Exact solutions of the Gerdjikov-Ivanov equation using Darboux transformations

0.47 MB·English

by  Halis Yilmaz

Exact solutions of the Gerdjikov-Ivanov equation using
Darboux transformations
Halis Yilmaz†1,2
1School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, UK
2Department of Mathematics, University of Dicle, 21280 Diyarbakir, Turkey
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February 2, 2015
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9 Abstract
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We study the Gerdjikov-Ivanov (GI) equation and present a standard Darboux transforma-

] tion for it. The solution is given in terms of quasideterminants. Further, the parabolic, soliton
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S and breather solutions of the GI equation are given as explicit examples.
.
n Keywords: Gerdjikov-Ivanov equation; Derivative nonlinear Schr¨odinger equation; Darboux
i
l transformation; Quasideterminants.
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[ 2010 Mathematics Subject Classification: 35C08, 35Q55, 37K10, 37K35

1
v
1 Introduction
4
3
6 The well known nonlinear Schr¨odinger (NLS) equation is one of the most important soliton equa-
7
tions. Extended versions of this equation with higher order nonlinearity have been proposed and
0
. studiedbyvariousauthors. Amongthem, therearethreecelebratedequationswithderivative–type
1
nonlinearities,whicharecalledthederivativeNLSequations(DNLS).OneistheKaup-Newellequa-
0
5 tion (DNLSI) [16]
1
v: iqt+qxx = i(|q|2q)x,
i
X
the second is the Chen–Lee–Liu equation (DNLSII) [2]
r
a
iq +q +i|q|2q = 0,
t xx x
while the third is the Gerdjikov–Ivanov equation (DNLSIII) [11]
1
iq +q +iq2q∗ + q3q∗2 = 0, (1.1)
t xx x 2
whereq∗ denotesthecomplexconjugateofq. TheNLSequationwithitscousintheDNLSequations
are completely integrable and play an important role in mathematical physics [1,3,14,15,17,26].
†E-mail: [email protected], [email protected]
1
It is known that these three equations may be transformed into each other by a chain of gauge
transformations and the method of gauge transformation can also be applied to some generalised
cases [4,18–20,27]. Therefore, in principle, the corresponding results for Chen-Lee–Lue and the
GI equations may be obtained from the Kaup-Newell equation. However, these transformations
involve very complicated integrals and it is not easy to obtain their explicit forms. So, even though
the three systems are related by gauge transformations it is more convenient to treat them each
separately.
In [24], the explicit quasideterminant solutions of the Kaup Newell equation (DNLSI) are pre-
sented via a standard Darboux transformation. In this paper, we study the Gerdjokov-Ivanov
equation (DNLSIII) to obtain explicit solutions by using a standard Darboux transformation. Dar-
boux transformations are an important tool for studying the solutions of integrable systems. They
provideauniversalalgorithmicproceduretoderiveexplicitexactsolutionsofintegrablesystems. In
recentyears, therehasbeensomeinterestinsolutionsoftheGerdjikov-Ivanovequationobtainedby
means of Darboux-like transformations [6,12,28]. These solutions are often written in terms of de-
terminants with a complicated structure, where the determinant representations of n-fold Darboux
transformations are obtained by stating and proving a sequence of theorems.
On the other hand, in the present paper, we present a systematic approach to the construction
of solutions of (1.1) by means of a standard Darboux transformation and written in terms of
quasideterminants [7,8]. Quasideterminants have various nice properties which play important
roles in constructing exact solutions of integrable systems [9,10,13,23–25].
This paper is organized as follows. In Section 1.1 below, we give a brief review on quasideter-
minants. In Section 3, we state a standard Darboux theorem for the Gerdjikov-Ivanov system. In
Sections 3.2 and 4, we present the quasideterminant solutions of the Gerdjikov-Ivanov equation by
using the Darboux transformation. Here, the quasideterminants are written in terms of solutions
of linear eigenvalue problems. In Section 5, particular solutions of the Gerdjikov-Ivanov equation
are given for both zero and non-zero seed solutions. The conclusion is given in the final Section 6.
