Geometry,
Integrability
and
IX
Quantization
Ninth International Conference on
Geometry, Integrability and Quantization
June 8–13, 2007, Varna, Bulgaria
Ivaïlo M. Mladenov, Editor
SOFTEX, Sofia 2008, pp 36–65
ON MULTICOMPONENT MKDV EQUATIONS ON SYMMETRIC
SPACES OF DIII-TYPE AND THEIR REDUCTIONS
VLADIMIR S. GERDJIKOV and NIKOLAY A. KOSTOV
Institute for Nuclear Research and Nuclear Energy
Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
Abstract. New reductions for the multicomponent modified Korteveg de
Vries (MMKdV) equations on the symmetric spaces of DIII-type are derived
using the approach based on the reduction group introduced by A. Mikhailov.
The relevant inverse scattering problem is studied and reduced to a RiemannHilbert problem. The minimal sets of scattering data Ti , i = 1, 2 which allow
one to reconstruct uniquely both the scattering matrix and the potential of the
Lax operator are defined. The effect of the new reductions on the hierarchy of
Hamiltonian structures of MMKdV and on Ti are studied. We illustrate our
results by the MMKdV equations related to the algebra g ' so(8) and derive
several new MMKdV-type equations using group of reductions isomorphic
to Z2 , Z3 , Z4 .
1. Introduction
The modified Korteweg de-Vries equation (MKdV) [27]
qt + qxxx + 6qx q 2 (x, t) = 0,
= ±1
(1)
has natural multicomponent generalizations (MMKdV) related to the symmetric
spaces [3]. They can be integrated by the ISM using the fact that they allow the
following Lax representation
d
Lψ ≡ i
+ Q(x, t) − λJ ψ(x, t, λ) = 0
(2)
dx
0 q
1 0
Q(x, t) =
,
σ3 =
(3)
p 0
0 −1
d
2
3
M ψ ≡ i + V0 (x, t) + λV1 (x, t) + λ V2 (x, t) − 4λ J ψ(x, t, λ)
dt
(4)
= ψ(x, t, λ)C(λ)
36
Multicomponent mKdV Equations and Their Reductions
V2 (x, t) = 4Q(x, t),
V1 (x, t) = 2iJQx + 2JQ2
V0 (x, t) = −Qxx − 2Q3 .
37
(5)
(6)
The corresponding MMKdV equations take the form
∂Q ∂ 3 Q
+ 3 Qx Q2 + Q2 Qx = 0.
(7)
+
3
∂t
∂x
The analysis in [2, 3, 11] reveals a number of important results. These include
the corresponding multicomponent generalizations of KdV equations and the generalized Miura transformations interrelating them with the generalized MMKdV
equations, two of their most important reductions as well as their Hamiltonians.
Our aim in this paper is to explore new types of reductions of the MMKdV equations. To this end we make use of the reduction group introduced by Mikhailov
[22, 24] which allows one to impose algebraic reductions on the coefficients of
Q(x, t) which will be automatically compatible with the evolution of MMKdV.
Similar problems have been analyzed for the N -wave type equations related to the
simple Lie algebras of rank 2 and 3 [16, 17] and the multicomponent NLS equations [18, 19]. Here we illustrate our analysis by the MMKdV equations related to
the algebras g ' so(2r) which are linked to the DIII-type symmetric spaces series. Due to the fact that the dispersion law for MNLS is proportional to λ2 while
for MMKdV it is proportional to λ3 the sets of admissible reductions for these two
NLEE equations differ substantially.
In the next Section 2 we give some preliminaries on the scattering theory for L, the
reduction group and graded Lie algebras. In Section 3 we construct the fundamental analytic solutions of L, formulate the corresponding Riemann-Hilbert problem
and introduce the minimal sets of scattering data Ti , i = 1, 2 which define uniquely
both the scattering matrix and the solution of the MMKdV Q(x, t). Some of these
facts have been discussed in more details in [18], others had to be modified and
extended so that they adequately take into account the peculiarities of the DIIItype symmetric spaces. In particular we modified the definition of the fundamental
analytic solution which lead to changes in the formulation of the Riemann-Hilbert
problem. In Section 4 we first briefly outline the hierarchy of Hamiltonian structures for the generic MMKdV equations. Next we list nontrivial examples of two
classes of reductions of the MMKdV equations related to the algebra so(8). The
first class is performed with automorphisms of so(8) that preserve J and the second
class uses automorphisms that map J into −J. While the reductions of first type
can be applied both to MNLS and MMKdV equations, the reductions of second
type can be applied only to MMKdV equations. Under them “half” of the members of the Hamiltonian hierarchy become degenerated [3, 9]. For both classes
of reductions we find examples with groups of reductions isomorphic to Z2 , Z3
and Z4 . We also provide the corresponding reduced Hamiltonians and symplectic
38
Vladimir S. Gerdjikov and Nikolay A. Kostov
forms and Poisson brackets. At the end of Section 4 we derive the effects of these
reductions on the scattering matrix and on the minimal sets of scattering data. The
last section contains some conclusions.
2. Preliminaries
In this section we outline some of the well known facts about the spectral theory
of the Lax operators of the type (2).
2.1. The Scattering Problem for L
Here we briefly outline the basic facts about the direct and the inverse scattering
problems [4,5,7,8,10,15,25,26,28,29] for the system (2) for the class of potentials
Q(x, t) that are smooth enough and fall off to zero fast enough for x → ±∞ for all
t. In what follows we treat DIII-type symmetric spaces which means that Q(x, t)
is an element of the algebra so(2r). In the examples below we take r = 4 and
g ' so(8). For convenience we choose the following definition for the orthogonal
algebras and groups
X ∈ so(2r) −→ X + S0 X T Ŝ0 = 0,
T ∈ SO(2r) −→ S0 T T Ŝ0 = T̂
where the “hat” denotes the inverse matrix T̂ ≡ T −1 and
r
0 s X
0
k+1
S0 ≡
(−1)
Ek,k̄ + Ek̄,k =
,
k̄ = 2r + 1 − k.
ŝ0 0
(8)
(9)
k=1
Here and below by Ejk we denote a 2r × 2r matrix with just one non-vanishing
and equal to 1 matrix element at j, k-th position: (Ejk )mn = δjm δkn . Obviously
S02 = 11.
The main tool for solving the direct and inverse scattering problems are the Jost
solutions which are fundamental solutions defined by their asymptotics at x →
±∞
lim ψ(x, λ)eiλJx = 11,
lim φ(x, λ)eiλJx = 11.
(10)
x→∞
x→−∞
Along with the Jost solutions we introduce
ξ(x, λ) = ψ(x, λ)eiλJx ,
ϕ(x, λ) = φ(x, λ)eiλJx
(11)
which satisfy the following linear integral equations
Z x
ξ(x, λ) = 11 + i
dye−iλJ(x−y) Q(y)ξ(y, λ)eiλJ(x−y)
Z∞x
ϕ(x, λ) = 11 + i
dye−iλJ(x−y) Q(y)ϕ(y, λ)eiλJ(x−y) .
(12)
(13)
−∞
These are Volterra type equations which, have solutions providing one can ensure
the convergence of the integrals in the right hand side. For λ real the exponential
39
Multicomponent mKdV Equations and Their Reductions
factors in (12) and (13) are just oscillating and the convergence is ensured by the
fact that Q(x, t) is rapidly vanishing for x → ∞.
Remark 1. It is an well known fact that if the potential Q(x, t) ∈ so(2r) then
the corresponding Jost solutions of equation (2) take values in the corresponding
group, i.e., ψ(x, λ), φ(x, λ) ∈ SO(2r).
The Jost solutions as whole can not be extended for im λ 6= 0. However, some of
their columns can be extended for λ ∈ C+ , others – for λ ∈ C− . More precisely
we can write down the Jost solutions ψ(x, λ) and φ(x, λ) in the following blockmatrix form
ψ(x, λ) = |ψ − (x, λ)i, |ψ + (x, λ)i ,
±
|ψ (x, λ)i =
ψ±
1 (x, λ)
ψ±
2 (x, λ)
φ(x, λ) = |φ+ (x, λ)i, |φ− (x, λ)i
!
,
φ±
1 (x, λ)
φ±
2 (x, λ)
±
|φ (x, λ)i =
!
(14)
where the superscript + and (respectively −) shows that the corresponding r × r
block-matrices allow analytic extension for λ ∈ C+ (respectively λ ∈ C− ).
Solving the direct scattering problem means given the potential Q(x) to find the
scattering matrix T (λ). By definition T (λ) relates the two Jost solutions
φ(x, λ) = ψ(x, λ)T (λ),
T (λ) =
+
a (λ) −b− (λ)
b+ (λ) a− (λ)
(15)
and has compatible block-matrix structure. In what follows we will need also the
inverse of the scattering matrix
ψ(x, λ) = φ(x, λ)T̂ (λ),
T̂ (λ) ≡
c− (λ) d− (λ)
−d+ (λ) c+ (λ)
(16)
where
c− (λ) = â+ (λ)(11 + ρ− ρ+ )−1 = (11 + τ + τ − )−1 â+ (λ)
(17a)
d− (λ) = â+ (λ)ρ− (λ)(11 + ρ+ ρ− )−1 = (11 + τ + τ − )−1 τ + (λ)â− (λ)
(17b)
c+ (λ) = â− (λ)(11 + ρ+ ρ− )−1 = (11 + τ − τ + )−1 â− (λ)
(17c)
+
−
+
− + −1
d (λ) = â (λ)ρ (λ)(11 + ρ ρ )
− + −1 −
= (11 + τ τ )
+
τ (λ)â (λ).
