the AKNS Equation

Jing Yu; Jingwei Han; Jingsong He

doi:10.1515/zna-2015-0407

Article Publicly Available

Determinant Representation of Binary Darboux Transformation for the AKNS Equation

Jing Yu , Jingwei Han and Jingsong He

Published/Copyright: November 26, 2015

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Zeitschrift für Naturforschung A Volume 70 Issue 12

Abstract

In this paper, the determinant representation of the n-fold binary Darboux transformation, which is a 2×2 matrix, for the Ablowitz–Kaup–Newell–Segur equation is constructed. In this 2×2 matrix, each element is expressed by (2n+1)-order determinants. When the reduction condition r=–q̅ is considered, we obtain one of binary Darboux transformations for the nonlinear Schrödinger (NLS) equation. As its applications, several solutions are constructed for the NLS equation. Especially, a new form of two-soliton is given explicitly.

Keywords: Binary Darboux Transformation; Darboux Transformation; Rogue Wave Solution; Soliton Solution

PACS Numbers: 02.30.Ik; 02.90.+p

1 Introduction

It is well known that the Ablowitz–Kaup–Newell–Segur (AKNS) system is very important in the field of soliton theory. Many important soliton equations [including nonlinear Schrödinger (NLS) equation, Korteweg–de Vries equation, modified Korteweg–de Vries equation, sine-Gordon equation, etc.] are deduced from the AKNS system by imposing a reduction condition on the potential. Especially, in very recent years, the NLS equation has been widely studied by many researchers in [1–5]. Furthermore, many solutions have been constructed for the NLS equation in these references.

Darboux transformation (DT) is an efficient way to construct explicit solutions for many integrable systems [6–10]. Recently, the determinant representation of DT for many integrable systems has been studied in [11–16]. The main advantage of the determinant representation of DT is that we can present directly the determinant expression of the n-soliton solution generated by the n-fold DT without the n-times iteration. Moreover, we can present an explicit expression of the eigenfunctions generated by the n-fold DT from the initial one. Furthermore, many solutions (soliton solutions, rogue wave solutions, and so on) are constructed by making use of the determinant representation of solutions. In [6], Matveev and Salle first proposed the binary Darboux transformation (BDT) of an integrable system. The main idea is that they construct DT for the linear system and the conjugate system. Then, they combine two DT together and find a two-fold DT (i.e. BDT). Then, several solutions were found for the Davey–Stewartson equation by the BDT [17]. Moreover, the BDT of some integrable equations has been constructed in [18–20]. A natural question arises: Does the determinant representation of BDT for the AKNS equation exist or not? Can we obtain new forms for interesting solutions (such as soliton solutions) by this determinant representation of BDT? All of these questions will be answered in this paper.

The paper is organised as follows. In the next section, we list the DT of the linear system and the adjoint linear system for the AKNS equation. By iterating two DT, we obtain the determinant representation of BDT. In Section 3, when the reduction condition r=−q̅ is imposed on the determinant representation, it reduces to the corresponding one of the NLS equations. As its application, we give several explicit forms of soliton and rogue wave solutions of the NLS equation starting from the zero seed solution and a period non-zero seed solution. Some conclusions and remarks are listed in the last section.

2 n-Fold BDT for the AKNS Equation

The AKNS equation

(1){qt = iqxx − 2iq2r,rt = −irxx + 2iqr2, (1)

is associated with both the following linear system

(2){ϕx = Uϕ, U = (−iλqriλ),ϕt = Vϕ, V = (−2iλ2 − iqr2qλ + iqx2rλ − irx2iλ2 + iqr), (2)

and the adjoint linear system

(3){ψx = Mψ, M = (iμ−r−q−iμ),ψt = Nψ, N = (2iμ2+iqr−2rμ+irx−2qμ−iqx−2iμ2−iqr). (3)

Here ϕ = (ϕ1ϕ2) is the eigenfunction associated with the eigenvalue λ, and ψ = (ψ1ψ2) is the adjoint eigenfunction associated with eigenvalue μ.

2.1 Two Types of DT of the AKNS Equation

Referring to [6, 11, 12, 17], we know that linear system (2) is transformed to

(4){ϕx[1] = U[1]ϕ[1],ϕt[1] = V[1]ϕ[1], (4)

under the following transformation:

(5)ϕ[1] = Taϕ, (5)

where U^[1] and V^[1], respectively, have the same forms as U and V, and the first type of Darboux matrix is defined by

(6)Ta = Ta(λ; λ1, λ2, f1, f2) = 1|Δa|((Ta)11(Ta)12(Ta)21(Ta)22), (6)

with

Δa = (f11f12f21f22),(Ta)11 = |p2λΔaξ^1|,(Ta)12 = |q20Δaξ^1|,(Ta)21 = |p20Δaξ^2|,(Ta)22 = |q2λΔaξ^2|.