1.1 Quasideterminants
In this short section we recall some of the key elementary properties of quasideterminants. The
reader is referred to the original papers [7,8] for a more detailed and general treatment.
The notion of a quasideterminant was first introduced by Gelfand and Retakh in [7] as a
straightforward way to define the determinant of a matrix with noncommutative entries. Many
equivalent definitions of quasideterminants exist, one such being a recursive definition involving
inverse minors. Let A = (a ) be an n×n matrix with entries over a usually non commutative ring
ij
|A| = a −rj(cid:0)Aij(cid:1)−1ci, (1.2)
ij ij i j
where rj represents the row vector obtained from ith row of A with the jth element removed, ci
i j
represents the column vector obtained from jth column of A with the ith element removed and Aij
is the (n−1)×(n−1) submatrix obtained by deleting the ith row and the jth column from A.
Quasideterminants can also be denoted by boxing the entry about which the expansion is made
(cid:12) (cid:12)
(cid:12) Aij ci (cid:12)
|A| = (cid:12) j (cid:12). (1.3)
ij (cid:12) rj a (cid:12)
(cid:12) i ij (cid:12)
2
If A is an n×n matrix over a commutative ring, then the quasideterminant |A| reduces to a ratio
ij
of determinants
detA
|A| = (−1)i+j . (1.4)
ij detAij
It should be noted that the expansion formula (1.2) is also valid in the case of block matrices
provided the matrix to be inverted is square.
In this paper, we will consider only quasideterminants that are expanded about a term in the
(cid:18) (cid:19)
A B
last column, most usually the last entry. For example considering a block matrix M = ,
C d
whereAisaninvertiblesquarematrixoverR ofarbitrarysizeandB,C arecolumnandrowvectors
overR ofcompatiblelengths,respectively,andd ∈ R,thequasideterminantofM isexpandedabout
d is defined by
(cid:12) (cid:12)
(cid:12)(cid:12) A B (cid:12)(cid:12) = d−CA−1B. (1.5)
(cid:12) C d (cid:12)
2 Gerdjikov-Ivanov equations
Let us consider the pair of Gerdjikov-Ivanov equations
1
iq +q +iq2r + q3r2 = 0, (2.1)
t xx x
2
1
ir −r +ir2q − q2r3 = 0, (2.2)
t xx x
2
where q = q(x,t) and r = r(x,t) are complex valued functions. Equations (2.1) and (2.2) reduce
to the Gerdjikov-Ivanov equation (1.1) for r = q∗ while the choice of r = −q∗ would lead to (1.1)
with the sign of the nonlinear term reversed.
The Lax pair for the Gerjiov-Ivanov system (2.1)–(2.2) is given by
1
L = ∂ +Jλ2−Rλ+ qrJ (2.3)
x
2
M = ∂ +2Jλ4−2Rλ3+qrJλ2+Uλ+W, (2.4)
t
where J, R and U are 2×2 matrices such that
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
i 0 0 q 0 −iq
J = , R = , U = x (2.5)
0 −i r 0 ir 0
x
and
(cid:18) −1 (rq −qr )− 1iq2r2 0 (cid:19)
W = 2 x x 4 . (2.6)
0 1 (rq −qr )+ 1iq2r2
2 x x 4
Here λ is an arbitrary complex number called the eigenvalue (or spectral parameter).
3
3 Darboux Theorem and Dimensional Reductions
Theorem 3.1 ( [5,21,22]). Consider the linear operator
n
(cid:88)
L = ∂ + u ∂i (3.1)
x i y
i=0
where u ∈ R, where R is a ring, in general non-commutative. Let G = θ∂ θ−1, where θ = θ(x,y)
i y
is an invertible eigenfunction of L, so that L(θ) = 0. Then
L˜ = GLG−1 (3.2)
has the same form as L:
n
(cid:88)
L˜ = ∂ + u˜ ∂i (3.3)
x i y
i=0
If φ is any eigenfunction of L then
φ˜= φ −θ θ−1φ (3.4)
x y
is an eigenfunction of L˜. In other words, if L(φ) = 0 then L˜(φ˜) = 0 where φ˜= G(φ).