(17d)
The diagonal blocks of T (λ) and T̂ (λ) allow analytic continuation off the real axis,
namely a+ (λ), c+ (λ) are analytic functions of λ for λ ∈ C+ , while a− (λ), c− (λ)
are analytic functions of λ for λ ∈ C− . We introduced also ρ± (λ) and τ ± (λ) the
multicomponent generalizations of the reflection coefficients (for the scalar case,
see [1, 6, 21])
ρ± (λ) = b± â± (λ) = ĉ± d± (λ),
τ ± (λ) = â± b∓ (λ) = d∓ ĉ± (λ).
(18)
40
Vladimir S. Gerdjikov and Nikolay A. Kostov
The reflection coefficients do not have analyticity properties and are defined only
for λ ∈ R.
From Remark 1 one concludes that T (λ) ∈ SO(2r), therefore it must satisfy the
second of the equations in (8). As a result we get the following relations between
c± , d± and a± , b±
c+ (λ) = sˆ0 a+,T (λ)s0 ,
c− (λ) = s0 a−,T (λ)ŝ0
d+ (λ) = −sˆ0 b+,T (λ)s0 ,
d− (λ) = −s0 b−,T (λ)ŝ0
(19)
and in addition we have
ρ+ (λ) = −ŝ0 ρ+,T (λ)s0 ,
ρ− (λ) = −s0 ρ−,T (λ)ŝ0
τ + (λ) = −s0 τ +,T (λ)ŝ0 ,
τ − (λ) = −ŝ0 τ −,T (λ)s0 .
(20)
Next we need also the asymptotics of the Jost solutions and the scattering matrix
for λ → ∞
lim φ(x, λ)eiλJx = lim ψ(x, λ)eiλJx = 11,
λ→−∞
λ→∞
−
lim a− (λ) = lim c+ (λ) = 11.
lim a+ (λ) = lim c (λ) = 11,
λ→∞
lim T (λ) = 11
λ→∞
λ→∞
λ→∞
(21)
λ→∞
The inverse to the Jost solutions ψ̂(x, λ) and φ̂(x, λ) are solutions to
i
dψ̂
− ψ̂(x, λ)(Q(x) − λJ) = 0
dx
(22)
satisfying the conditions
lim e−iλJx φ̂(x, λ) = 11.
lim e−iλJx ψ̂(x, λ) = 11,
x→∞
x→−∞
(23)
Now it is the collections of rows of ψ̂(x, λ) and φ̂(x, λ) that possess analytic properties in λ
ψ̂(x, λ) =
±
hψ̂ (x, λ)| =
hψ̂ + (x, λ)|
hψ̂ − (x, λ)|
−
!
,
± ±1 ±
(s±1
0 ψ 2 , s0 ψ 1 )(x, λ),
φ̂(x, λ) =
±
hφ̂ (x, λ)| =
hφ̂ (x, λ)|
+
hφ̂ (x, λ)|
!
(24)
± ∓1 ±
(s∓1
0 ψ 2 , s0 ψ 1 )(x, λ).
Just like the Jost solutions, their inverse (24) are solutions to linear equations (22)
with regular boundary conditions (23) and therefore they have no singularities on
the real axis λ ∈ R. The same holds true also for the scattering matrix T (λ) =
ψ̂(x, λ)φ(x, λ) and its inverse T̂ (λ) = φ̂(x, λ)ψ(x, λ), i.e.,
a+ (λ) = hψ̂ + (x, λ)|φ+ (x, λ)i,
a− (λ) = hψ̂ − (x, λ)|φ− (x, λ)i
(25)
c− (λ) = hφ̂− (x, λ)|ψ − (x, λ)i
(26)
as well as
c+ (λ) = hφ̂+ (x, λ)|ψ + (x, λ)i,
Multicomponent mKdV Equations and Their Reductions
41
are analytic for λ ∈ C± and have no singularities for λ ∈ R. However they may
become degenerate (i.e., their determinants may vanish) for some values λ±
j ∈ C±
of λ. Below we briefly analyze the structure of these degeneracies and show that
they are related to discrete spectrum of L.
2.2. The Reduction Group of Mikhailov
The reduction group GR is a finite group which preserves the Lax representation (2), i.e., it ensures that the reduction constraints are automatically compatible with the evolution. GR must have two realizations: i) GR ⊂ Aut g and ii)
GR ⊂ Conf C, i.e., as conformal mappings of the complex λ-plane. To each
gk ∈ GR we relate a reduction condition for the Lax pair as follows [24]
Ck (L(Γk (λ))) = ηk L(λ),
Ck (M (Γk (λ))) = ηk M (λ)
(27)
where Ck ∈ Aut g and Γk (λ) ∈ Conf C are the images of gk and ηk = 1 or −1
depending on the choice of Ck . Since GR is a finite group then for each gk there
exist an integer Nk such that gkNk = 11.
More specifically the automorphisms Ck , k = 1, . . . , 4 listed above lead to the
following reductions for the potentials U (x, t, λ) and V (x, t, λ) of the Lax pair
U (x, t, λ) = Q(x, t) − λJ,
V (x, t, λ) =
2
X
λk Vk (x, t) − 4λ3 J
(28)
k=0
of the Lax representation
1) C1 (U † (κ1 (λ))) = U (λ),
C1 (V † (κ1 (λ))) = V (λ)
(29)
2) C2 (U T (κ2 (λ))) = −U (λ),
C2 (V T (κ2 (λ))) = −V (λ)
(30)
∗
3) C3 (U (κ1 (λ))) = −U (λ),
4) C4 (U (κ2 (λ))) = U (λ),
∗
C3 (V (κ1 (λ))) = −V (λ)
(31)
C4 (V (κ2 (λ))) = V (λ).
(32)
The condition (27) is obviously compatible with the group action.
2.3. Cartan-Weyl Basis and Weyl Group for so(2r)
Here we fix the notations and the normalization conditions for the Cartan-Weyl
generators of g ' so(2r), see e.g. [20]. The root system ∆ of this series of
simple Lie algebras consists of the roots ∆ ≡ {±(ei − ej ), ±(ei + ej )} where
1 ≤ i < j ≤ r. We introduce an ordering in ∆ by specifying the set of positive
roots ∆+ ≡ {ei − ej , ei + ej } for 1 ≤ i < j ≤ r. Obviously all roots have the
same length equal to 2.
42
Vladimir S. Gerdjikov and Nikolay A. Kostov
We introduce the basis in the Cartan subalgebra by hk ∈ h, k = 1, . . . , r where
{hk } are the Cartan elements dual to the orthonormal basis {ek } in the root space
Er . Along with hk we introduce also
Hα =
r
X
(α, ek )hk ,
α∈∆
(33)
k=1
where (α, ek ) is the scalar product in the root space Er between the root α and ek .
The basis in so(2r) is completed by adding the Weyl generators Eα , α ∈ ∆.
The commutation relations for the elements of the Cartan-Weyl basis are given
in [20]
[hk , Eα ] = (α, ek )Eα ,
[Eα , E−α ] = Hα
(
[Eα , Eβ ] =
Nα,β Eα+β
0
for α + β ∈ ∆
for α + β ∈
/ ∆ ∪ {0}.
(34)
We will need also the typical 2r-dimensional representation of so(2r). In order
to have the Cartan generators represented by diagonal matrices we modified the
definition of orthogonal matrix, see (8). Using the matrices Ejk defined after equation (9) we get
hk = Ekk − Ek̄k̄ ,
Eei −ej = Eij − (−1)i+j Ej̄ ī
Eei +ej = Eij̄ − (−1)i+j Ej̄ ī ,
E−α = EαT
(35)
where k̄ = 2r + 1 − k.
Pr
We will
denote
by
~
a
=
k=1 ek the r-dimensional vector dual to J ∈ h where
P
J = rk=1 hk . If the root α ∈ ∆+ is positive (negative) then (α, ~a) ≥ 0 ((α, ~a) <
0 respectively). The normalization of the basis is determined by
E−α = EαT ,
hE−α , Eα i = 2,
N−α,−β = −Nα,β .
(36)
The root system ∆ of g is invariant with respect to the Weyl reflections Sα which
act on the vectors ~y ∈ Er specified by the formula
2(α, ~y )
α,
α ∈ ∆.
(37)
Sα ~y = ~y −
(α, α)
All Weyl reflections Sα form a finite group Wg known as the Weyl group. On
the root space this group is isomorphic to Sr ⊗ (Z2 )r−1 where Sr is the group of
permutations of the basic vectors ej ∈ Er . Each of the Z2 groups acts on Er by
changing simultaneously the signs of two of the basic vectors ej .