Here p₂=(1, 0), q₂=(0, 1), fk = (fk1fk2) is a solution for (2) with λ=λ_k and ξ^k = (λ1f1kλ2f2k) (k=1, 2). It is not difficult to find that the new solutions q^[1] and r^[1] are respectively given as follows:

(7)q[1] = q + 2i|Δa|(Ta)12, r[1] = r − 2i|Δa|(Ta)21. (7)

Similarly, the adjoint linear system (3) is transformed to

(8){ψx[1] = M[1]ψ[1],ψt[1] = N[1]ψ[1], (8)

by the second type of DT [6]

(9)ψ[1] = Tbψ, (9)

where M^[1] and N^[1], respectively, have the same forms as M and N, and the Darboux matrix is defined by

(10)Tb = Tb(μ; μ1, μ2, g1, g2) = 1|Δb|((Tb)11(Tb)12(Tb)21(Tb)22), (10)

with

Δb = (g11g12g21g22),(Tb)11 = |p2μΔbη^1|,(Tb)12 = |q20Δbη^1|,(Tb)21 = |p20Δbη^2|,(Tb)22 = |q2μΔbη^2|.

Here gk = (gk1gk2) is a solution for (3) with μ=μ_k and η^k = (μ1g1kμ2g2k) (k=1, 2). Meanwhile, the new solutions q^[1] and r^[1] are respectively given by

(11)q[1]=q − 2i|Δb|(Tb)21, r[1] = r + 2i|Δb|(Tb)12. (11)

Thus, we respectively list DTs of the linear system (2) and the adjoint linear system (3) for the AKNS equation (1), which are essential elements for the following construction of BDT for the AKNS equation (1).

2.2 One-fold BDT for the AKNS Equation

In this subsection, we will focus on constructing the determinant representation of one-fold BDT for the AKNS equation (1). To this end, we first give the following lemma.

Lemma 1Set eigenvalues μ_k =λ_k, and then the corresponding adjoint eigenfunctions g_k1=f_k2and g_k2=−f_k1 (k=1, 2).

Making use of Lemma 1, we have the following conclusion.

Theorem 1If μ_k =λ_k (k=1, 2) in (10), then Tb = (Ta∗)T. Meanwhile, If λ_k =μ_k (k=1, 2) in (6), then Ta = (Tb∗)T. Here, “*” and “T” denotes adjoint matrix and transposition of a matrix, respectively.

Proof. Substituting μ_k =λ_k , g_k1=f_k2, and g_k2=−f_k1 into (10), we find that |Δ_b |=|Δ_a |, (T_b )₁₁=(T_a )₂₂, (T_b )₁₂=−(T_a )₂₁, (T_b )₂₁=−(T_a )₁₂, (T_b )₂₂=(T_a )₁₁, and then Tb = (Ta∗)T. Similarly, Ta = (Tb∗)T can be proved by a direct calculation. □

Thus, we obtain the transformed eigenfunction and adjoint eigenfunction

(12)fi[1] = Ta(λi; λ1, λ2, f1, f2)fi,gi[1] = (Ta∗(μi; λ1, λ2, f1, f2))Tgi, (12)

respectively. Similarly, for the second type DT, we have

(13)fi[1] = (Tb∗(λi; λ1, λ2, g1, g2))Tfi,gi[1] = Tb(μi; λ1, λ2, g1, g2)gi. (13)

In what follows, we choose the following chain of DT:

(14)ϕx = Uϕ→ϕ[1] = Ta(λ; λ1, λ2, f1, f2)ϕϕx[1] = U[1]ϕ[1]→ϕ[2] = (Tb∗(λ; μ1, μ2, g1[1], g2[1]))Tϕ[1]ϕx[2] = U[2]ϕ[2], (14)

where

gk[1] = (Ta∗(λ; λ1, λ2, f1, f2))T|λ = μkgk = μkgk + 1|Δa|(−λ2f11f22gk1 + λ1f12f21gk1 − λ2f12f22gk2 + λ1f12f22gk2λ2f11f21gk1 − λ1f11f21gk1 + λ2f12f21gk2 − λ1f11f22gk2),

with k=1, 2.

Remark 1The one-fold BDT of the AKNS equation(1) was first proposed by Matveev in [6]. Here, the above chain of DT is a counterpart of Matveev’s result [6]. In this paper, the chain of DT expressed by a spectral problem is much more precise than the one of [6]. Moreover, we will give its determinant representation, which has never been studied before.

Theorem 2The one-fold BDT for the AKNS equation(1) can be expressed as

(15)T2 = (Tb∗(λ; μ1, μ2, g1[1], g2[1]))TTa(λ; λ1, λ2, f1, f2) = Ta(λ; μ1, μ2, g1[1], g2[1])Ta(λ; λ1, λ2, f1, f2) = 1|W4|(|p4λ2W4ξ4||q40W4ξ4||p40W4η4||q4λ2W4η4|), (15)

where

W4 = (f11f12λ1f11λ1f12f21f22λ2f21λ2f22g12−g11μ1g12−μ1g11g22−g21μ2g22−μ2g21,),p₄=(1, 0, λ, 0), q₄=(0, 1, 0, λ), ξ4 = (λ12f11λ22f21μ12g12μ22g22),η4 = (λ12f12λ22f22−μ12g11−μ22g21).

