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ISSN 2074-1871 Уфимский математический журнал. Том 13. № 2 (2021). С. 121-140.
ON MKDV EQUATIONS RELATED ТО
KAC-MOODY ALGEBRAS A^ AND A5
J1) л кт л (2)
V.S. GERDJIKOV
Abstract. We outline the derivation of the mKdV equations related to the Kac-Moody algebras and A^2. First we formulate their Lax representations and provide details how they can be obtained from generic Lax operators related to the algebra sl(6) by applying proper Mikhailov type reduction groups Z^. Here h is the Coxeter number of the relevant Kac-Moody algebra. Next we adapt Shabat's method for constructing the fundamental analytic solutions of the Lax operators L. Thus we are able to reduce the direct and inverse spectral problems for L to Riemann-Hilbert problems (RHP) on the union of 2h rays lv. They leave the origin of the complex A-plane partitioning it into equal angles n/h. To each lu we associate a subalgebra qu which is a direct sum of s/(2)-subalgebras. In this way, to each regular solution of the RHP we can associate scattering data of L consisting of scattering matrices Tv £ Qv and their Gauss decompositions. The main result of the paper states how to find the minimal sets of scattering data Tk, k = 1, 2, from To and T\ related to the rays lo and l\. We prove that each of the minimal sets 71 and 72 allows one to reconstruct both the scattering matrices Tv, v = 0,1,... 2h and the corresponding potentials of the Lax operators L.
Keywords: mKdV equations, Kac-Moody algebras, Lax operators, minimal sets of scattering data.
Mathematics Subject Classification:! 7B07. 35P25, 35Q15, 35Q53
1. Introduction
This paper is a continuation of a series of papers on Kax-Moodv algebras and mKdV equations [14], [15], [16], [17], [18] and two recent papers [19], [13]. There we derived explicitly the system of mKdV equations related to several particular choices of Kac-Moody algebras, including some twisted ones like d[s\ s = 1, 2, 3, A^ and A^2^.
The next natural steps to be considered are to develop the direct and inverse scattering method for the relevant Lax operators and to construct their reflectionless potentials and, as a consequence, soliton solutions to the mKdV systems. The methods for doing this have been already developed in [7], [20], [8], [21], [22], [23], [39]. This is why it will not be difficult to specify the construction of the fundamental analytic solutions (FAS) [32], [33] of the relevant Lax operators and to formulate the corresponding Riemann-Hilbert problem (RHP). In constructing the soliton solutions, the most effective method known to us is the dressing Zakharov-Shabat method [37], [38].
The structure of the paper is as follows. In Section 2 we outline preliminary known results about the structure of the Lax operators for the case of A^1 and Kac-Moodv algebras and for the recursion operators, see [13]. Section 3 is devoted to the fundamental analytic solutions (FAS) and to the Riemann-Hilbert problems for both cases. In Section 4 we introduce the
V.S. Gerdjikov, On mKdV equations related to Kac-Moody algebras a^1 and a^k © V.S. Gerdjikov. 2021.
The reported study by V.S. Gerdjikov was funded in part by the Bulgarian National Science Foundation under contract KP-06N42-2. Submitted April 12, 2021.
minimal sets of scattering data and show by these set we can reconstruct both the potential and the sewing functions of the EHP, In the appendices we discuss some algebraic details of the structure of Kac-Moody algebras.
2. Preliminaries
2.1. Lax representations: A^ case. We suppose that the readers are familiar with the theory of simple Lie algebras and Kac-Moody algebras, see [3], [24], [4] and their applications in the studies of integrable nonlinear evolution equations [5], [6]. Details about the bases and the gradings of the Kac-Moody algebras are given in the appendices. Here we consider a nonlinear evolution equation with a simplest nontrivial dispersion law, which is /mKdv(A) = X3K.
In this section, following our previous papers, we define the Lax pairs whose potentials are elements of the A^ and A^ algebras for the mKdV equations. They represent the third nontrivial member in the hierarchy of soliton equations related to these algebras. The results presented here are derived in [16], [14] for and in [13], [19] for
We consider a Lax pair that is polynomial in the spectral parameter A:
d
Lif> =( %— + Q(x,t) - XJ)^ = 0
d
Mip =( i^ + Vo(x,t) + XVi(x,t) + \2V2(x,t) - X3K )ip = -X3^K.