One may introduce also an action of the Weyl group on the Cartan-Weyl basis,
namely [20]
Sα (Hβ ) ≡ Aα Hβ A−1
α = HSα β
Sα (Eβ ) ≡ Aα Eβ A−1
α = nα,β ESα β ,
nα,β = ±1.
(38)
Multicomponent mKdV Equations and Their Reductions
43
The matrices Aα are given (up to a factor from the Cartan subgroup) by
Aα = eEα e−E−α eEα HA
(39)
where HA is a conveniently chosen element from the Cartan subgroup such that
2 = 11. The formula (39) and the explicit form of the Cartan-Weyl basis in the
HA
typical representation will be used in calculating the reduction condition following
from (27).
2.4. Graded Lie Algebras
One of the important notions in constructing integrable equations and their reductions is the one of graded Lie algebra and Kac-Moody algebras [20]. The standard
construction is based on a finite order automorphism C ∈ Aut g, C N = 11. The
eigenvalues of C are ω k , k = 0, 1, . . . , N − 1, where ω = exp(2πi/N ). To each
eigenvalue there corresponds a linear subspace g(k) ⊂ g determined by
n
g(k) ≡ X ; X ∈ g,
o
C(X) = ω k X .
(40)
N −1
Then g = ⊕ g(k) and the grading condition holds
k=0
h
i
g(k) , g(n) ⊂ g(k+n)
(41)
where k +n is taken modulo N . Thus to each pair {g, C} one can relate an infinitedimensional algebra of Kac-Moody type b
gC whose elements are
X(λ) =
X
Xk λk ,
Xk ∈ g(k) .
(42)
k
The series in (42) must contain only finite number of negative (positive) powers of
λ and g(k+N ) ≡ g(k) . This construction is a most natural one for Lax pairs and we
will see that due to the grading condition (41) we can always impose a reduction
on L(λ) and M (λ) such that both U (x, t, λ) and V (x, t, λ) ∈ b
gC . In the case of
symmetric spaces N = 2 and C is the Cartan involution. Then one can choose the
Lax operator L in such a way that
Q ∈ g(1) ,
J ∈ g(0)
(43)
(0) consists of all elements of g
as it is the case in (2). Here the subalgebra gP
commuting with J. The special choice of J = rk=1 hk taken above allows us to
split the set of all positive roots ∆+ into two subsets
+
∆+ = ∆+
0 ∪ ∆1 ,
∆+
0 = {ei − ej }i<j ,
∆+
1
∆+
1 = {ei + ej }i<j .
(44)
Obviously the elements α ∈
have the property α(J) = (α, ~a) = 2, while the
elements β ∈ ∆+
have
the
property
β(J) = (β, ~a) = 0.
0
44
Vladimir S. Gerdjikov and Nikolay A. Kostov
3. The Fundamental Analytic Solutions and the Riemann-Hilbert
Problem
3.1. The Fundamental Analytic Solutions
The next step is to construct the fundamental analytic solutions (FAS) χ± (x, λ)
of (2). Here we slightly modify the definition in [18] to ensure that χ± (x, λ) ∈
SO(2r). Thus we define
χ+ (x, λ) ≡ |φ+ i, |ψ + ĉ+ i (x, λ) = φ(x, λ)S + (λ) = ψ(x, λ)T − (λ)D+ (λ)
(45)
χ− (x, λ) ≡ |ψ − ĉ− i, |φ− i (x, λ) = φ(x, λ)S − (λ) = ψ(x, λ)T + (λ)D− (λ)
where the block-triangular functions S ± (λ) and T ± (λ) are given by
+
S (λ) =
−
S (λ) =
11 d− ĉ+ (λ)
0
11
0
−d+ ĉ− (λ) 11
11
−
,
0
+ +
b â (λ) 11
11 −b− â− (λ)
0
11
T (λ) =
+
,
T (λ) =
11
(46)
.
The matrices D± (λ) are block-diagonal and equal
D+ (λ) =
+
a (λ)
0
ĉ+ (λ)
0
D− (λ) =
,
−
ĉ (λ)
0
0
a− (λ)
.
(47)
The upper scripts ± here refer to their analyticity properties for λ ∈ C± .
In view of the relations (19) it is easy to check that all factors S ± , T ± and D±
take values in the group SO(2r). Besides, since
+
−
−
+
T (λ) = T − (λ)D+ (λ)Ŝ (λ) = T + (λ)D− (λ)Ŝ (λ)
(48)
T̂ (λ) = S + (λ)D̂+ (λ)T̂ (λ) = S − (λ)D̂− (λ)T̂ (λ)
we can view the factors S ± , T ± and D± as generalized Gauss decompositions
(see [20]) of T (λ) and its inverse.
The relations between c± (λ), d± (λ) and a± (λ), b± (λ) in equation (17) ensure
that equations (48) become identities. From equations (45), (46) we derive
χ+ (x, λ) = χ− (x, λ)G0 (λ),
G0 (λ) =
11
τ−
τ+
11 +
τ −τ +
χ− (x, λ) = χ+ (x, λ)Ĝ0 (λ)
,
Ĝ0 (λ) =
11 +
τ +τ −
−τ −
(49)
−τ +
11
(50)
valid for λ ∈ R. Below we introduce
X ± (x, λ) = χ± (x, λ)eiλJx .
(51)
45
Multicomponent mKdV Equations and Their Reductions
Strictly speaking it is X ± (x, λ) that allow analytic extension for λ ∈ C± . They
have also another nice property, namely their asymptotic behavior for λ → ±∞ is
given by
lim X ± (x, λ) = 11.
(52)
λ→∞
Along with X ± (x, λ) we can use another set of FAS X̃ ± (x, λ) = X ± (x, λ)D̂± ,
which also satisfy equation (52) due to the fact that
lim D± (λ) = 11.
(53)
λ→∞
The analyticity properties of X ± (x, λ) and X̃ ± (x, λ) for λ ∈ C± along with equation (52) are crucial for our considerations.
3.2. The Riemann-Hilbert Problem
The equations (49) and (50) can be written down as
X + (x, λ) = X − (x, λ)G(x, λ),
λ∈R
(54)
where
G(x, λ) = e−iλJx G0 (λ)eiλJx .
(55)
Likewise the second pair of FAS satisfy
X̃ + (x, λ) = X̃ − (x, λ)G̃(x, λ),
λ∈R
(56)
with
−iλJx
G̃(x, λ) = e
G̃0 (λ)e
iλJx
,
G̃0 (λ) =
11 + ρ− ρ+ ρ−
ρ+
11
.
(57)
Equation (54) (respectively equation (56)) combined with (52) is known in the
literature [12] as a Riemann-Hilbert problem (RHP) with canonical normalization.
It is well known that RHP with canonical normalization has unique regular solution
while the matrix-valued solutions X0+ (x, λ) and X0− (x, λ) in (54), obeying (52)
are called regular if det X0± (x, λ) does not vanish for any λ ∈ C± .
Let us now apply the contour-integration method to derive the integral decompositions of X ± (x, λ). To this end we consider the contour integrals
J1 (λ) =
1
2πi
I
γ+
dµ
1
X + (x, µ) −
µ−λ
2πi
I
γ−
dµ
X − (x, µ)
µ−λ
(58)
and
1
dµ
1
dµ
X̃ + (x, µ) −
X̃ − (x, µ)
2πi γ+ µ − λ
2πi γ− µ − λ
where λ ∈ C+ and the contours γ± are shown in Fig. 1.
J2 (λ) =
I
I
(59)
46
Vladimir S. Gerdjikov and Nikolay A. Kostov
6
k
+;1
Y
--
-
1
;
Figure 1. The contours γ± = R ∪ γ±∞ .
Each of these integrals can be evaluated by Cauchy residue theorem. The result for
λ ∈ C+ are
J1 (λ) = X + (x, λ) +
N
X
Res
+
j=1 µ=λj
N
X + (x, µ) X
X − (x, µ)
+
Res
−
µ−λ
µ−λ
j=1 µ=λj
N
X̃ + (x, µ) X
X̃ − (x, µ)
Res
Res
J2 (λ) = X̃ (x, λ) +
+
.
+
−
µ−λ
µ−λ
j=1 µ=λj
j=1 µ=λj
+
N
X
(60)
(61)
The discrete sums in the right hand sides of equations (60) and (61) naturally provide the contribution from the discrete spectrum of L. For the sake of simplicity
we assume that L has a finite number of simple eigenvalues λ±
j ∈ C± and for
additional details see [18]. Let us clarify the above statement. For the 2 × 2
Zakharov-Shabat problem it is well known that the discrete eigenvalues of L are
provided by the zeroes of the transmission coefficients a± (λ), which in that case
are scalar functions. For the more general 2r × 2r Zakharov-Shabat system (2)
the situation becomes more complex because now a± (λ) are r × r matrices. The
±
discrete eigenvalues λ±
j now are the points at which a (λ) become degenerate
and their inverse develop pole singularities. More precisely, we assume that in the
±
±
±
±
vicinities of λ±
j a (λ), c (λ) and their inverse â (λ), ĉ (λ) have the following
decompositions in Taylor series
± ±
±
± ±
±
a± (λ) = a±
j + (λ − λj )ȧj + · · · , c (λ) = cj + (λ − λj )ċj + · · · (62)
±
â (λ) =
â±
j
λ − λ±
j
ˆ± + · · · ,
+ ȧ
j
±
ĉ (λ) =
ĉ±
j
λ − λ±
j
ˆ± + · · ·
+ ȧ
j
(63)
Multicomponent mKdV Equations and Their Reductions
47
±
±
±
where all the leading coefficients a±
j , âj , cj , ĉj are degenerate matrices such
that
±
± ±
±
± ±
â±
ĉ±
(64)
j aj = aj âj = 0,
j cj = cj ĉj = 0.