The new solutionsq^[2]andr^[2]generated byT₂are

(16){q[2] = q − 2i|W4|(t2)12,r[2] = r − 2i|W4|(t2)21, (16)

with

(t2)12 = |f11f12λ1f11λ12f11f21f22λ2f21λ22f21g12−g11μ1g12μ12g12g22−g21μ2g22μ22g22|,(t2)21 = |f11f12λ1f12λ12f12f21f22λ2f22λ22f22g12−g11−μ1g11−μ12g11g22−g21−μ2g21−μ22g21|.

Proof. According to the construction of DT, it is easy to find T2|λ = λkfk = 0 and (T2∗)T|λ=μkgk = 0 (k = 1, 2). By solving the above four algebraic systems, we get elements of T₂. Setting U[2]=U|(q → q[2], r → r[2]), then T_2,x +T₂U=U^[2]T₂ implies (q^[2], r^[2]) in (16). □

Theorem 2 shows that the kernel of the one-fold BDT T₂ is expressed by {f₁, f₂, g₁, g₂}, which is different from the case of two-fold DT T₂. It is trivial to know that the kernel of T₂ is expanded by {f_i , i=1, 2, 3, 4}.

2.3 n-Fold BDT of the AKNS Equation

The main task is to establish the determinant representation of the n-fold BDT for the AKNS equation (1) in this subsection. By iteration T₂ for n times, we get the following chain of BDT, which is equivalently expressed by a complicated operator:

T2n = T2n × ⋯ × T8 × T6 × T4 × T2,

where

T2 : (Tb∗(λ; μ1, μ2, g1[1], g2[1]))TTa(λ; λ1, λ2, f1, f2),T4 : (Tb∗(λ; μ3, μ4, g3[3], g4[3]))TTa(λ; λ3, λ4, f3[2], f4[2]),T6 : (Tb∗(λ; μ5, μ6, g5[5], g6[5]))TTa(λ; λ5, λ6, f5[4], f6[4]),T8 : (Tb∗(λ; μ7, μ8, g7[7], g8[7]))TTa(λ; λ7, λ8, f7[6], f8[6]),⋮T2n : (Tb∗(λ; μ2n − 1, μ2n, g2n − 1[2n − 1], g2n[2n − 1]))TTa(λ; λ2n − 1, λ2n, f2n − 1[2n − 2], f2n[2n − 2]).

By a similar calculation of T₂, we get the determinant representation of T_2n .

Theorem 3The Darboux matrix of the n-fold BDT for the AKNS equation(1) can be expressed by

(17)T2n = 1|W4n|(|p4nλ2nW4nξ4n||q4n0W4nξ4n||p4n0W4nη4n||q4nλ2nW4nη4n|), (17)

where

W4n = (f11f12λ1f11λ1f12⋯λ12n−1f11λ12n−1f12f21f22λ2f21λ2f22⋯λ22n−1f21λ22n−1f22⋮⋮⋮⋮⋮⋮f2n−1,1f2n−1,2λ2n−1f2n−1,1λ2n−1f2n−1,2⋯λ2n−12n−1f2n−1,1λ2n−12n−1f2n−1,2f2n,1f2n,2λ2nf2n,1λ2nf2n,2⋯λ2n2n−1f2n,1λ2n2n−1f2n,2g12−g11μ1g12−μ1g11⋯μ12n−1g12−μ12n−1g11g22−g21μ2g22−μ2g21⋯μ22n−1g22−μ22n−1g21⋮⋮⋮⋮⋮⋮g2n−1,2−g2n−1,1μ2n−1g2n−1,2−μ2n−1g2n−1,1⋯μ2n−12n−1g2n−1,2−μ2n−12n−1g2n−1,1g2n,2−g2n,1μ2ng2n,2−μ2ng2n,1⋯μ2n2n−1g2n,2−μ2n2n−1g2n,1),

p4n = (1, 0, λ, 0, ⋯, λ2n − 1, 0), q4n = (0, 1, 0, λ, ⋯, 0, λ2n − 1),

ξ4n = (λ12nf11, λ22nf21, ⋯, λ2n−12nf2n−1,1, λ2n2nf2n,1, μ12ng12, μ22ng22, ⋯, μ2n−12ng2n−1,2, μ2n2ng2n,2)T,