(2.1)
The zero-curvature condition [L, M] = 0 leads to a polynomial of fourth order in A, which has to vanish identically. The Kac-Moodv algebra is graded by the Coxeter automorphism C1, see Appendix A below) The basis we use reads as
T(fc) - V^ ^ C
■JS — t3,3+sLV1 ^3,3+s, t3,3+s
3 = 1
j(k) tM] _ (, —ms_ , ,-kl\ 1(k+m) j(k) j(m) _ , — sm j(k+m)
■JS , Jl — U1 ) 'Js+l , Js Jp — U1 Js+p ,
m-1 - (j(sk)y.
The potential coefficients of the Lax pair are defined as
{
1 if j + s ^ 6, -1 if j + s > 6,
Q(x, t)=Y, 13 (x, t)J(°\ Mx, t)=Y, v(1\x, t)J¡
3 = 1
6
=
1=1
5
(1)/
J =x
(1)
(2),
t(2)
V2(x, t)=J2 v¡2)(x, t)jf\ Vo(x, t)=J2 vi0)(x, t)J((0), K =J( 1=1
=
1=1
(0)
(0)
A 3)
(2.2)
The condition [L,M] = 0 leads us to a set of recurrent relations, see [20], [22], [9], which allow us to determine V(k\x, t) in terms of the potential Q(x, t) and its ^-derivatives. By using the choices for Q, J and K from (2,2) we get:
Q
/0 qi q2 q3 q4
- Q5 0 qi q2 q:i q4
- Q4 - 95 0 qi q2 ®
- Q3 - Q4 - Q5 0 qi q2
- Q2 - Q3 - Q4 - Q5 0 qi \-qi - q2 - q3 - <?4 - ® 0/
J = diag (1, j5,j4,j3,j2,j), K = diag (1, -1,1,-1,1, -1),
where u = e These equations admit the following Hamiltonian formulation:
dqi dt
d / SH
dx\ Sq6-,
5H \ Sqe-i) '
The Hamiltonian density is: H
- 32 945 + 2[ ^
dx dx
idqs V \dx )
,8^0 dqi I O 2^2 . , . dq5\
+ 8V3 -2q2q3^~ + 2% — + (qiq2 + q4q5)^~ + --2^4 —
\ ox ox ox ox ox J
+ 2- 24(qiq5 + ^4)gf + 16(q\ - 3qiqj - 3q2,q5 + q50)q3 + 24(^q2 - q4q5)2-
2.2. Lax representations: Af2 case. Here we formulate the main results of a recent paper [13], see also [11], [12], [14], [15], [18], The grading used here is described in Appendix B, It uses the Coxeter automorphism C2 and splits A5 into 10 subspaees. The dispersion laws of the nonlinear evolution equation to A52) are odd functions in A; therefore, NLS-tvpe equations here are not allowed. Thus, we are left with fmKdv(A) = A3K. The Lax pair is of the form
L =idx + Q(x, t) - AJ,
M =idt + V(0) (x, t) + AV(1) (x, t) + A2 V(2) (x, t) - A3K,
where
Q(x, t) E 0(o), V(fc)(x, t) E g(k), K E g(3) J K
J = diag (u4,u2, 1, 0,u6,u8), K = 20 J where u2 = e^ and choose
J e 0
(i)
Q
S ^
3=i
(0)
0 Qi Qa Q2 - Qi - q3\
- Qi 0 Qi - Q2 - qa - qa
- qa - Qi 0 Q2 - qa - qi
- q2 Q2 - q2 0 2 - q2
Qi Qs Qa - Q2 0 - qi
\ Qs Qs Qi 2 qi 0
Then we solve the recurrent relations obtaining the following result:
Vf
p
S3
(p)
p = 2,1,0, V = Vf + v 1;4J,
3=i
and obtain explicit expressions for vp;j in terms of qj and their x-derivatives, for details see [14] [13], The equations of motion
% dx
ÔV 0;
0 ; 3
dx '
3 = 1, 2, 3,
can be cast in Hamiltonian form as follows:
d V0;
d3l = JL Œ
dx dx 5qj (x) dx
0;j
3 = 1, 2, 3,
where
and