3.1 Dimensional reduction of Darboux transformation
Here, we describe a reduction of the Darboux transformation from (2+1) to (1+1) dimensions.
We choose to eliminate the y-dependence by employing a ‘separation of variables’ technique. The
reader is referred to the paper [25] for a more detailed treatment. We make the ansatz
φ = φr(x,t)eλy, (3.5)
θ = θr(x,t)eΛy, (3.6)
where λ is a constant scalar and Λ an N × N constant matrix and the superscript r denotes
reduced functions, independent of y. Hence in the dimensional reduction we obtain ∂i (φ) = λiφ
y
and ∂i (θ) = θΛi and so the operator L and Darboux transformation G become
y
n
(cid:88)
Lr = ∂ + u λi, (3.7)
x i
i=0
Gr = λ−θrΛ(θr)−1, (3.8)
where θr is a matrix eigenfunction of Lr such that Lr(θr) = 0, with λ replaced by the matrix Λ,
that is,
n
(cid:88)
θr + u θrΛi = 0. (3.9)
x i
i=0
Below we omit the superscript r for ease of notation.
4
3.2 Iteration of reduced Darboux Transformations
In this section we shall consider iteration of the Darboux transformation and find closed form
expressions for these in terms of quasideterminants.
LetLbeanoperator, forminvariantunderthereducedDarbouxtransformationG = λ−θΛθ−1
discussed above.
Let φ = φ(x,t) be a general eigenfunction of L such that L(φ) = 0. Then
φ˜ = G (φ)
θ
= λφ−θΛθ−1φ
(cid:12) (cid:12)
(cid:12) θ φ (cid:12)
= (cid:12) (cid:12)
(cid:12) θΛ λφ (cid:12)
(cid:12) (cid:12)
is an eigenfunction of L˜ = G LG−1 so that L˜(φ˜) = λφ˜. Let θ for i = 1,...,n, be a particular
θ θ i
set of invertible eigenfunctions of L so that L(θ ) = 0 for λ = Λ , and introduce the notation
i i
Θ = (θ ,...,θ ). To apply the Darboux transformation a second time, let θ = θ and φ = φ be
1 n [1] 1 [1]
(cid:0) (cid:1)
a general eigenfunction of L = L. Then φ = G φ and θ = φ | are eigenfunctions
[1] [2] θ[1] [1] [2] [2] φ→θ2
for L = G L G−1.
[2] θ[1] [1] θ[1]
In general, for n ≥ 1, we define the nth Darboux transform of φ by
φ = λφ −θ Λ θ−1φ , (3.10)
[n+1] [n] [n] n [n] [n]
in which
θ = φ | .
[k] [k] φ→θk
For example,
(cid:12) (cid:12)
(cid:12) θ φ (cid:12)
φ = λφ−θ Λ θ−1φ = (cid:12) 1 (cid:12),
[2] 1 1 1 (cid:12) θ Λ λφ (cid:12)
(cid:12) 1 1 (cid:12)
φ = λφ −θ Λ θ−1φ
[3] [2] [2] 2 [2] [2]
(cid:12) (cid:12)
(cid:12) θ1 θ2 φ (cid:12)
(cid:12) (cid:12)
= (cid:12) θ1Λ1 θ2Λ2 λφ (cid:12).
(cid:12) (cid:12)
(cid:12) θ Λ2 θ Λ2 λ2φ (cid:12)
(cid:12) 1 1 2 2 (cid:12)
After n iterations, we get
(cid:12) θ θ ... θ φ (cid:12)
(cid:12) 1 2 n (cid:12)
(cid:12) (cid:12)
θ Λ θ Λ ...θ Λ λφ
(cid:12) 1 1 2 2 n n (cid:12)
φ = (cid:12)(cid:12) θ1Λ21 θ2Λ22...θnΛ2n λ2φ (cid:12)(cid:12). (3.11)
[n+1] (cid:12) . . . . (cid:12)
(cid:12) . . . . (cid:12)
. . ... . .