η4n = (λ12nf12, λ22nf22, ⋯, λ2n−12nf2n−1,2, λ2n2nf2n,2, −μ12ng11, −μ22ng21, ⋯,−μ2n−12ng2n−1,1, −μ2n2ng2n,1)T,

and the new solutionsq^[2n]andr^[2n]are given by

(18){q[2n] = q − 2i|Ω12||W4n|,r[2n] = r − 2i|Ω21||W4n|, (18)

where

Ω12 = (f11f12λ1f11λ1f12⋯λ12n−1f11λ12nf11f21f22λ2f21λ2f22⋯λ22n−1f21λ22nf21⋮⋮⋮⋮⋮⋮f2n−1,1f2n−1,2λ2n−1f2n−1,1λ2n−1f2n−1,2⋯λ2n−12n−1f2n−1,1λ2n−12nf2n−1,1f2n,1f2n,2λ2nf2n,1λ2nf2n,2⋯λ2n2n−1f2n,1λ2n2nf2n,1g12−g11μ1g12−μ1g11⋯μ12n−1g12μ12ng12g22−g21μ2g22−μ2g21⋯μ22n−1g22μ22ng22⋮⋮⋮⋮⋮⋮g2n−1,2−g2n−1,1μ2n−1g2n−1,2−μ2n−1g2n−1,1⋯μ2n−12n−1g2n−1,2μ2n−12ng2n−1,2g2n,2−g2n,1μ2ng2n,2−μ2ng2n,1⋯μ2n2n−1g2n,2μ2n2ng2n,2),

Ω21 = (f11f12λ1f11λ1f12⋯λ12n−1f12λ12nf12f21f22λ2f21λ2f22⋯λ22n−1f22λ22nf22⋮⋮⋮⋮⋮⋮f2n−1,1f2n−1,2λ2n−1f2n−1,1λ2n−1f2n−1,2⋯λ2n−12n−1f2n−1,2λ2n−12nf2n−1,2f2n,1f2n,2λ2nf2n,1λ2nf2n,2⋯λ2n2n−1f2n,2λ2n2nf2n,2g12−g11μ1g12−μ1g11⋯−μ12n−1g11−μ12ng11g22−g21μ2g22−μ2g21⋯−μ22n−1g21−μ22ng21⋮⋮⋮⋮⋮⋮g2n−1,2−g2n−1,1μ2n−1g2n−1,2−μ2n−1g2n−1,1⋯−μ2n−12n−1g2n−1,1−μ2n−12ng2n−1,1g2n,2−g2n,1μ2ng2n,2−μ2ng2n,1⋯−μ2n2n−1g2n,1−μ2n2ng2n,1).

Remark 2The new solutions {q^[2n], r^[2n]} are expressed by functions {fk, gk}k = 12n, which are different from the case of [12]. For more detail, we can refer to Eqs.(12) and(13) in [12].

3 Particular Solutions of the NLS Equation

In this section, we shall consider some particular solutions of the NLS equation by means of the determinant representation of T_2n . The NLS equation is in the form of

(19)iqt + qxx + 2|q|2q = 0, (19)

which can be derived from the AKNS equation (1) under the reduction condition r=−q̅, where symbol “q̅” denotes the complex conjugation of q.

In order to keep a reduction condition r[2n] = −q[2n]¯, we have to select λ2k = λ2k − 1¯,f2k = (−f2k − 1,2¯f2k − 1,1¯) for the linear system (2), and μ2k = μ2k − 1¯,g2k = (−g2k − 1,2¯g2k − 1,1¯) for the adjoint linear system (3). Meanwhile, under the above selection, (18) becomes the following form:

(20)q[2n] = q − 2i|Q4n||W˜4n|, (20)

where

Q4n=(f11f12λ1f11λ1f12⋯λ12n−1f11λ12nf11−f12¯f11¯−λ1¯ f12¯λ1¯ f11¯⋯−λ12n−1¯f12¯−λ12n¯f12¯⋮⋮⋮⋮⋮⋮f2n−1,1f2n−1,2λ2n−1f2n−1,1λ2n−1f2n−1,2⋯λ2n−12n−1f2n−1,1λ2n−12nf2n−1,1−f2n−1,2¯f2n−1,1¯−λ2n−1¯ f2n−1,2¯λ2n−1¯ f2n−1,1¯⋯−λ2n−12n−1¯f2n−1,2¯−λ2n−12n¯f2n−1,2¯g12−g11μ1g12−μ1g11⋯μ12n−1g12μ12ng12g11¯g12¯μ1¯ g11¯μ1¯ g12¯⋯μ12n−1¯g11¯μ12n¯g11¯⋮⋮⋮⋮⋮⋮g2n−1,2−g2n−1,1μ2n−1g2n−1,2−μ2n−1g2n−1,1⋯μ2n−12n−1g2n−1,2μ2n−12ng2n−1,2g2n−1,1¯g2n−1,2¯μ2n−1¯ g2n−1,1¯μ2n−1¯ g2n−1,2¯⋯μ2n−12n−1¯g2n−1,1¯μ2n−12n¯g2n−1,1¯),