In addition we have more relations such as
±
ˆ± ±
â±
j ȧj + ȧj aj = 11,
±
ˆ± ±
ĉ±
j ċj + ċj cj = 11
(65)
that are needed to ensure the identities â± (λ)a± (λ) = 11, ĉ± (λ)c± (λ) = 11, etc
for all values of λ.
The assumption that the eigenvalues are simple here means that we have considered
±
only first order pole singularities of â±
j (λ) and ĉj (λ). After some additional considerations we find that the “halfs” of the Jost solutions |ψ ± (x, λ)i and |φ± (x, λ)i
satisfy the following relationships for λ = λ±
j
±
±
|ψj± (x)ĉ±
j i = ±|φj (x)τj i,
±
±
±
|φ±
j (x)âj i = ±|ψj (x)ρj i
(66)
±
±
±
where |ψj± (x)i = |ψ ± (x, λ±
j )i, |φj (x)i = |φ (x, λj )i
± ±
± ±
ρ±
j = ĉj dj = bj âj ,
± ±
±
τj± = â±
j bj = dj ĉj
(67)
±
and the additional coefficients b±
j and dj are constant r×r nondegenerate matrices
which, as we shall see below, are also part of the minimal sets of scattering data
needed to determine the potential Q(x, t).
These considerations allow us to calculate explicitly the residues in equations (60),
(61) with the result
+
(|0i, |φ+
X + (x, µ)
j (x)τj i)
Res
,
=
µ−λ
λ+
µ=λ+
j −λ
j
Res
µ=λ+
j
−
(|φ−
X − (x, µ)
j (x)τj i, |0i)
,
=−
µ−λ
λ+
j −λ
(|ψj+ (x)ρ+
X̃ + (x, µ)
j i, |0i)
Res
=
+
+
µ−λ
λj − λ
µ=λj
(68)
−
−
(|0i, |ψj (x)τj− i)
X̃ (x, µ)
Res
=−
µ−λ
λ+
µ=λ+
j −λ
j
where |0i stands for a collection of r columns whose components are all equal to
zero.
We can also evaluate J1 (λ) and J2 (λ) by integrating along the contours. In integrating along the infinite semi-circles of γ±,∞ we use the asymptotic behavior of
X ± (x, λ) and X̃ ± (x, λ) for λ → ∞. The results are
∞
1
J1 (λ) = 11 +
2πi −∞
Z ∞
1
J2 (λ) = 11 +
2πi −∞
Z
dµ
φ(x, µ)eiµJx K(x, µ)
µ−λ
dµ
ψ(x, µ)eiµJx K̃(x, µ)
µ−λ
K(x, µ) = e−iµJx K0 (µ)eiµJx ,
K̃(x, µ) = e−iµJx K̃0 (µ)eiµJx
(69)
(70)
(71)
48
Vladimir S. Gerdjikov and Nikolay A. Kostov
K0 (µ) =
0
τ + (µ)
−
τ (µ)
0
,
K̃0 (µ) =
0
ρ+ (µ)
−
ρ (µ)
0
(72)
where in evaluating the integrands we made use of equations (15), (17), (54)
and (56).
Equating the right hand sides of (60) and (69), and (61) and (70) we get the following integral decomposition for X ± (x, λ)
X + (x, λ) = 11 +
X − (x, λ) = 11 +
1
2πi
Z ∞
1
2πi
Z ∞
−∞
−∞
N X − (x)K (x)
X
dµ
1,j
j
(73)
X − (x, µ)K1 (x, µ) +
−
µ−λ
λj − λ
j=1
N X + (x)K (x)
X
dµ
2,j
j
(74)
X − (x, µ)K2 (x, µ) −
+
µ−λ
λ
−
λ
j
j=1
where Xj± (x) = X ± (x, λ±
j ) and
−iλ−
j Jx
K1,j (x) = e
0 ρ+
j
τj− 0
!
iλ−
j Jx
e
−iλ+
j Jx
, K2,j (x) = e
0 τj+
ρ−
0
j
!
+
eiλj Jx .
(75)
Equations (73), (74) can be viewed as a set of singular integral equations which are
equivalent to the RHP. For the MNLS these were first derived in [23].
We end this section by a brief explanation of how the potential Q(x, t) can be
recovered provided we have solved the RHP and know the solutions X ± (x, λ).
First we take into account that X ± (x, λ) satisfy the differential equation
dX ±
+ Q(x, t)X ± (x, λ) − λ[J, X ± (x, λ)] = 0
(76)
dx
which must hold true for all λ. From equation (52) and also from the integral
equations (73), (73) one concludes that X ± (x, λ) and their inverse X̂ ± (x, λ) are
regular for λ → ∞ and allow asymptotic expansions of the form
i
X ± (x, λ) = 11 +
∞
X
λ−s Xs (x),
X̂ ± (x, λ) = 11 +
s=1
∞
X
λ−s X̂s (x).
(77)
s=1
Inserting these into equation (76) and taking the limit λ → ∞ we get
Q(x, t) = lim λ(J − X ± (x, λ)J X̂ ± (x, λ)]) = [J, X1 (x)].
λ→∞
(78)
3.3. The Minimal Set of Scattering Data
Obviously, given the potential Q(x) one can solve the integral equations for the
Jost solutions which determine them uniquely. The Jost solutions in turn determine
uniquely the scattering matrix T (λ) and its inverse T̂ (λ). But the potential Q(x)
contains r(r − 1) independent complex-valued functions of x. Thus it is natural
49
Multicomponent mKdV Equations and Their Reductions
to expect that at most r(r − 1) of the coefficients in T (λ) for λ ∈ R will be
independent and the rest must be functions of those.
The set of independent coefficients of T (λ) are known as the minimal set of scattering data. As such we may use any of the following two sets Ti ≡ Ti,c ∪ Ti,d
T1,c ≡ ρ+ (λ), ρ− (λ),
±
T1,d ≡ ρ±
j , λj
n
oN
n
oN
λ∈R ,
T2,c ≡ τ + (λ), τ − (λ),
λ∈R ,
T1,d ≡ τj± , λ±
j
j=1
(79)
j=1
where the reflection coefficients ρ± (λ) and τ ± (λ) were introduced in equation (17),
±
±
λ±
j are (simple) discrete eigenvalues of L and ρj and τj characterize the norming
constants of the corresponding Jost solutions.
Remark 2. A consequence of equation (20) is the fact that S ± (λ), S ± (λ) ∈
SO(2r). These factors can be written also in the form
S ± (λ) = exp 
 X
α∈∆+
1




τα± (λ)E±α  ,

T ± (λ) = exp 
 X
α∈∆+
1
ρ±
α (λ)E±α  . (80)

Taking into account that in the typical representation we have E±α E±β = 0 for all
roots α, β ∈ ∆+
1 we find that
X
τα+ (λ)E±α
=
α∈∆+
1
0 τ + (λ)
0
0
0 ρ+ (λ)
0
0
,
X
τα− (λ)E−α
0
0
−
τ (λ) 0
0
0
−
ρ (λ) 0
=
α∈∆+
1
(81)
X
α∈∆+
1
ρ+
α (λ)E±α =
,
X
α∈∆+
1
ρ−
α (λ)E−α =
where ∆+
1 is a subset of the positive roots of so(2r) defined below in Subsection 3.3. The formulae (81) ensure that the number of independent matrix elements
of τ + (λ) and τ − (λ) (respectively, ρ+ (λ) and ρ− (λ)) equals 2|∆+
1 | = r(r − 1)
which coincides with the number of independent functions of Q(x).
The reflection coefficients ρ± (λ) and τ ± (λ) are defined only on the real λ-axis,
while the diagonal blocks a± (λ) and c± (λ) (or, equivalently, D± (λ)) allow analytic extensions for λ ∈ C± . From the equations (17) there follows that
a+ (λ)c− (λ) = (11 + ρ− ρ+ (λ))−1 ,
a− (λ)c+ (λ) = (11 + ρ+ ρ− (λ))−1 (82)
c− (λ)a+ (λ) = (11 + τ + τ − (λ))−1 ,
c+ (λ)a− (λ) = (11 + τ − τ + (λ))−1 . (83)
50
Vladimir S. Gerdjikov and Nikolay A. Kostov
Given T1 (respectively, T2 ) we determine the right hand sides of (82) (respectively (83)) for λ ∈ R. Combined with the facts about the limits
lim a+ (λ) = lim c− (λ) = lim a− (λ) = lim c+ (λ) = 11
λ→∞
λ→∞
λ→∞
λ→∞
(84)
each of the relations (82), (83) can be viewed as a RHP with canonical normalization. Such RHP can be solved explicitly in the one-component case (provided we
know the locations of their zeroes) by using the Plemelj-Sokhotsky formulae [12].