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) θ1Λn1 θ2Λn2 ...θnΛnn λnφ (cid:12)
4 Constructing Solutions for Gerdjikov-Ivanov Equation
InthissectionwedeterminethespecificeffectoftheDarbouxtransformationG = λ−θΛθ−1 onthe
2×2 Lax operators L,M given by (2.3),(2.4). Here θ is a eigenfunction satisfying L(θ) = M(θ) = 0
5
with 2×2 matrix eigenvalue Λ. By supposing that L is transformed to a new operator L˜, say, we
calculate that the effect of the Darboux transformation L˜ = GLG−1 is such that
R˜ = R−(cid:2)J,θΛθ−1(cid:3) (4.1)
and
1
R˜θΛθ−1−θΛθ−1R+ J(q˜r˜−qr) = 0, (4.2)
2
(cid:0)θΛθ−1(cid:1) + 1 (cid:2)J(cid:0)θΛθ−1(cid:1)q˜r˜−θΛθ−1Jqr(cid:3) = 0. (4.3)
x 2
From (4.2), we see that θΛθ−1 must be an anti-diagonal matrix, antidiag(a,b), say, and then from
(4.3) the multiplication of the anti-diagonal terms must be constant (ab = constant). Guided by
this, we choose
(cid:18) (cid:19)
1 0
Λ = λ. (4.4)
0 −1
Finally, the condition θΛθ−1 = antidiag(a,b) leads to the requirement that the matrix θ has
the structure
(cid:18) (cid:19)
θ θ
θ = 11 12 , (4.5)
θ θ
21 22
where θ θ +θ θ = 0.
11 22 12 21
For notational convenience, we introduce a 2 × 2 matrix P = (p ) (i,j = 1,2) such that
ij
R = [J,P], and hence
(cid:18) (cid:19)
1 p q
P = 11 . (4.6)
2i −r p
22
From (4.1), since R = [J,P], we have
P˜ = P −θΛθ−1 (4.7)
which can be written in a quasideterminant structure as
(cid:12) (cid:12)
P˜ = P +(cid:12)(cid:12) θ I2 (cid:12)(cid:12). (4.8)
(cid:12) θΛ 02 (cid:12)
We rewrite (4.7) as
P = P −θ Λ θ−1 (4.9)
[2] [1] [1] 1 [1]
where P = P, P = P˜, θ = θ = θ, Λ = Λ and λ = λ . Then after n repeated Darboux
[1] [2] [1] 1 1 1
transformations, we have
P = P −θ Λ θ−1 (4.10)
[n+1] [n] [n] n [n]
6
in which θ = φ | . We express P in quasideterminant form as
[k] [k] φ→θk [n+1]
(cid:12) θ θ ... θ 0 (cid:12)
(cid:12) 1 2 n 2 (cid:12)
(cid:12) (cid:12)
θ Λ θ Λ ...θ Λ 0
(cid:12) 1 1 2 2 n n 2 (cid:12)
(cid:12) . . . . (cid:12)
(cid:12) .. .. ... .. .. (cid:12)
P = P +(cid:12) (cid:12). (4.11)
[n+1] (cid:12)(cid:12) θ1Λ1n−2 θ2Λ2n−2...θnΛnn−2 02 (cid:12)(cid:12)
(cid:12)(cid:12) θ1Λ1n−1 θ2Λ2n−1...θnΛnn−1 I2 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) θ Λn θ Λn...θ Λn 0 (cid:12)
1 1 2 2 n n 2
We now express each θ , Λ as a 2×2 matrix
i i
(cid:18) (cid:19) (cid:18) (cid:19)
φ φ 1 0
θ = 2i−1 2i , Λ = λ (4.12)
i ψ ψ i 0 −1 i
2i−1 2i
so that
(cid:18) φ (−1)kφ (cid:19)
θ Λk = 2i−1 2i λk (4.13)
i i ψ (−1)kψ i
2i−1 2i
for positive integers i,k = 1,...,n. Here the relation φ ψ +φ ψ = 0 holds.