and

W˜4n=(f11f12λ1f11λ1f12⋯λ12n−1f11λ12n−1f12−f12¯f11¯−λ1¯ f12¯λ1¯ f11¯⋯−λ12n−1¯f12¯λ12n−1¯f11¯⋮⋮⋮⋮⋮⋮f2n−1,1f2n−1,2λ2n−1f2n−1,1λ2n−1f2n−1,2⋯λ2n−12n−1f2n−1,1λ2n−12n−1f2n−1,2−f2n−1,2¯f2n−1,1¯−λ2n−1¯ f2n−1,2¯λ2n−1¯ f2n−1,1¯⋯−λ2n−12n−1¯f2n−1,2¯λ2n−12n−1¯f2n−1,1¯g12−g11μ1g12−μ1g11⋯μ12n−1g12−μ12n−1g11g11¯g12¯μ1¯ g11¯μ1¯ g12¯⋯μ12n−1¯g11¯μ12n−1¯g12¯⋮⋮⋮⋮⋮⋮g2n−1,2−g2n−1,1μ2n−1g2n−1,2−μ2n−1g2n−1,1⋯μ2n−12n−1g2n−1,2−μ2n−12n−1g2n−1,1g2n−1,1¯g2n−1,2¯μ2n−1¯ g2n−1,1¯μ2n−1¯ g2n−1,2¯⋯μ2n−12n−1¯g2n−1,1¯μ2n−12n−1¯g2n−1,2¯).

Formula (20) provides a way to obtain higher order soliton and rogue wave solutions.

3.1 Soliton Solutions of the NLS Equation

If we take the seed solution q=0 of (19), for the linear system (2), the eigenfunction is given by

(21)ϕ(λ) = (e−iλx − 2iλ2teiλx+2iλ2t), (21)

and for the adjoint linear system (3), the adjoint eigenfunction is given by

(22)ψ(μ) = (eiμx + 2iμ2te−iμx−2iμ2t). (22)

Here λ and μ are the spectral parameters of the linear system (2) and its adjoint linear system (3), respectively.

Furthermore, if we take λ_k =ξ_k +iη_k (k=1, 3, 5, …, 2n−1), α_k =η_kx+4ξ_kη_kt, βk = ξkx + 2(ξk2 − ηk2)t, the odd-th eigenfunctions

fk = ϕ(λk) = (eαk − iβke−αk + iβk):=(fk1fk2)

are (21) with λ=λ_k and the even-th eigenfunctions

fk = ϕ(λk) = (−e−αk − 1 − iβk − 1eαk − 1 + iβk − 1) := (fk1fk2) (k = 2, 4, 6, …, 2n).

Similarly, if we take μ_k =ζ_k +iν_k (k=1, 3, 5, …, 2n−1), γ_k =ν_kx+4ζ_kν_kt, δk = ζkx + 2(ζk2 − νk2)t, the odd-th adjoint eigenfunctions

gk = ψ(μk) = (e−γk + iδkeγk − iδk) := (gk1gk2)

are (22) with μ=μ_k and the even-th eigenfunctions

gk = ψ(μk) = (−eγk − 1 + iδk − 1e−γk − 1 − iδk − 1) := (gk1gk2) (k = 2, 4, 6, …, 2n).

Substituting different eigenfunctions f_k and g_k into (20), we obtain the soliton solutions of the NLS equation (19). Next, we shall give several soliton solutions of the NLS equation (19).

Case 1 (one-soliton solution). If we take λ₁=ξ₁+iη₁ and λ2 = λ1¯, then the corresponding eigenfunctions f₁ and f₂ can be expressed as f1 = (f11f12) = (eα1 − iβ1e−α1 + iβ1) and f2 = (f21f22) = (−e−α1 − iβ1eα1 + iβ1). Moreover, we choose μ₂=μ₁ be any real number. Then, by (21), the eigenfunctions of adjoint system be expressed as g1 = (g11g12) = (e−γ1 + iδ1eγ1 − iδ1) and g2 = (g21g22) = (−eγ1 + iδ1e−γ1 − iδ1). Substituting f₁, f₂ and g₁, g₂ into (20) with n=1, we obtain the one-soliton solution of the NLS equation (19),

(23)q[2](x, t) = 2η1cosh(2η1(x + 4ξ1t))e2iθ, (23)

where θ = (−ξ1x − 2(ξ12 − η12)t), which is nothing but a standard one-soliton constructed in [21] and other early references in [6]. The profile of |q^[2]|² with λ₁=0.1+0.7i and its density are plotted in Figure 1. It is trivial to know from (23) that q^[2] is independent of μ₂=μ₁=ζ₁.

Figure 1:

The profile of one-soliton (left) and its denisty plot (right).

Remark 3The same one-soliton solution of the NLS equation (19) can be easily obtained from a 2×2 determinant expression [12]. We remark here that a standard one-soliton solution is obtained from a 4×4 determinant expression, which indicates that not only even-th but also odd-th soliton solutions can be obtained by formula (20). Later, we shall show that a three-soliton solution can also be constructed by n-fold BDT.