These zeroes are in fact the discrete eigenvalues of L. One possibility to make use
of these facts is to take log of the determinants of both sides of (82) which leads to
A+ (λ) + C − (λ) = − ln det(11 + ρ− ρ+ (λ),
λ∈R
(85)
where
A± (λ) = ln det a± (λ),
C ± (λ) = ln det c± (λ).
Then Plemelj-Sokhotsky formulae allows us to recover
A(λ) =
i
2π
Z ∞
−∞
A± (λ)
and
(86)
C ± (λ)
N
X
λ − λ+
dµ
j
ln det(11 + ρ− ρ+ (µ)) +
ln
−
µ−λ
λ
−
λ
j
j=1
(87)
where A(λ) = A+ (λ) for λ ∈ C+ and A(λ) = −C − (λ) for λ ∈ C− . In deriv±
±
ing (87) we have also assumed that λ±
j are simple zeroes of A (λ) and C (λ).
Let us consider the reduction condition (29) with C1 from the Cartan subgroup
2 = 11. Then
C1 = diag(B+ , B− ) where the diagonal matrices B± are such that B±
we get the following constraints on the sets T1,2
ρ− (λ) = (B− ρ+ (λ)B+ )† ,
+
†
ρ−
j = (B− ρj B+ ) ,
+ ∗
λ−
j = (λj )
(88)
τ − (λ) = (B+ τ + (λ)B− )† ,
τj− = (B+ τj+ B− )† ,
+ ∗
λ−
j = (λj )
(89)
where j = 1, . . . , N . For more details see Subsection 4.4 and Subsection 4.5.
Remark 3. For certain reductions such as, e.g. Q = −Q† the generalized Zakharov-Shabat system L(λ)ψ = 0 can be written down as an eigenvalue problem
Lψ = λψ(x, λ) where L is a self-adjoint operator. The continuous spectrum of
L fills up the whole real λ-axis thus “leaving no space” for discrete eigenvalues.
Such Lax operators have no discrete spectrum and the corresponding MNLS or
MMKdV equations do not have soliton solutions.
From the general theory of RHP [12] one may conclude that (82), (83) allow unique
solutions provided the number and types of the zeroes λ±
j are properly chosen.
Thus we can outline a procedure which allows one to reconstruct not only T (λ)
and T̂ (λ) and the corresponding potential Q(x) from each of the sets Ti , i = 1, 2:
i) Given T2 (respectively T1 ) solve the RHP (82) (respectively (83)) and construct a± (λ) and c± (λ) for λ ∈ C± .
51
Multicomponent mKdV Equations and Their Reductions
ii) Given T1 we determine b± (λ) and d± (λ) as
b± (λ) = ρ± (λ)a± (λ),
d± (λ) = c± (λ)ρ± (λ)
(90)
d± (λ) = τ ± (λ)c± (λ).
(91)
or if T2 is known then
b± (λ) = a± (λ)τ ± (λ),
iii) The potential Q(x) can be recovered from T1 by solving the RHP (54) and
using equation (78).
Another method for reconstructing Q(x) from Tj uses the interpretation of the ISM
as generalized Fourier transform, see [1, 13, 21].
4. Finite Order Reductions of MMKdV Equations
In order that the potential Q(x, t) be relevant for a DIII-type symmetric space it
must be of the form
Q(x, t) =
X
(qα (x, t)Eα + pα (x, t)E−α )
(92)
α∈∆+
1
or, equivalently
Q(x, t) =
X
qij (x, t)Eei +ej + pij (x, t)E−ei −ej .
(93)
1≤i<j≤r
4.1. Hamiltonian Formulations for the Generic MMKdV Type Equations
Let us, before going into the non-trivial reductions, briefly discuss the Hamiltonian
formulations for the generic (i.e., non-reduced) MMKdV type equations. It is well
known (see [18] and the numerous references therein) that the class of these equations is generated by the so-called recursion operator Λ = 1/2(Λ+ + Λ− ) which
act on generic block-off-diagonal matrix valued function Z(x) by
Z x
dZ
dy
[Q(y),
Z(y)]
.
(94)
+
Q(x),
Λ± Z = i ad−1
J
dx
±∞
Any nonlinear evolution equation (NLEE) integrable via the inverse scattering
method applied to the Lax operator L (2) can be written in the form
∂Q
i ad−1
+ 2f (Λ)Q(x, t) = 0
(95)
J
∂t
where the function f (λ) is known as the dispersion law of this NLEE. The generic
MMKdV equation is a member of this class and is obtained by choosing f (λ) =
−4λ3 . If Q(x, t) is a solution to (95) then the corresponding scattering matrix
satisfy the linear evolution equation
dT
+ f (λ)[J, T (λ, t)] = 0
(96)
i
dt
52
Vladimir S. Gerdjikov and Nikolay A. Kostov
and vice versa. In particular from (96) there follows that a± (λ) and c± (λ) are
time-independent and therefore can be considered as generating functionals of integrals of motion for the NLEE.
If no additional reduction is imposed one can write each of the equations in (95)
in Hamiltonian form. The corresponding Hamiltonian and symplectic form for the
MMKV equation are given by
(0)
HMMKdV
Ω(0)
1 ∞
=
dx tr(JQx Qxx ) − 3 tr(JQ3 Qx )
4 −∞
Z
h
i
1 ∞
−1
−1
=
dx tr adJ δQ(x) ∧0 J, adJ δQ(x)
i −∞
Z
1 ∞
dx tr(JδQ(x) ∧0 δQ(x)).
=
2i −∞
Z
(97)
(98)
The Hamiltonian can be identified as proportional to the fourth coefficient I4 in the
asymptotic expansion of A+ (λ) (84) over the negative powers of λ
A+ (λ) =
∞
X
iIk λ−k .
(99)
k=1
This series of integrals of motion is known as the principal one. The first three of
these integrals take the form
I1 =
1
4
Z ∞
−∞
dx tr(Q2 (x, t)),
I2 = −
i
4
Z ∞
−∞
dx tr(Q ad−1
J Qx )
1 ∞
dx tr(QQxx + 2Q4 )
I3 = −
8 −∞
Z
1 ∞
dx tr(JQx Qxx ) − 3 tr(JQ3 Qx ) .
I4 =
32 −∞
Z
(100)
We will remind also another important result, namely that the gradient of Ik is
expressed through Λ as
1
∇QT (x) Ik = − Λk−1 Q(x, t).
2
(101)
Then the Hamiltonian equations written through Ω(0) and the Hamiltonian vector
field XH (0) in the form
Ω(0) (·, XH (0) ) + δH (0) = 0
(102)
for H (0) given by (97) coincides with the MMKdV equation.
An alternative way to formulate Hamiltonian equations of motion is to introduce
along with the Hamiltonian the Poisson brackets on the phase space M which is
53
Multicomponent mKdV Equations and Their Reductions
the space of smooth functions taking values in g(0) and vanishing fast enough for
x → ±∞, see (93). These brackets can be introduced by
{F, G}(0) = i
Z ∞
−∞
h
i
dx tr ∇QT (x) F, J, ∇QT (x) G
.
(103)
Then the Hamiltonian equations of motions
dqij
dpij
= {qij , H (0) }(0) ,
= {pij , H (0) }(0)
(104)
dt
dt
with the above choice for H (0) again give the MMKdV equation.
Along with this standard Hamiltonian formulation there exist a whole hierarchy of
them. This is a special property of the integrable NLEE. The hierarchy is generated
again by the recursion operator and has the form
(m)
HMMKdV = −8I4+m
(105)
h
i
1 ∞
−1
m
dx tr ad−1
J,
Λ
ad
δQ(x)
.
Ω
=
J δQ(x) ∧
J
0
i −∞
Of course there is also a hierarchy of Poisson brackets
(m)
Z
{F, G}(m) = i
Z ∞
−∞
h
i
dx tr ∇QT (x) F, J, Λ−m ∇QT (x) G
.
(106)
(107)
For a fixed value of m the Poisson bracket {·, ·}(m) is dual to the symplectic form
Ω(m) in the sense that combined with a given Hamiltonian they produce the same
equations of motion. Note that since Λ is an integro-differential operator in general
it is not easy to evaluate explicitly its negative powers. Using this duality one can
avoid the necessity to evaluate negative powers of Λ.
Then the analogs of (102) and (104) take the form
Ω(m) (·, XH (m) ) + δH (m) = 0
dqij
dpij
= {qij , H (−m) }(m) ,
= {pij , H (−m) }(m)
dt
dt
where the hierarchy of Hamiltonians is given by
H (m) = −4
X
fk Ik+1−m .