2i−1 2i 2i 2i−1
Let
(cid:18) φ(n) (cid:19)
Θ(n) = (θ Λn,...,θ Λn) = , (4.14)
1 1 n n ψ(n)
where
φ(n) = (λnφ ,(−λ )nφ ,...,λnφ ,(−λ )nφ ),
1 1 1 2 n 2n−1 n 2n
ψ(n) = (λnψ ,(−λ )nψ ,...,λnψ ,(−λ )nψ )
1 1 1 2 n 2n−1 n 2n
denote 1×2n row vectors. Thus, (4.11) can be rewritten as
(cid:12) (cid:12)
(cid:12) Θ(cid:98) E (cid:12)
P = P +(cid:12) (cid:12), (4.15)
[n+1] (cid:12) θ(n) 0 (cid:12)
(cid:12) 2 (cid:12)
(cid:16) (cid:17)
where Θ(cid:98) = θiΛji−1 and E = (e2n−1,e2n) denote 2n×2n and 2n×2 matrices respectively,
i,j=1,...,n
where e represents a column vector with 1 in the ith row and zeros elsewhere. Hence, we obtain
i
 (cid:12) (cid:12) (cid:12) (cid:12) 
(cid:12)(cid:12) Θ(cid:98) e2n−1 (cid:12)(cid:12) (cid:12)(cid:12) Θ(cid:98) e2n (cid:12)(cid:12)
 (cid:12) φ(n) 0 (cid:12) (cid:12) φ(n) 0 (cid:12) 
 (cid:12) (cid:12) (cid:12) (cid:12) 
 
P[n+1] = P + . (4.16)
 (cid:12) (cid:12) (cid:12) (cid:12) 
 (cid:12)(cid:12) Θ(cid:98) e2n−1 (cid:12)(cid:12) (cid:12)(cid:12) Θ(cid:98) e2n (cid:12)(cid:12) 
(cid:12) ψ(n) 0 (cid:12) (cid:12) ψ(n) 0 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
By comparing with (4.6), we immediately see that q and r can be expressed as quaside-
[n+1] [n+1]
terminants, namely,
(cid:12) (cid:12) (cid:12) (cid:12)
q = q+2i(cid:12)(cid:12) Θ(cid:98) e2n (cid:12)(cid:12), r = r−2i(cid:12)(cid:12) Θ(cid:98) e2n−1 (cid:12)(cid:12). (4.17)
[n+1] (cid:12) φ(n) 0 (cid:12) [n+1] (cid:12) ψ(n) 0 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
7
We now consider the linear eigenvalue problems L(Φ ) = M(Φ ) = 0, where the operators L, M
i i
are given in (2.3)-(2.4) and Φ denotes n distinct eigenfunctions as
i
(cid:18) (cid:19)
φ
Φ = i (i = 1,...,n). (4.18)
i ψ
i
Thus, the pair q and r are written with respect to n, where n is an odd (n = 2k−1) or
[n+1] [n+1]
even number (n = 2k), and k ∈ N is a positive integer.