Case 2 (two-soliton solution). Just as in Case 1, choosing μ₁=ζ₁+iν₁1 and μ2 = μ1¯, the corresponding eigenfunctions of the adjoint system are given by g1 = (g11g12) = (e−γ1 + iδ1eγ1 − iδ1) and g2 = (g21g22) = (−eγ1 + iδ1e−γ1 − iδ1). Substituting f₁, f₂ and g₁, g₂ into (20) with n=1, we have the determinant expression

(24)q[2] = −2i|eα1 − iβ1e−α1 + iβ1(ξ1 + iη1)eα1 − iβ1(ξ1 + iη1)2eα1 − iβ1−e−α1 − iβ1eα1 + iβ1−(ξ1 − iη1)e−α1 − iβ1−(ξ1 − iη1)2e−α1 − iβ1eγ1 − iδ1−e−γ1 + iδ1(ζ1 + iν1)eγ1 − iδ1(ζ1 + iν1)2eγ1 − iδ1e−γ1 − iδ1eγ1 + iδ1(ζ1 − iν1)e−γ1 − iδ1(ζ1 − iν1)2e−γ1 − iδ1||eα1 − iβ1e−α1 + iβ1(ξ1 + iη1)eα1 − iβ1(ξ1 + iη1)e−α1 + iβ1−e−α1 − iβ1eα1 + iβ1−(ξ1 − iη1)e−α1 − iβ1(ξ1 − iη1)eα1 + iβ1eγ1 − iδ1−e−γ1 + iδ1(ζ1 + iν1)eγ1−iδ1−(ζ1 + iν1)e−γ1 + iδ1e−γ1 − iδ1eγ1 + iδ1(ζ1 − iν1)e−γ1 − iδ1(ζ1 − iν1)eγ1 + iδ1|. (24)

Expanding the determinants in q^[2], we get a two-soliton solution of the NLS equation (19),

(25)q[2] = −4iP2(x, t)Q2(x, t), (25)

where

P2(x, t) = iν1[(η12 − ν12) − (ξ1 − ζ1)2]e−2iδ1cosh2α1 + iη1[(η12 − ν12) + (ξ1 − ζ1)2]e−2iβ1cosh2γ1+2(ξ1 − ζ1)η1ν1[e−2iβ1sinh(2γ1) + e−2iδ1sinh(2α1)],

Q2(x,t) = [(ξ1 − ζ1)2 + (ν1 + η1)2]cosh(2α1 − 2γ1)+ [(ξ1 − ζ1)2 + (ν1 − η1)2]cosh(2α1 + 2γ1) + 4ν1η1cos(2β1 − 2δ1).

with δ1 = ζ1x + 2(ζ12 − ν12)t. For the two-soliton, the profile (left panel of Figure 2) of |q^[2]|² with λ₁=0.5+0.4i and μ₁=−0.5−0.4i on the (x, t)-plane and its density (right panel of Figure 2) are plotted. It is easy to observe that the denominator in (25) contains a cosine term, which is different from other results of the two-soliton solution of the NLS equation (19). For detail, we can refer to [6, 22]. So, we claim here that q^[2] in (25) is a new form of a two-soliton solution.

Figure 2:

The profile of two-soliton (left) and its density plot (right).

To understand the two-soliton better we shall analyse the properties of |q^[2]|² in the following. The height of |q^[2]|² at the point (0, 0) is 4(ν₁−η₁)², which indicates that the amplitude will be changed through the interaction of two single solitons. We point out that there are two simultaneous trajectories on the (x, t)-plane. The trajectories are two straight lines as general two-soliton; the equations of the two lines are x=−4ξ₁t and x=−4ζ₁t with two different phase shifts, respectively. Although we do not write two phase shifts explicitly, they are verified by a remarkable change of the intercepts of two lines in Figure 2 (left panel).

Case 3 (four-soliton solution). Set n=2 in (20), the numerator and denominator of q^[4] are respectively 8×8 determinant, which contain eight parameters. Due to the complexity, we can only give the profile of q^[4] with λ₁=0.5−0.3i, λ₃=0.1+0.3i, μ₁=1+0.6i, and μ₃=−1+0.5i in Figure 3. From Figure 3, it is easy to observe that q^[4] is nothing but four-soliton. The height of |q^[4]|² at the point (0, 0) is 4(ν₁+η₁)², which is distinct with the case of q^[2].

Figure 3:

The profile of four-soliton (left) and its density plot (right).

We shall remark that under the reduction condition γ₄=γ₃, a real number, the three-soltion can also be obtained.

3.2 Rogue Wave Solution of the NLS Equation

Assuming that q=ce^iρ is a periodic seed solution of (19), where ρ=ax+bt, we can obtain the relation

(26)c2 = 12(b + a2). (26)

Relation (26) reduces the number of independent parameters from three to two, and then we may presume a, c∈R and c≠0 without any loss of generality.

Substituting q=ce^iρ into the linear system (2), and using the method of superposition principle, the eigenfunction ϕ(λ) associated with λ is given by

(27)ϕ(λ) = (ϕ1(λ)ϕ2(λ)) = (cei(12ρ + d(λ)) + i(λ + a2 + c1(λ))ei(12ρ − d(λ))ce−i(12ρ + d(λ)) + i(λ + a2 + c1(λ))ei(− 12ρ + d(λ))), (27)

where c1(λ) = c2 + (λ + a2)2,d(λ)=c₁(λ)x+c₂(λ)t+c₁(λ)Φ, c₂(λ)=(2λ−a)c₁(λ), Φ = ∑k = 02nskεk, sk = ξk + iηk and n denotes the number of the steps of the multi-fold BDT. For the eigenfunction ϕ(λ) in (27), there exists parameter λ0 = −a2 + ic such that ϕ(λ₀)=0.