(108)
(109)
(110)
k
The equations (108) and (109) with the Hamiltonian
H (m) given by (110) will
P
produce the NLEE (95) with dispersion law f (λ) = k fk λk for any value of m.
Remark 4. It is a separate issue to prove that the hierarchies of symplectic structures and Poisson brackets have all the necessary properties. This is done using the
spectral decompositions of the recursion operators Λ± which are known also as
the expansions over the “squared solutions” of L. We refer the reader to the review
papers [14, 18] where he/she can find the proof of the completeness relation for
54
Vladimir S. Gerdjikov and Nikolay A. Kostov
the “squared solutions” along with the proof that any two of the symplectic forms
introduced above are compatible.
In the next two subsections we display new reductions of the MMKdV equations.
4.2. Class A Reductions Preserving J
The class A reductions can be applied also to the NLS type equations. The corresponding automorphisms C preserve J, i.e., C −1 JC = J and are of the form
C −1 U † (x, λ∗ )C = U (x, λ),
U (x, λ) = Q(x, t) − λJ
(111)
where J is an element of the Cartan subalgebra dual to the vector e1 + e2 + e3 + e4 .
In the typical representation of so(8) U (x, λ) takes the form
U (x, t, λ) =
q14
 q24
q(x, t) = 
 q34
0

λ11 q(x, t)
p(x, t) −λ11
q13 q12
0
q23 0
q12 
,
0 q23 −q13 
q34 −q24 q14

(112)
p14 p24 p34
0
 p13 p23
0
p
34 

p(x, t) = 
 p12 0 p23 −p24  .
0 p12 −p13 p14


Remark 5. The automorphisms that satisfy C −1 JC = J naturally preserve the
eigensubspaces of adJ ; in other words their action on the root space maps the
±
±
subsets of roots ∆±
1 onto themselves: C∆1 = ∆1 .
We list here several inequivalent reductions of the ZS system. In the first one we
choose C = C0 to be an element of the Cartan subgroup
C0 = exp πi
4
X
!
sk hk
(113)
k=1
where sk take the values 0 and 1. This condition means that C02 = 11, so this will
be a Z2 -reduction, or involution. Then the first example of Z2 -reduction is
C0−1 Q† (x, t)C0 = Q(x, t)
(114)
or in components
∗
pij = ij qij
,
ij = i j ,
j = eπisj = ±1.
(115)
Obviously j takes values ±1 depending on whether sj equals 0 or 1.
The next examples of Z2 -reduction correspond to several choices of C as elements
of the Weyl group eventually combined with the Cartan subgroup element C0
C1 = Se1 −e2 Se3 −e4 C0
(116)
55
Multicomponent mKdV Equations and Their Reductions
where Sei −ej is the Weyl reflection related to the root ei − ej . Again we have a
Z2 -reduction, or an involution
C1−1 Q† (x, t)C1 = Q(x, t).
(117)
Written in components it takes the form (12 = 34 = 1)
∗
∗
∗
p12 = −q12
,
p24 = −23 q13
,
p23 = −13 q14
∗
p14 = −13 q23
,
∗
p13 = −23 q24
,
(118)
∗
p34 = −q34
.
The corresponding Hamiltonian and symplectic form take the form
8I3 = −
Z ∞
−∞
∗
∗
∗
∗
dx (∂x q12
∂x q12 + ∂x q34
∂x q34 + 13 (q23
∂x q14 + ∂x q14
∂x q23 )
∗
∗
+ 23 (∂x q24
∂x q13 + ∂x q13
∂x q24 )) +
Z ∞
−∞
∗
∗
dx (12 q12
q12 + q34
q34
(119)
∗
∗
∗
∗
+ 13 (q23
q14 + q24
q13 ) + 23 (q14
q23 + q13
q24 ))2
Z ∞
+
(0)
Ω
−∞
∞
dx |q13 q24 + q12 q34 − q14 q23 |2
1
∗
∗
∗
=
dx (δq12
∧ δq12 + 13 (q23
∧ δq14 + δq14
∧ δq23 )
i −∞
∗
∗
∗
+ 23 (δq24
∧ δq13 + δq13
∧ δq24 ) + δq34
∧ δq34 ).
Z
(120)
Another inequivalent examples of Z2 -reduction corresponds to
C2 = Se1 −e2 C0 .
(121)
The involution is
C2−1 Q† (x, t)C2 = Q(x, t)
or in components it takes the form
∗
∗
∗
p12 = −12 q12
,
p24 = −13 q14
,
p23 = −14 q13
∗
p14 = −23 q24
,
∗
p13 = −24 q23
,
(122)
(123)
∗
p34 = −34 q34
.
As a consequence we get
8I3 = −
Z ∞
−∞
∗
∗
∗
dx (12 ∂x q12
∂x q12 + 34 ∂x q34
∂x q34 + 14 ∂x q13
∂x q23
∗
∗
∗
+ 23 ∂x q24
∂x q14 + 24 ∂x q23
∂x q13 + 13 ∂x q14
∂x q24 )
Z ∞
+
−∞
2
dx (12 |q12 | + 34 |q34 | +
∗
+ 13 q14
q24 )2 + 12 34
Ω
(0)
2
Z ∞
−∞
∗
23 q24
q14
+
∗
24 q23
q13
(124)
+
∗
14 q13
q23
dx |q13 q24 + q12 q34 − q14 q23 |2
1 ∞
∗
∗
=
dx (12 δq12 ∗ ∧δq12 + 34 q34
∧ δq34 + 23 δq24
∧ δq14
i −∞
∗
∗
∗
+ 14 δq13
∧ δq23 + 24 δq23
∧ δq13 + 13 δq14
∧ δq24 ).
Z
(125)
56
Vladimir S. Gerdjikov and Nikolay A. Kostov
Next we consider a Z3 -reduction generated by C3 = Se1 −e2 Se2 −e3 which also
maps J into J. It splits each of the sets ∆±
1 into two orbits which are
(O)±
1 = {±(e1 + e2 ), ±(e2 + e3 ), ±(e1 + e3 )}
(126)
(O)±
2 = {±(e1 + e4 ), ±(e2 + e4 ), ±(e3 + e4 )}.
In order to be more efficient we make use of the following basis in g(0)
Eα(k) =
2
X
ω −kp C3−k Eα C3k ,
Fα(k) =
p=0
2
X
ω −kp C3−k Fα C3k
(127)
p=0
where ω = exp(2πi/3) and α takes values e1 + e2 and e1 + e4 . Obviously
C3−1 Eα(k) C3 = ω k Eα(k) ,
C3−1 Fα(k) C3 = ω k Fα(k) .
(0)
(0)
(128)
(k)
(3−k)
In addition, since ω ∗ = ω −1 we get (Eα )† = Fα and (Eα )† = Fα
k = 1, 2. Then we introduce the potential
Q(x, t) =
3 X
X
(k)
qα(k) (x, t)Eα(k) + p(k)
.
α (x, t)Fα
for
(129)
k=0 α
In view of equation (128) the reduction condition (29) leads to the following relations between the coefficients
(0)
(0)
(k)
(3−k) ∗
p12 = (q12 )∗ ,
p12 = ω k (q12
(0)
p14
(k)
p14
) ,
(k)
(3−k) ∗
q12 = ω k (p12
)
(130)
=
(0)
(q14 )∗ ,
=ω
k
(3−k)
(q14 )∗ ,
(k)
q14
=ω
k
(3−k)
(p14 )∗
where k = 1, 2. It is easy to check that from the conditions (130) there follows
(k)
(k)
(k)
(k)
p12 = q12 = p14 = q14 = 0. So we are left with only one pair of independent
(0)
(0)
(0) (0)
functions q12 and q14 and their complex conjugate p14 , q14 .
Similarly the reduction (30) leads to
(0)
(0)
q12 = −ω 3−k (q12
(0)
(0)
p12 = −ω 3−k (p12
q12 = −(q12 )∗ ,
(k)
(3−k) ∗
q14 = −ω 3−k (q14
(k)
(3−k) ∗
p14 = −ω 3−k (p14
) ,
(k)
(3−k) ∗
(k)
(3−k) ∗
)
(131)
p12 = −(p12 )∗ ,
) ,
(k)
)
(k)
where k = 1, 2. Again from the conditions (131) it follows that p12 = q12 =
(k)
(k)
p14 = q14 = 0. So we are left with two pairs of purely imaginary independent
(0) (0)
(0) (0)
functions: q12 , q14 and p12 , q14 .
The corresponding Hamiltonian and symplectic forms are obtained from the slightly more general formulae below by imposing the constraints (130) and (131). Here
for simplicity we skip the upper zeroes in qij and pij
57
Multicomponent mKdV Equations and Their Reductions
HMMKdV =
1
6
Z ∞
−∞
1
−
12
4
8I3 =
3
dx ∂x2 q12 ∂x p12 − ∂x q12 ∂x2 p12 + ∂x2 q14 ∂x p14 − ∂x q14 ∂x2 p14
(132)
Z ∞
dx
−∞
2
p212 q12
∂x
−
2
p212 ∂x q12
Z ∞
+
8
dx (∂x q12 ∂x p12 + ∂x q14 ∂x p14 ) −
9
−∞
2
q14
∂x p214
Z ∞
−∞
−
2
p212 ∂x q12
2 2
2 2
dx q14
p14 + q12
p12
(133)
Ω(0) =
Z
4 ∞
dx (δq14 ∧ δp14 + δq12 ∧ δp12 )
3 −∞
i.e., in this case we get two decoupled mKdV equations.