In the case of n odd (n = 2k−1)
(cid:12) (cid:12)
ψ ψ ... ψ 0
(cid:12) 1 2 n (cid:12)
(cid:12) (cid:12)
(cid:12) φ1λ1 φ2λ2 ... φnλn 0 (cid:12)
(cid:12)(cid:12) ψ1λ21 ψ2λ22 ... ψnλ2n 0 (cid:12)(cid:12)
(cid:12) . . . . (cid:12)
q[n+1] = q+2i(cid:12)(cid:12) .. .. .. .. (cid:12)(cid:12), (4.19)
(cid:12)(cid:12) φ1λn1−2 φ2λ2n−2 ... φnλnn−2 0 (cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) ψ1λn1−1 ψ2λn2−1 ... ψnλnn−1 1 (cid:12)(cid:12)(cid:12)
(cid:12) φ λn φ λn ... φ λn 0 (cid:12)
1 1 2 2 n n
(cid:12) (cid:12)
φ φ ... φ 0
(cid:12) 1 2 n (cid:12)
(cid:12) (cid:12)
(cid:12) ψ1λ1 ψ2λ2 ... ψnλn 0 (cid:12)
(cid:12)(cid:12) φ1λ21 φ2λ22 ... φnλ2n 0 (cid:12)(cid:12)
(cid:12) . . . . (cid:12)
r[n+1] = r−2i(cid:12)(cid:12) .. .. .. .. (cid:12)(cid:12). (4.20)
(cid:12)(cid:12) ψ1λ1n−2 ψ2λn2−2 ... ψnλnn−2 0 (cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) φ1λ1n−1 φ2λn2−1 ... φnλnn−1 1 (cid:12)(cid:12)(cid:12)
(cid:12) ψ λn ψ λn ... ψ λn 0 (cid:12)
1 1 2 2 n n
For n = 1, we obtain a pair of new solutions for the Gerdjikov-Ivanov system (2.1)-(2.2)
(cid:12) (cid:12)
(cid:12) ψ1 1 (cid:12)
q[2] = q+2i(cid:12)(cid:12) φ1λ1 0 (cid:12)(cid:12)
φ
1
= q−2iλ , (4.21)
1
ψ
1
(cid:12) (cid:12)
(cid:12) φ1 1 (cid:12)
r[2] = r−2i(cid:12)(cid:12) ψ1λ1 0 (cid:12)(cid:12)
ψ
1
= r+2iλ , (4.22)
1
φ
1
where Φ = (φ ,ψ )T is a solution of the eigenvalue problems L(Φ ) = M(Φ ) = 0.
1 1 1 1 1
8
In the case of n even (n = 2k)
(cid:12) (cid:12)
φ φ ... φ 0
(cid:12) 1 2 n (cid:12)
(cid:12) (cid:12)
(cid:12) ψ1λ1 ψ2λ2 ... ψnλn 0 (cid:12)
(cid:12)(cid:12) φ1λ21 φ2λ22 ... φnλ2n 0 (cid:12)(cid:12)
(cid:12) . . . . (cid:12)
q[n+1] = q+2i(cid:12)(cid:12) .. .. .. .. (cid:12)(cid:12), (4.23)
(cid:12)(cid:12) φ1λn1−2 φ2λ2n−2 ... φnλnn−2 0 (cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) ψ1λn1−1 ψ2λn2−1 ... ψnλnn−1 1 (cid:12)(cid:12)(cid:12)
(cid:12) φ λn φ λn ... φ λn 0 (cid:12)
1 1 2 2 n n
(cid:12) (cid:12)
ψ ψ ... ψ 0
(cid:12) 1 2 n (cid:12)
(cid:12) (cid:12)
(cid:12) φ1λ1 φ2λ2 ... φnλn 0 (cid:12)
(cid:12)(cid:12) ψ1λ21 ψ2λ22 ... ψnλ2n 0 (cid:12)(cid:12)
(cid:12) . . . . (cid:12)
r[n+1] = r−2i(cid:12)(cid:12) .. .. .. .. (cid:12)(cid:12). (4.24)
(cid:12)(cid:12) ψ1λ1n−2 ψ2λn2−2 ... ψnλnn−2 0 (cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) φ1λ1n−1 φ2λn2−1 ... φnλnn−1 1 (cid:12)(cid:12)(cid:12)
(cid:12) ψ λn ψ λn ... ψ λn 0 (cid:12)
1 1 2 2 n n
For n = 2, we have
(cid:12) (cid:12)
(cid:12) φ1 φ2 0 (cid:12)
(cid:12) (cid:12)
q[3] = q+2i(cid:12) ψ1λ1 ψ2λ2 1 (cid:12) , (4.25)
(cid:12) (cid:12)
(cid:12) φ λ2 φ λ2 0 (cid:12)
1 1 2 2
(cid:12) (cid:12)
(cid:12) ψ1 ψ2 0 (cid:12)
(cid:12) (cid:12)
r[3] = r−2i(cid:12) φ1λ1 φ2λ2 1 (cid:12) . (4.26)
(cid:12) (cid:12)
(cid:12) ψ λ2 ψ λ2 0 (cid:12)
1 1 2 2
Thus, we obtain a pair of new solutions for the system (2.1)-(2.2), namely
q = q−2i(cid:0)λ2−λ2(cid:1) φ1φ2 , (4.27)
[3] 1 2 λ ψ φ −λ φ ψ
1 1 2 2 1 2
r = r+2i(cid:0)λ2−λ2(cid:1) ψ1ψ2 , (4.28)
[3] 1 2 λ φ ψ −λ ψ φ
1 1 2 2 1 2
where Φ = (φ ,ψ )T is a solution of the eigenvalue problems L(Φ ) = M(Φ ) = 0 (i = 1,2).