At the same time, for the adjoint linear system (3), the adjoint eigenfunction ψ(μ) associated with μ is given by

ψ(μ) = (ψ1(λ)ψ2(λ)) = (cei(− 12ρ + d(μ)) + i(λ + a2 − c1(μ))e−i(12ρ + d(μ))−cei(12ρ − d(μ)) − i(μ + a2 − c1(μ))ei(12ρ + d(μ))).

Obviously, λ0 = −a2 + ic is also a zero point of ψ(μ).

For different eigenvalues, we need to introduce 4n eigenfunctions by fk = ϕ(λk) = (fk1(λ)fk2(λ)) associated with eigenvalue λ_k and gk = ψ(μk) = (gk1(μ)gk2(μ)) associated with eigenvalue μ_k , k=1, …, 2n. Substituting all eigenfunctions f_k and g_k into (20), higher order rogue waves can be generated via the process of eigenvalue degeneration λ_k →λ₀ and μ_k →λ₀, respectively. In (20), letting W˜4n = (wij)4n × 4n, Q4n = (qij)4n × 4n and by means of the Taylor expansion, we can get the determinant expression of a higher order rogue wave of BDT [14],

(28)qrw[2n] = ceiρ − 2i|Q″4n||W″4n|, (28)

where

Q″4n = (∂ni∂εni|ε = 0qij(λ0 + ε))4n × 4n,

W″4n4n = (∂ni∂εni|ε = 0wij(λ0 + ε))4n × 4n.

Here ni = [i + 12], [x] denotes the floor function of x.

Case 4 (second-order rogue wave solution). When n=1, substituting f₁, f₂, g₁, g₂, and a=0 into (28), then a second-order rogue wave solution of the NLS equation (19) is

(29)qrw[2] = −ce2ic2tP1(x, t) + iP2(x, t)Q2(x, t), (29)

where

P1(x, t) = −3072c8t2(s03x + x3s0 + c2s02t2 + c2x2t2) + 144c4(s04 + x4 − s12)+ 5760c6t2(x2 + s02) + 576c4s0x(x2 + s02) − 64c6(x6 + s06)− 768c8t2(x4 + s04)− 384c6s0x(x4 + s04)+ 1152c6s1t(x2 + s02) − 960c6s02x2(x2 + s02)+ 180c2(s02 + x2) − 4096c12t6 + 360c2s0x+ 1872c4t2 + 8448c8t4 + 288c4s1t + 2304c6s0s1xt− 1536c8s1t3 + 864c4s02x2 − 1280c6s03x3+ 11520c6s0xt2 − 4608c8s02x2t2 − 6144c10s0xt4 − 45,

P2(x, t) = 12288c8t3(s0x + c2t2) − 1152c4(s0s1x + x2t + s02t) + 2304c4t(c2ts1 − s0x)+ 768c6t(x4 + s04) − 576c4s1(x2 + s02) + 6144c8t3(x2 + s02) + 3072c6s0xt(x2 + s02)+ 4608c6s02x2t − 720tc2 − 144s1c2 + 1536c6t3,

Q2(x, t) = 3072c8t2(c2t2 + s0x)(x2 + s02) − 1152c6t(t + s1)(x2 + s02) + 108c2(x2 + s02) + 960c6s02x2(x2 + s02)+ 192c4s0x(x2 + s02) + 48c4(x4 + s04) + 768c8t2(x4 + s04) + 384c6s0x(x4 + s04) + 64c6(x6 + s06) − 2304c6s0xt(t + s1)+ 4608c8s02t2x2 + 216c2s0x+ 4096c12t6 + 6144c10t4s0x + 1584c4t2 + 6912c8t4 + 144c4s12+ 864c4s1t + 1536c8s1t3 + 288c4s02x2+ 1280c6s03x3 + 9.

The rogue wave solution qrw[2] is the same as the result obtained by general DT [14]. In (29), if we choose the parameters a=0, c=0.5, s₀=0, and s₁=0, then the profile of a fundamental pattern for |qrw[2]|2 is as shown in Figure 4. Moreover, if we take the parameters a=0, c=0.7, s₀=0, and s₁=1000, then the corresponding profile of a triangle pattern for |qrw[2]|2 is as shown in Figure 5.

Figure 4:

The fundamental pattern of a second-order rogue wave (left) and its density plot (right).

Figure 5:

The triangle pattern of a second-order rogue wave (left) and its density plot (right).

4 Conclusions and Remarks

In this paper, we have given the determinant representation (18) of solutions for the AKNS equation (1). When the reduction condition r=−q̅ is considered, we obtain the corresponding result of the NLS equation (19). According to the idea of [12], we express the solutions in (20) of NLS by using the determinant representation of the n-fold BDT. Furthermore, by the special choice of the eigenfunction f_k associated with eigenvalue λ_k and the adjoint eigenvalue g_k associated with μ_k , several solutions can be constructed by (20), such as soliton solutions and rogue wave solutions. To illustrate our method of BDT, solutions of four specific cases are discussed by analytical formulae and figures.