H
2 + (^)' - 10 (t)2 - (3VS- 5) (
+ 20 (c-ql + c+qiq3 - 2c+q%) ^ ~ 20 (c2 QiQa + c+qj - 2c-q\)
'dqs
dx
+ 40 (-c-q3 + c+qi) qi^ + 20q24 + 40q!q3(qj - q\) + 60(qjql - qjql - q\ql)
x
2
<i = ,/2 + ^,
-=
f
V5 •
Let us now repeat the calculations using the second type of grading, see equation (B.2), In this equation, potential takes diagonal form while J becomes the sum of admissible roots. We can do the grading using an alternative choice of the Coxeter automorphism given by (B.3), (B.4). This gives:
Q(x, t) = is^Uj (x, t)e+, J
3=1
V <0'(x, t) = ±-»Vet,
v + V
-li + e3i + 2 e23 + 2 e15,
3 = 1
v (1)(x, t) = v^e+1 + v^e+i + +1 ^e-
(2.3)
v(i)(x,t) = -v?e+1 - v^e+i - 1V¡l)eu,
K = 5 J3
where
(i)
(1)
- 5z(u1 +Ui + u ),
(i)
5 Ui
(i)
du1
vf =10 ( U1Ui - -x
v^1 =5 ( u\ - u{ + U1Ui + uiu3 + (u3 + ui - U1) ) + v2',
+ V(21),
x
- 5 (U1 - Ui - U ) ,
(1)
=5 ( u\ - U - u1 + u1ui + — (u3 + 2u2 - u^ ) + v\
d
dx
(1)
v^1 =2ui + 2ui - 3ui - 5u1ui + — (5u1 - 4u2 - 3u3)
d
d x
For V(0)(x, t) we find:
(0)
di u1 3 i i
-H -5dxi + 3u1 -^x(3ui + u3) - 2u1 + 3u1(u2 + u3)
(0) . i =
di d u d u1
(4u2 + 3u3) + 3u2—--9^ —--+ 6U3
(0) = • ^ = l dxi
d xi di
d x
d x
d ui
u1
i3
du3 d x d ui
,
- 2ui + 3ui(u1 + ui)
(u3 + 3u2) - 6u3- - - 3u2^~ - 2u\ + 3u3(u1 + U)
d x
x
d x
(2.4)
1
2
3
1
Finally, the set of mKdV equations takes the form: du 1
dt dU2 ~dt
du3
)
d ( d2ui d 3 2 2\
Z~dx V "dx2 + 3uidfa (3u2 + U3) - 2u1 + 3ui(u2 + u3)
^TT(4u2 + 3u3) + 3u2du3 - 9uidu1 dx\dx2 dx dx
d u3 3 2 2
+ 6u^"dx— 2u2 + 3u2(u1
+ u3)
d 2 d u2 d ui d u2 3 2 2
— ( (u3 + 3u2) — 6u3 —--?^u1—--3u2 —--2u3 + 3u3(u1 + u2) I.
dt dx\ dx2 dx dx
These equations acquire Hamiltonian form:
du% _ d Î5H \ dt dx \ Sui )
where the Hamiltonian is
dvo-i dx
H _ £ K -1 |>4 + 3 £ E «?«■» + |(
=1 3=1
i<3
2
(dr) -1( dr) + du^f 9u - 3u3i
V dx ) 2\ dx ) dx \2 1 y
3 d u3 2 2
+ 2 ttx3 (u2+u2) - 3
d u2
d x
)(
du3
x
Using the second type of grading is in fact equivalent to the first one. One can check that the two types of gradings are related by a similarity transformations of the form:
Wq 1QWQ _ Q,
w0 1Jwq
J.
Wq
1
75
( 1 1 1 1 1 \
2 2 2 2 6 2 2 2
£ to | £22 1 0
£ to to -4 1 1 0
£22 £4 1 0 £ to to 4
£2 1 0 4
1 11 -75 1 1
Effectively we find that u^d qs are related linearly as follows:
u1
c q1 + c+q3,
^10 + 275 10 '
1
u2
-tQ2'
u3
-c q1 + c+q3
V5 V10 - 275
10
2.3. Recursion relations and recursion operators Afc. Our aim here is to describe the hierarchies of equations in terms of the recursion operators Ak. The idea is to treat the compatibility conditions as recurrent relations which will be solved using the recursion operators, see [19], [13], [14], [15], [18], The initial condition reads as
V2 _adj1[K'Q].