i i i i i
Reduction
(cid:18) (cid:19)
φ
The eigenfunction Φ = k associated with the eigenvalue λ has the following relations when
k ψ k
k
we choose the reduction r = q∗ :
[n+1] [n+1]
ψ = φ∗ for real λ , (4.29)
k k k
ψ = φ∗ when λ = λ∗ (k (cid:54)= l), (4.30)
k l k l
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where k ∈ N. There are many ways to guarantee the reduction r = q∗ for the n-fold
[n+1] [n+1]
Darboux transformations when n > 2. In the present paper we will restrict ourselves to the
reductions r = q∗ and r = q∗ . For the one-fold Darboux transformation, the reduction
[2] [2] [3] [3]
r = q∗ implies
[2] [2]
ψ = φ∗ for λ ∈ R. (4.31)
1 1 1
Furthermore, for the two-fold Darboux transformation, in order that r = q∗ , the eigenfunctions
[3] [3]
Φ = (φ ,ψ )T and Φ = (φ ,ψ )T with the eigenvalues λ ,λ , either of the following conditions
1 1 1 2 2 2 1 2
hold:
ψ = φ∗, ψ = φ∗ for λ ,λ ∈ R or (4.32)
1 1 2 2 1 2
ψ = φ∗, ψ = φ∗ for λ∗ = λ . (4.33)
1 2 2 1 2 1
5 Particular solutions
Let us consider the spectral problem L(Φ) = M(Φ) = 0 with eigenvalue λ, where Φ = (φ,ψ)T and
L,M are given by (2.3)-(2.4) so that
1
Φ +JΦλ2−RΦλ+ qrJΦ = 0, (5.1)
x
2
Φ +2JΦλ4−2RΦλ3+qrJΦλ2+UΦλ+WΦ = 0. (5.2)
t
5.1 Solutions for the vacuum
For q = r = 0, the above equations transform into the first-order linear system
Φ +JΦλ2 = 0 (5.3)
x
Φ +2JΦλ4 = 0 (5.4)
t
which has solution
φk = e−iλ2k(x+2λ2kt), ψk = eiλ2k(x+2λ2kt), (5.5)
where k ∈ N.
Case 1 (n = 1)
For one single Darboux transformation, due to the required reduction r = q∗, we must take λ
1
to be real and ψ1 = φ∗1. By substituting φ1 = e−iλ21(x+2λ21t) and ψ1 = eiλ21(x+2λ21t) into (4.21), we
obtain a new solution q for the GI equation (1.1) as
[2]
q[2] = −2iλ1e−2iλ21(x+2λ21t), (5.6)
where r = q∗ . This, of course, is not a soliton but a periodic solution. It is obvious that
[2] [2]
|q |2 = constant so that it satisfies a linear equation iq +q = 0 obtained from (1.1). However,
[2] t xx
this is not an interesting solution obtained by the use of the Darboux transformation.
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