Compared with other results [6, 21, 23] of the DT for the NLS equation (19), a novel form of a two-soliton solution in Case 2, which contains a cosine term, has never been appeared in former references. Moreover, a form of a second-order rogue wave solution has been provided in Case 4. Due to the extreme complexity of a breather solution, we have to point out that the breather solution has not been given by a simple and explicit formula in this paper, although it can be plotted according to (20) with the help of the eigenfunction and adjoint eigenfunction associated with a periodic seed solution. We believe that there exists a new form of breather solutions for the NLS equation (19) generated by BDT. Furthermore, we remark that the BDT method may be useful for the other integrable equations, which will be considered in our future paper.

Corresponding author: Jingsong He, School of Science, Ningbo University, Ningbo, Zhejiang, 315211, P.R. China, E-mail: hejingsong@nbu.edu.cn; jshe@ustc.edu.cn

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11001069, 11271210, 10671187, and 61273077, and Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ12A01002.

References

[1] X. L. Wang, W. G. Zhang, B. G. Zhai, and H. Q. Zhang, Commun. Theor. Phys. 58, 531 (2012).10.1088/0253-6102/58/4/15Search in Google Scholar

[2] M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 110, 064105 (2013).10.1103/PhysRevLett.110.064105Search in Google Scholar

[3] L. C. Zhao and J. Liu, Phys. Rev. E 87, 013201 (2013).10.1103/PhysRevE.87.013201Search in Google Scholar

[4] Q. L. Zha, Phys. Lett. A 377, 855 (2013).10.1016/j.physleta.2013.01.044Search in Google Scholar

[5] E. Yomba, Phys. Lett. A 377, 167 (2013).10.1016/j.physleta.2012.11.049Search in Google Scholar

[6] V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons, Springer-Verlag, Berlin 1991.10.1007/978-3-662-00922-2Search in Google Scholar

[7] V. B. Matveev, Lett. Math. Phys. 3, 213 (1979).10.1007/BF00405295Search in Google Scholar

[8] V. B. Matveev, Lett. Math. Phys. 3, 217 (1979).10.1007/BF00405296Search in Google Scholar

[9] V. B. Matveev and Salle M A. Lett. Math. Phys. 3, 425 (1979).10.1007/BF00397217Search in Google Scholar

[10] V. B. Matveev, Lett. Math. Phys. 3, 503 (1979).10.1007/BF00401932Search in Google Scholar

[11] G. Neugebauer and R. Meinel, Phys. Lett. 100A, 467 (1984).10.1016/0375-9601(84)90827-2Search in Google Scholar

[12] J. S. He, L. Zhang, Y. Cheng, and Y. S. Li, Sci. China Ser. A: Math. 49, 1867 (2006).10.1007/s11425-006-2025-1Search in Google Scholar

[13] S. W. Xu, J. S. He, and L. H. Wang, J. Phys. A: Math. Theor. 44, 305203 (2011).10.1088/1751-8113/44/30/305203Search in Google Scholar

[14] J. S. He, H. R. Zhang, L. H. Wang, K. Porsezian, and A. S. Fokas, Phys. Rev. E 87, 052914 (2013).Search in Google Scholar

[15] S. W. Xu and J. S. He, J. Math. Phys. 53, 063507 (2012).10.1063/1.4726510Search in Google Scholar

[16] J. W. Han, J. Yu, and J. S. He, Modern Phys. Lett. B 27, 1350216 (2013).10.1142/S0217984913502163Search in Google Scholar

[17] C. H. Gu, H. S. Hu, and Z. X. Zhou, Darboux Transformations in Integrable Systems, Theory and Their Applications to Geometry, Springer, Dordrecht 2005.Search in Google Scholar

[18] S. B. Leble, M. A. Salle, and A. V. Yurov, Inverse Problems 8, 207 (1992).10.1088/0266-5611/8/2/004Search in Google Scholar

[19] L. M. Ling and Q. P. Liu, J. Phys. A: Math. Theor. 43, 434023 (2010).10.1088/1751-8113/43/43/434023Search in Google Scholar

[20] L. M. Ling and Q. P. Liu, J. Math. Phys. 52, 053513 (2011).10.1063/1.3589285Search in Google Scholar

[21] J. S. He, M. Ji, and Y. S. Li, Chin. Phys. Lett. 24, 2157 (2007).Search in Google Scholar

[22] D. Y. Chen, Introduction of Soliton, Science Press, Beijing 2006 (in Chinese).Search in Google Scholar

[23] B. L. Guo, L. M. Ling, and Q. P. Liu, Phys. Rev. E 85, 026607 (2012).10.1103/PhysRevE.85.026607Search in Google Scholar PubMed

Received: 2015-7-16

Accepted: 2015-10-25

Published Online: 2015-11-26

Published in Print: 2015-12-1

Articles in the same Issue

https://doi.org/10.1515/zna-2015-0407

Keywords for this article

Binary Darboux Transformation; Darboux Transformation; Rogue Wave Solution; Soliton Solution