We note that the operator ad j acting on each element X G 0 by the rule ad JX _ [J, X] has a non-trivial kernel and therefore, it could be inverted only if X belongs to its image. Hence, while solving the recurrent relations, we need to split each Vs into, roughly speaking, 'diagonal' and 'off-diagonal' parts:
Vs _ vf + vd,
2
where Vf e Im ad j and Vd is such that ad JVf = 0, Then we have:
Vn-1(x, t) = Vh(x, t) = £ ^¡J-)qp(x, t)e(ri~1)•
p=1 ®p(J)
1
off-diagonal parts. Evaluating the Killing form of this equation with Hlh-si, we obtain:
wSl (x, t) = — d-1 ([Q,VS1 ], Ul-si) + const, csi = (ni1, U\~si) •
si
0
can easily work out the more general cases when some of these constants do not vanish. The off-diagonal part of the third equation in (2,4) gives:
idxVS + [Q, Vl]f +[Q,wsUS1 ] = [J,Vs-1],
i.e.
V- = ad -1 (idxVf + [Q, Vf ]f + [Q, WsHs1 0 = A si Vs • Thus, we have obtained an integro-differential operator Asi which acts on each Z = Zf e si)
i as
A siZ = ad-1[idxZ + [(Q, Z]f + c-[Q, nsli]d-1 {[(Q, Z], nh1-si^ •
1
in (2.3):
V/-1 =ad -1 (icXVf + [Q, Vf]f) = AoVf, AoZ =ad-1 (idxZ + [Q,Z]f) •
A0
Now we can study the hierarchies related to A^, Since the Coxeter number is 6 and the exponents are 1, 2, 3 4, 5, the results are as follows:
n =6no + 1 dtQ =dx (An°Q(x, t)), f(X) =XNin{1),
n =6no + a dtQ =dx (Ara°Aa-1 • • • A^-1[Ha1, Q(x, t)]) , f(X) =XN*H[a),
where Na = 6n0 + a, a = 1, 2, • • •, ^d A = A1AiA3A2A5A0.
In the same way we can study the hierarchies related to A^, Here the Coxeter nu mber is 10 and the exponents are 1, 3, 5 7, 9. The results are
n =10no + 1 dtQ =dx (An°Q(x, t)),
n =10no + 3 dtQ =dx (Ara° A^ad fHf ,Q(x, t)]j ,
n =10no + 5 dtQ =dx (An°A1AoA3Ao%d-1[Hf\Q(x,t)]} ,
n =10no + 7 dtQ =dx (An° A1AoA3AoA5Ao%d ^1[H(i),Q(x, i)]) ,
n =10no + 9 dtQ =dx [An°A1A0A3A0A5A0A7A0ad ^[^^(x, i)]) ,
where A = A1A0A3A0A5A0A7A0A9A0 and the dispersion laws are given by f, (X) = X10n°+n>Hal
(i)
nj = 2j - 1, being the exponents of A5 \
3. Riemann-Hilbert problem
3.1. General aspects. The general methods for constructing the FAS of the Lax operators were proposed in the pioneer papers by A.B, Shabat [32], [33], in which he constructed the FAS of a class of n x n Lax operators of type (2,1) with J = diag (a\, . . . ,an) assuming that the eigenvalues of J are real and are taken in the descending order. The continuous spectrum of such L operator with a fast decaying potential Q fills up the real axis in the complex A-plane, One of the corresponding FAS x+(x, A) admits an analytic extension into the upper half plane C+; the other one x-(x, A) is analytic in the lower half plane C_ and on the real axis they are related linearly:
X+(x,t,A) = x_(x,t,A)Ga(t, A), (3.1)
where the sewing function G(t, A) is expressed bv the Gauss factors of the corresponding scattering matrix, A simple transformation from x±(x,A) to (x, A) = x±(x,A)etXJx allows one to reformulate RHP (3,1) as follows:
£+(x,t,A) = C(x,t,A)G(x,t,A), G(x,t, A) = e_lXJxG0(t,A)elXJx. (3.2)
An advantage of RHP (3.2) is that it allows canonical normalization in the form Hmx^£±(x,t, A) = 1.