The Z4 -reduction generated by C4 = Se1 −e2 Se2 −e3 Se3 −e4 also maps J into J. It
splits each of the sets ∆±
1 into two orbits which are
(O)±
1 = {±(e1 + e2 ), ±(e2 + e3 ), ±(e3 + e4 ), ±(e1 + e4 )}
(134)
(O)±
2 = {±(e1 + e3 ), ±(e2 + e4 )}.
Again we make use of a convenient basis in g(0)
Eα(k) =
3
X
i−kp C4−k Eα C4k ,
Fα(k) =
p=0
3
X
i−kp C4−k Fα C4k
(135)
p=0
where α takes values e1 + e2 and e1 + e3 . Obviously
C4−1 Eα(k) C4 = ik Eα(k) ,
(0)
(Eα )†
(0)
Fα ,
C4−1 Fα(k) C4 = ik Fα(k) .
(k)
(Eα )†
and in addition,
=
=
k = 1, 2, 3. Then we introduce the potential
Q(x, t) =
3 X
X
(4−k)
Fα
and
(k)
(Eα )∗
(136)
=
(k)
qα(k) (x, t)Eα(k) + p(k)
.
α (x, t)Fα
(4−k)
Eα
for
(137)
k=0 α
In view of equation (136) the reduction condition (29) leads to the following relations between the coefficients
(0) ∗
p(0)
α = (qα ) ,
k (4−k) ∗
p(k)
) ,
α = i (qα
qα(k) = ik (pα(4−k) )∗
(138)
for k = 1, 2, 3. Here pα , qα coincide with p12 , q12 (respectively p13 , q13 ) for
α = e1 + e2 (respectively α = e1 + e3 ). Analogously the reduction (30) gives
qα(0) = −(qα(0) )∗ ,
(0) ∗
p(0)
α = −(pα )
qα(k) = −ik (qα(4−k) )∗ ,
k (4−k) ∗
p(k)
)
α = −i (pα
(k)
(139)
(k)
(k)
for k = 1, 2, 3. Both conditions (138) and (139) lead to p12 = q12 = p14 =
(k)
(0)
(2)
(0)
(2)
q14 = 0 for k = 1, 3. In addition, it comes up that E13 = E13 = F13 = F13 . So
58
Vladimir S. Gerdjikov and Nikolay A. Kostov
(0)
(0)
(2)
(2)
we are left with only two pairs of independent functions p12 , q12 and p12 , q12 . We
provide below slightly more general formulae for the corresponding Hamiltonian
and symplectic form which are obtained by imposing the constraints (138) or (139);
(0)
(0)
again for simplicity of notations we skip the upper zeroes in qij and pij and
(2)
(2)
replace qij and pij by q̃ij and p̃ij
H MMKdV =
3
−
32
1
4
Z ∞
dx ∂x2 q12 ∂x p12 − ∂x q12 ∂x2 p12 + ∂x2 q̃12 ∂x p̃12 − ∂x q̃12 ∂x2 p̃12
−∞
Z ∞
dx
−∞
∂x (p212 ) + ∂x (p̃212 )
2
2
+ ∂x (q12
) + ∂x (q̃12
)
2
2
q12
+ q̃12
+ p12 p̃12 ∂x (q12 q̃12 ) (140)
p212 + p̃212 + q12 q̃12 ∂x (p12 p̃12 )
Z ∞
8I3 = 2
dx (∂x q12 ∂x p12 + ∂x q̃12 ∂x p̃12 )
−
−∞
Z ∞
−∞
Ω(0) = 2
Z ∞
−∞
dx (q12 p12 + q̃12 p̃12 )2 + (q12 p̃12 + q̃12 p12 )2
dx (δq12 ∧ δp12 + δ q̃12 ∧ δ p̃12 ) .
(141)
(142)
Now we get two specially coupled mKdV-type equations.
4.3. Class B Reductions Mapping J into −J
The class B reductions of the Zakharov-Shabat system change the sign of J, i.e.,
C −1 JC = −J; therefore we must have also λ → −λ.
Remark 6. Note that adJ has three eigesubspaces Wa , a = 0, ±1 corresponding
to the eigenvalues 0 and ±2. The automorphisms that satisfy C −1 JC = −J
naturally preserve the eigensubspace W0 but map W−1 onto W1 and vice versa.
−
In other words their action on the root space maps the subset of roots ∆+
1 onto ∆1
±
∓
and vice versa: C∆1 = ∆1 .
Here we first consider
C5 = Se1 −e2 Se1 +e2 Se3 −e4 Se3 +e4 C0 .
(143)
Obviously the product of the above four Weyl reflections will change the sign of
J. Its effect on Q in components reads
∗
qij = −ij qij
,
pij = −ij p∗ij ,
ij = i j
(144)
i.e., some of the components of Q become purely imaginary, others may become
real depending on the choice of the signs j .
59
Multicomponent mKdV Equations and Their Reductions
There is no Z3 reduction for the so(8) MMKdV that maps J into −J. So we go
directly to the Z4 -reduction generated by C6 = Se1 +e2 Se2 +e3 Se3 +e4 which maps
J into −J. The orbits of C3 are
(O)±
1 = {±(e1 + e2 ), ∓(e2 + e3 ), ±(e3 + e4 ), ∓(e1 + e4 )}
(O)±
2 = {±(e1 + e3 ), ∓(e2 + e4 )}.
(145)
Again we make use of a convenient basis in g(0)
Eα(k) =
3
X
i−kp C6−k Eα C6k ,
Fα(k) =
p=0
3
X
i−kp C6−k Fα C6k
(146)
p=0
where α takes values e1 + e2 and e1 + e3 . Obviously
C6−1 Eα(k) C6 = ik Eα(k) ,
(0)
(0)
(k)
C6−1 Fα(k) C6 = ik Fα(k)
(4−k)
where again (Eα )† = Fα , (Eα )† = Fα
1, 2, 3. Then we introduce the potential
Q(x, t) =
3 X
X
(k)
(147)
(4−k)
and (Eα )∗ = Eα
(k)
qα(k) (x, t)Eα(k) + p(k)
.
α (x, t)Fα
for k =
(148)
k=0 α
In view of equation (147) the reduction condition (29) leads to the following relations between the coefficients
(0) ∗
p(0)
α = (qα ) ,
k (4−k) ∗
p(k)
) ,
α = i (qα
qα(k) = ik (pα(4−k) )∗ ,
k = 1, 2, 3 (149)
while the reduction (30) gives
qα(0) = −(qα(0) )∗ ,
qα(k) = −ik (qα(4−k) )∗ ,
k = 1, 2, 3
(0) ∗
p(0)
α = −(pα ) ,
k (4−k) ∗
p(k)
) ,
α = −i (pα
k = 1, 2, 3
(150)
where α takes values e1 + e2 and e1 + e3 . From the conditions (149) it follows
(k)
(k)
(k)
(k)
that p12 = q12 = p13 = q13 = 0 for k = 1, 3. In addition, however, it
(0)
(2)
(0)
(2)
comes up that E13 = E13 = F13 = F13 . So we are left with only two pairs
(0) (0)
(2) (2)
of independent functions p12 , q12 and p12 , q12 . We provide below slightly more
general formulae for the corresponding Hamiltonian and symplectic form which
are obtained by imposing the constraints (149) or (150) and again for simplicity
(2)
(2)
we skip the upper zeroes in qij and pij and replace q12 and p12 by q̃12 and p̃12
60
Vladimir S. Gerdjikov and Nikolay A. Kostov
1
4
H MMKdV =
3
−
32
Z ∞
Z ∞
dx
−∞
−∞
dx ∂x2 q12 ∂x p12 − ∂x q12 ∂x2 p12 + ∂x2 q̃12 ∂x p̃12 − ∂x q̃12 ∂x2 p̃12
2
∂x (p212 ) + ∂x (q̃12
)
2
− ∂x (q12
) + ∂x (p̃212 )
2
q12
+ p̃212 + ∂x (q12 p̃12 ) p12 q̃12 (151)
2
p212 + q̃12
− ∂x (p12 q̃12 ) q12 p̃12
Z ∞
(∂x q12 ∂x p12 + ∂x q̃12 ∂x p̃12 )
8I3 = 2
−
Ω(0) = 2
−∞
Z ∞
Z
−∞
∞
−∞
(152)
2
2
dx (q12 p12 + q̃12 p̃12 ) + (q12 p̃12 + q̃12 p12 )
dx (δq12 ∧ δp12 + δ q̃12 ∧ δ p̃12 ) .