Shabat and Zakharov developed further these ideas by discovering a deep relation between RHP (3.2) and the corresponding pair of Lax operators. They proved a theorem [37], [38] stating that if £±(x, t, A) satisfy RHP (3,2) and the sewing function G(x,t,A) has a proper x—dependence, then the corresponding x±(x,t,A) is FAS of the relevant Lax pair,
A next important step was that they devised a method of deriving a special class of singular solutions to the RHP, Today it is known as the Zakharov-Shabat dressing method [37], [38], [31], It has several formulations and is one of the best known methods for constructing the multi-soliton solutions of the integrable nonlinear linear evolution equation. Later Shabat's
Q J
simple Lie algebras g [10],
A further progress in this direction was made by Beals and Coifman [2] who treated the n x n J
the Shabat's case was that the continuous spectrum of L filled up a set of ravs lp, which splitted the complex A-plan e C into several sec tors Qp. In each of these sectors, Beals and Coifman succeeded to construct FAS ^p(x, A) Let us assume that the sectors Qp and share the rav lp, then we have a set of relations like
£p(x,t,A) = £a(x,t,A)Gp(x,t,A), Gp(x, t, A) = e_iXJxGp0(t,A)eiXJx,
where lp = Qp fl p = 1, 2,..., which is a generalized RHP. Zakharov-Shabat theorem mentioned above and the dressing method can easily be extended to such generalized RHP. And of course, the results of Beals and Coifman were generalized also to the case when Q(x, t) and J took values in any simple Lie algebra 0 [23], [22], [21].
Let us also mention briefly how the analvtieity properties of (x,t,A) are proved. Since xv(x, t, A) are fundamental solutions of the above operators L and M, then (x,t,A) are fundamental solutions of the related operators:
Lxu = i-jx + Q(x, t)&(x, t, A) - A[J, ] = 0,
Bf 2 (3.3)
Mx„ = i-^ + V(x, t, A)^(x, t, A) - A3[K, ^] = 0, V(x, t,A) = ^ Vp(x, t)Ap.
p=0
We already made special choices for both Q(x, t) and J using two different specific gradings of A5 ~ s/(6). Each of these choices can be viewed as a realization of Mikhailov reduction group
128
V.S. gerdjikov
Zh [27]:
C(Q(x, t) — XJ) = Q(x, t) — XuJ, C(V(x,t,X) — X3K) = V(x,t,Xu) — X3u3K, (3.4)
with a properly chosen Coxeter automorphism C such that Ch = 1 and h is the Coxeter number. In other words, the Lax pairs with Zh reductions of Mikhailov type [27] provide an
J
the potentials Q(x, t) — XJ and V(x, t, X) — X3K of these Lax pairs take values in a Kac-Moody algebras, which are based on the simple Lie algebras graded bv Coxeter automorphisms [3], [6], [5], [25], [4].
The derivation of the FAS of equation (3.3) is based on the set of integral equations which incorporate also the asymptotic behavior of ^ (x,t, X) as x ^ ±<x. These equations have the form, see [2], [23], [22] ,"[21]:
(^(x,t,X))kj =6kj + i / dy (Q(y, t)&(у,t,\))kj e-iX(Jk-JjХ*-У\
J—те
for Л E and Im X(Jk — Jj) ^ 0,
V
(3.5)
(&(x, t, X))kj =i / dy (Q(y, t)^(y,t,X))kj e—iX(Jk—Jj
J те
for X E Qv and Im X(Jk — Jj) > 0,
V
where the index v in the inequalities in (3.5) means that we restrict X E
Roughly speaking, our first task in analyzing the integral equations (3.5) is to determine
X
Normally these lines constitute the continuous spectrum of L. They would be determined by Im X(Jk — Jj) = 0, which can be written in the form:
lmXa(J) = 0, (3.6)
where a = ek — ej is a root of A5. The set of equations (3.6), where a runs over the root system Д of A5, are simple algebraic equations. Their solutions are collected in Table 1 for A^ and in Table 3 for Af\ Thus, we establish that the continuous spectrum of L fills up all rays Iv = arg X = vn/h, и = 0,1,..., 2h — 1.
Lemma 3.1. To each, pair of rays Iv U12h—v there corresponds a subalgebra gv С s 1(6), which in the case of A^ is isomorphic either to si(2) ф s 1(2) or to si(2) ф s 1(2) ф s 1(2). In the case of A^2 it is isomorphic eith er to si (2) ф s 1(2) or to si (2).
a — a
solution. It remains to confirm that any two non-proportional roots related to each pair of rays lv U 12h—v are mutually orthogonal. Inspecting Table 1, we prove the lemma for A51\ Similarly, inspecting Table 3, we prove the lemma for Af2. The proof is complete. □
Theorem 3.1. The solution (x,t, X) of eq. (3.5) is an analytic function of X for X E Qv. In addition,
С (^ (x,t,X)) = Си+2(x,t,Xw). (3.7)
Idea of the proof. The solutions of the conditions Im X(Jk — Jj) ^ 0 for X E Qv in the case of
V
A51'1 are listed in Table 2 as the subsets 8+. All other roots of A5 for X e Qv satisfy the condition Im X(Jk — Jj) > 0. As a result, it is easy to see that the exponential factors in equation (3.5)
V
decrease exponentially for all x and X e Qp. In particular, this means that the integrals converge for each X e which guarantees the existence of ^(x,t, X).
Let us now consider the integral equations for the derivatives ^^(x,t,X). The integrands
x
Л
k 15 14
л
Jh
По
1o
П19
119
Рис, 1: Continuous spectrum of the Lax operators and contours of the RHP for A^ (left panel) and 42) (right panel).
l
3
l
0
I
9
lv 1 0 и I 6 h U I 7
a ±(e 1 - eA), ±(e2 - e 3), ±(e5 - 6) ±(e 1 - e3), ±(e4 - e б)
lv 1 2 U l8 1 3 U I 9
a ±(e 1 - e2), ±(e3 - еб), ±(e 4 - e5) ±(e 2 - eб), ±(e3 - e5)
lv 14 U /10 U 11
a ±(e 1 - eб), ±(e2 - e5), ±(e3 - 4) ±(e 1 - ), ±( 2 - e 4)
Таблица 1: The roots of A^ related to the rays lv, и = 0,..., 11, see the left panel of Figure 1,
in the right hand side, which means that (x,t, A) possesses the derivatives of all orders with respect to A in the sector This is one of the basic properties of the analytic functions. Finally, equation (3,7) follows directly from Mikhailov reduction condition (3,4), □
The corresponding generalized RHP can be written as follows:
& (x, t, A) = Cv_1(x,t,A)Gv (x, t, A), Gv (x, t, A) = e_lXJxGv0(t, A) etAJ", (3.8)
where A e lv and the rays lv are determine d as argA = vx/h, u = 0,... 2h — 1, and h is the Coxeter number. The sector is determined by the rSiVS Iv cUid I S66 Figure 1, In fact, A.V, Mikhailov, developing his ideas on the reduction groups in |27|, came very close to such formulation of the RHP,
A
(2)
vectors Eij can easily be expressed in terms of the root vectors of Ag taking into account the
hv
hq (e 1 - e 4), (e 2 - ез), -(65 - e6) (e 1 - e5), (e2 - 64)
H1 (e 1 - e5), (e2 - 64) (e 1 - e e), (e2 - 65), (e3 - 64)
П2 (e 1 - ее), (e2 - e5), (e3 - 64) (e2 - e e), (e3 - e5)
П3 (e2 - e е), (ез - 65) -(e 1 - e 2), (e3 - e e), (e4 - 65)
П4 -(e 1 - e 2), (e3 - e 6), (e4 - e,5) -(e 1 - 63), (e4 - ее)
П5 -(e2 - e 3), (e4 - e e) -(e2 - e 3), (e 5 - ее),-(e 1 - 64)
Таблица 2: The root subsystems of A5^ related to the seetors Qv, v = 0,..., 11,
see the left panel of Figure 1.
lv a l0 U ¿1Q ±(e3 - e 4) h U /11 ±(e 1 - e 2), ±(e3 - 65) l2 U /12 ±(e4 - e 5) l3 U /13 ±(e2 - e 5),±(e3 - e е)
L a l4 U /14 ±(e2 - e4) l5 U /15 ±(e 1 - e5), ±(e2 - ее) l е U Z 1е ±(e4 - ее) l7 U h7 ±(e 1 - e е),±(e2 - e 3)
lu a l 8 U 118 ±(e 1 - e 4) l9 U /19 ±(e 1 - e 3), ±(e5 - ее)
Таблица 3: The roots of A5 related to the rays lv, и = 0,..., 19 with J = diag (^2,^2, -1, 0, see the right panel of Figure 1 and Remark 3.1.