(153)
Now we get four specially coupled mKdV-type equations given by
3
3
∂t q0 + ∂x3 q0 + (q0 q2 + p0 p2 )∂x p2 + (p2 q2 + q0 p0 )∂x q0 = 0
2
2
3
3
∂t q2 + ∂x3 q2 + (q2 p2 + q0 p0 )∂x q2 + (q0 q2 + p0 p2 )∂x p0 = 0
2
2
∂t p 0 +
∂x3 p0
3
3
+ (q2 p2 + q0 p0 )∂x p0 + (q0 q2 + p0 p2 )∂x q2 = 0
2
2
(154)
3
3
∂t p2 + ∂x3 p2 + (q2 q0 + p0 p2 )∂x q0 + (q0 p0 + q2 p2 )∂x p2 = 0
2
2
where we use for simplicity
q12 = q0 ,
q̃12 = q2 ,
p12 = p0 ,
p̃12 = p2
(155)
and with second reduction (149) p0 = q0∗ and p2 = −q2∗ we have
3
3
∂t q0 + ∂x3 q0 − (q0 q2 − q0∗ q2∗ )∂x q2∗ − (q2∗ q2 − q0 q0∗ )∂x q0 = 0
2
2
∂t q2 +
∂x3 q2
3
3
− (q2 q2∗ − q0 q0∗ )∂x q2 + (q0 q2 − q0∗ q2∗ )∂x q0∗ = 0.
2
2
(156)
Obviously in the system (156) we can put both q0 , q2 real with the result
3
∂t q0 + ∂x3 q0 + (q02 − q22 )∂x q0 = 0
2
∂ t q2 +
∂x3 q2
3
+ (q02 − q22 )∂x q2 = 0.
2
(157)
61
Multicomponent mKdV Equations and Their Reductions
(0)
(0)
(2)
(2)
∨
∨
The reduction (150) means that qα = iq0∨ , pα = ip∨
0 , qα = q2 , pα = ip2 with
∨
∨
real valued pi , qi , i = 0, 2. Thus we get
3 ∨ ∨
3
∨
∨
∨ ∨
∨
∂t q0∨ + ∂x3 q0∨ + (q0∨ q2∨ + p∨
0 p2 )∂x p2 + (p2 q2 − q0 p0 )∂x q0 = 0
2
2
3 ∨ ∨
3
∨ ∨
∨
∨ ∨
∨
∂t q2∨ + ∂x3 q2∨ + (q2∨ p∨
2 − q0 p0 )∂x q2 − (q0 q2 + p0 p2 )∂x p0 = 0
2
2
3 ∨ ∨
3 ∨ ∨
3 ∨
∨ ∨
∨
∨ ∨
∨
∂t p∨
0 + ∂x p0 + (q2 p2 − q0 p0 )∂x p0 + (q0 q2 + p0 p2 )∂x q2 = 0
2
2
3 ∨ ∨
3 ∨ ∨
3 ∨
∨ ∨
∨
∨ ∨
∨
∂t p∨
2 + ∂x p2 − (q2 q0 + p0 p2 )∂x q0 − (q0 p0 − q2 p2 )∂x p2 = 0.
2
2
(158)
The class B reductions also render all the symplectic forms and Hamiltonians in the
hierarchy real-valued. They allow to render the corresponding systems of MMKdV
equations into ones involving only real-valued fields. “Half” of the Hamiltonian
structures do not survive these reductions and become degenerate. This holds true
for all symplectic forms Ω(2m) and integrals of motion I2m with even indices.
However the other “half” of the hierarchy with Ω(2m+1) and integrals of motion
I2m+1 remains and provides Hamiltonian properties of the MMKdV.
4.4. Effects of Class A Reductions on the Scattering Data
Let in this subsection all automorphisms
Ci are of class A. Therefore acting on the
P
root space they preserve the vector rk=1 ek which is dual to J, and as a consequence, the corresponding Weyl group elements map the subset of roots ∆+
1 onto
itself.
Remark 7. An important consequence of this is that Ci will map block-uppertriangular (respectively block-lower-triangular) matrices like in equation (46) into
matrices with the same block structure. The block-diagonal matrices will be mapped again into block-diagonal ones.
From the reduction conditions (29)–(32) one gets, in the limit x → ∞ that
a)
C1 ((κ1 (λ)J)† ) = λJ,
b) C2 ((κ2 (λ)J)T ) = −λJ
c)
C3 ((κ3 (λ)J)∗ ) = λJ,
d)
(159)
C4 ((κ4 (λ)J)) = λJ.
Using equation (159) and Ci (J) = J one finds that
a) κ1 (λ) = λ∗ ,
b) κ2 (λ) = −λ,
c) κ3 (λ) = −λ∗ ,
d) κ4 (λ) = λ.
(160)
62
Vladimir S. Gerdjikov and Nikolay A. Kostov
It remains to take into account that the reductions (29)–(32) for the potentials of L
lead to the following constraints on the scattering matrix T (λ)
a) C1 (T † (λ∗ )) = T̂ (λ),
b)
C2 (T T (−λ)) = T̂ (λ)
c) C3 ((T ∗ (−λ∗ )) = T (λ),
d)
C4 (T (λ)) = T (λ).
(161)
These results along with remark 7 lead to the following results for the generalized
Gauss factors of T (λ)
C1 (S +,† (λ∗ )) = Sˆ− (λ),
b) C2 (S +,T (−λ)) = Sˆ− (λ),
a)
c)
C3 (S
±,∗
∗
±
(−λ )) = S (λ),
d) C4 (S ± (λ)) = S ± (λ),
C1 (T −,† (λ∗ )) = Tˆ+ (λ)
C (T −,T (−λ)) = Tˆ+ (λ)
2
C3 (T ±,∗ (−λ∗ )) = T ± (λ)
(162)
C4 (T ± (λ)) = T ± (λ)
and
a) C1 (D+,† (λ∗ )) = Dˆ− (λ),
b) C2 (D+,T (−λ)) = Dˆ− (λ)
c) C3 (D±,∗ (−λ)) = D± (λ),
d) C4 (D± (λ)) = D± (λ).
(163)
4.5. Effects of Class B Reductions on the Scattering Data
In this subsection all automorphisms
Ci are of class B.PTherefore acting on the root
Pr
space they map the vector k=1 ek dual to J into − rk=1 ek . As a consequence,
−
the corresponding Weyl group elements map the subset of roots ∆+
1 onto ∆1 ≡
+
−∆1 .
Remark 8. An important consequence of this is that Ci will map block-uppertriangular into block-lower-triangular matrices like in equation (46) and vice versa.
The block-diagonal matrices will be mapped again into block-diagonal ones.
Now equation (159) with Ci (J) = −J leads to
a) κ1 (λ) = −λ∗ ,
∗
c) κ3 (λ) = λ ,
b) κ2 (λ) = λ
d) κ4 (λ) = −λ.
(164)
The reductions (29)–(32) for the potentials of L lead to the following constraints
on the scattering matrix T (λ)
a) C1 (T † (−λ∗ )) = T̂ (λ),
b) C2 (T T (λ)) = T̂ (λ)
c) C3 ((T ∗ (λ∗ )) = T (λ),
d) C4 (T (−λ)) = T (λ).
(165)
Multicomponent mKdV Equations and Their Reductions
63
Then along with Remark 8 we find the following results for the generalized Gauss
factors of T (λ)
a)
b)
C1 (S ±,† (−λ∗ )) = Sˆ± (λ),
C (S ±,T (λ)) = Sˆ± (λ),
2
−
2
c)
C3 (S
(λ )) = S (λ),
C3 (T −,∗ (λ∗ )) = T + (λ)
d)
C4 (S + (−λ)) = S − (λ),
C4 (T − (−λ)) = T + (λ)
+,∗
∗
C1 (T ±,† (−λ∗ )) = Tˆ± (λ)
C (T ±,T (λ)) = Tˆ± (λ)
(166)
and
a)
C1 (D±,† (−λ∗ )) = Dˆ± (λ),
b) C2 (D±,T (λ)) = Dˆ± (λ)
c)
C3 (D+,∗ (λ∗ )) = D− (λ),
d) C4 (D+ (−λ)) = D− (λ).
(167)
5. Conclusions
In conclusion, we have considered the reduced multicomponent MKdV equations
associated with the DIII-type symmetric spaces using Mikhailov’s reduction group.
Several examples of such nontrivial reductions leading to new MMKdV systems
related to the so(8) Lie algebra are given. In particular we provide examples with
reduction groups isomorphic to Z2 , Z3 , Z4 and derive their effects on the scattering matrix, the minimal sets of scattering data and on the hierarchy of Hamiltonian
structures. These results can be generalized also for other types of symmetric
spaces.
Acknowledgements
It is our pleasure to thank Professor Gaetano Vilasi for useful discussions. We
also thank the Bulgarian Science Foundation for partial support through contract
# F-1410. One of us (VG) would like to thanks the Physics Department of the
University of Salerno for the kind hospitality and the Italian Istituto Nazionale di
Fisica Nucleare for financial support. He is also grateful to the organizers of the
Kiev conference for their hospitality and acknowledges financial support by an
ICTP grant.
References
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