Î+ s- Hu s-
hq (e 1 - e 4) -(e 1 - e3), -(e 5 - ее) H1 (e 1 - e 5), (e2 - 64) -(e 1 - e 4)
П2 -(e 1 - e 4) -(e 1 - ее), -(e2 - 63) H3 -(e 1 - e е), -(e2 - e 3) -(e4 - e е)
П4 Не -(e4 - e е) -(e2 - e 4) -(e 1 - e 5), -(e 2 - ее) -(e2 - e 5), - (e3 - e е) H5 H7 -(e 1 - e 5), -(e2 - e е) -(e2 - e 5), -(e3 - e е) -(e2 - e 4) -(e4 - e 5)
^8 -(e4 - e 5) (e 1 - e 2), -(e3 - e 5) H9 (e 1 - e2), -(e3 - 65) -(e3 - e 4)
Таблица 4: The root subsystems of A5 related to the sectors П^, и = 0,..., 9, see the left panel of Figure 1 and Remark 3.1.
relations (B.l) from Appendix B. Indeed,
E%, = ^¿t + ), Ei, = ^(¿i - ), Efl = £+-,
°3 2 3^ 2 l3 l3 J J 33
where 1 ^ i < j ^ 3 and k = 7 — k.
It is obvious that all the information about the scattering data of L (or L) is hidden in the sewing functions Gv(x,t,X). For the Lax operators we are considering it is not possible to introduce Jost solutions without imposing additional severe restrictions on Q(x, t), such as tending to 0 as x ^ ±œ faster than each exponential e~clxl for each positive c, or even assuming that Q(x, t) has a compact support. However, we can use the limits of (x,i,À) as x ^ ±œ>
and A G lv. They are given by [22], [21]:
lim elXJxXu(x,t,A) = S+(t,A), х^-те A G Iveг0,
lim e*XJxx.(x,t, A) = T—(t,A)D+(A), A G Iveг0,
lim elXJxx.-i(x,t,A) = S—(t,A), х^-те A G Ive-г0
lim e u Jxx.-i(x,t,A) = T+(t,A)D—(A), х^те A G Ive-г0
where и = 0,1,..., 2h — 1 and S±, T± and D± are of the form
(3.9)
S±(X) =exp ( £ s±(X)E±a T±(X) =exp ( ^ r±(X)E±a D±(X) =exp ( ^ d±JX)Ha
\a£S+
Remark 3.2. Formally one can introduce an analogue of the scattering 'matrix for each pair of rays Iv U lh+v as follows:
Tv(t, A) = T-(t, A)D+(A)S+(t, A) = T+(t, A)D-(A)S-(t, A), A G h
(3.10)
Note that Tv(t, X) belongs to the subgroup Qv C SL(6) whose root system is 5+ U 5-. Then T±(t,X), S±(t, X) and D±(X) can be regarded as the Gauss factors of Tv(t,X). Another peculiar fact is that to each, sector Q,v we relate a specific ordering of the root systems, i. e. specific choice of the positive and negative roots, see [22], [21].
(3.11)
-T ±
i-T.- — A3[K,T±(t, A)] =0,
-t
-Q ±
г-^ — A3[K,S±(t, A)] =0,
D
±
-
0,
-
- T.
— A [K, T. (t, A)] =0.
ii) The function D+ (A) (respectively, Dv (A)) is analytic in A G (respectively, in A G Qv-i). They are generating functionals of the integrals of 'motion for the mKdV hierarchy.
Доказательство, i) We multiply the second equation in (3.3) by etXJx and take the limits for x ^ œ and x ^ — <x. Takin into account equation (3.9) and the fact that Q(x, t) and V(x, t, A) vanish fast enough as x ^ ±œ, we easily obtain the equations (3.11).
ii) The analvtieity properties of D±(A) were proven in [21] for generic Kae-Moodv algebras. As generating functionals of the integrals of motion, it is more convenient to consider d±a(A). Their asymptotic expansions
те
" -p I
d±,a(A) = J2A-pI%
p=i
provide integrals of motion llPi whose densities are local in Q(x, t), i.e. depend only on Q(x, t) and its x-derivatives. The proof is complete. □
4. Minimal set of scattering data
Here we reformulate the basic results of [22], [21] for the specific Kac-Moody algebras used above. It is natural to expect that these sets are expressed in terms of the sewing functions of the RHP, Our considerations are relevant only for the cases when the solution of the RHP is regular. This means that the spectra of the corresponding Lax operators contain no discrete eigenvalues,
4.1. The case. We introduce two minimal sets of scattering data for the Kac-Moodv algebra as follows, see Table 2:
71 = {s±:a(A, t), a e S+,A e l0] u {s±.a(\, t), a e S+, \ e h],
72 = {T±a(A, t), a e A e lQ] u {r±a(\, t), a e A e /1].
Theorem 4.1. Assume that the potential of the Lax operator (2.1) Q(x, t) is a Schwartz-type function of x and is such, that the corresponding RHP is regular. Then each of the minimal sets Ti, i = 1,2 determines uniquely:
i) all sewing functions Gv(x, t, A) for v = 0,1,..., 11;
ii) all scattering matrices Tv, v = 0,1,..., 11; Hi) 71 ~ 72;
iv) the potential Q(x, t).
Idea of the proof. The fact that the solution of the RHP is regular means that the corresponding Lax operator L has no discrete eigenvalues. In other words, the functions D°(A) have neither zeroes nor poles in their regions of analytieitv,
i) Let us now demonstrate that the sets 7^, k = 0,1 allow us to construct all S±(A, t) and T±(A, t). It is obvious that
S± =exp (S0;UE±(e!-e4) + So;23E±(e2—3) + S0;56ET(e5-ea}) , T0 =exp (7(0;i4E±(e 1 -£4) + T0;23E±(e2-e3) + T0;56EV(e5-e6}) ,
S± =exp (s°i3Eo(e 1-£3) + S0046E±(e4-e6)) , T1° =exp (T10i3E±(e 1 —3) + T1046E±(e4-e6)) .
Note that the reduction condition (3,7) on the FAS reflects also on their asvmptoties for x —y as follows:
Cv (S±(x, t, A)) =S0ou (x,t,Auv), Cv (S±0(x,t,A)) =S±+l(x,t,Auv),
^ (T±(x,t,A)) =T0 (x,t,Auv), Cv (T±(x, t, A)) =T±+i(x, t, Auju),
for v = 0,1..., 11. Thus, we have recovered all S±(A, t) and T±(A, t).
ii) It remains to recover D+(A) and D-(A) (or d±a(A)) using the fact that they are analytic
functions of A in the sec tor Qv and Qv_i; respectively. In addition, it follows from equation
-
d+a — d-;a (1 — Su,aSu,-a), A e^v, a e,
dt;a -d-;a =ln (1 - T+aT-a) , A elv, a e5+,
= 0, 1 . . . , 11 = 0, 1
d+a - d-a =ln (1 - 4«s-^J, A e ^ a e{ei - e4, e2 - eз,-(e5 - e6)],
d+a - d-;a =ln (1 - s+as-a), A e li, a e {ei - e3, e4 - e6],
and similar expressions in terms of r+a and T-_a, k = 0,1,
iii) Comparing the asvmptoties (3,9) of the FAS for x ^ we easily find that the sewing functions GVi0 in (3,8) are given by:
Gkfi(X, t) = S_(X, t)S+(X, t) = D-(X)T+ (X, t)Tj_(X, t)D+ (X), X E lk,k = 0,1.
Thus we know the left hand side of the relation:
D-(X)Gk,o(X, t)D +(X, t) = T+(X, t)Tj_(X, t), k = 0,1, (4.1)
Tk (X, )
Gauss factors, which has unique solution. This means that knowing ^ we can recover T2. Quite analogously one can prove that knowing 72 we can uniquely re cover T\.
iv) The RHP has unique regular solution. Suppose we have constructed the solution (x, t, X) in the sector Then we recover the potential from the well known relation:
Q(x, t) = lim X (J - tuJC1^^, X)) .
C(Q(x, t)) = Q(x, t).
□
4.2. The Af2 case. We introduce two minimal sets of scattering data for the A^ Kae-Moodv algebra as follows,
see Table 4:
Ti = {4JX, t), a E S+, X E Zo> U {sf.a(X, t), a E S+, X E h], % = {r±a(X, t), a E X E lo]U {r±a(X, t), a E 5+, X E h].
Theorem 4.2. Assume that the potential Q(x, t) in Lax operator (2.1) is a Schwartz-type x
Ti, i = 1, 2 determines uniquely:
i) all sewing functions Gv(x, t, X) for u = 0,1,..., 19;
ii) all scattering matrices Tv, u = 0,1,..., 19; Hi) Ti
iv) the potential Q(x, t).
Idea of the proof. The fact that the solution of the RHP is regular means that the corresponding Lax operator L has no discrete eigenvalues. In other words, the functions D±(X) have neither zeroes nor poles in their regions of analvtieity,
i) Let us now demonstrate that the sets %, k = 0,1 allow us to construct all S±(X, t) and T±(X, t). It is obvious that
S± = exP (s±.uE±(ee4)) , T0± = exP (T±uE±(ei-e4)) ,
S± =exp (sf;i5E±(ei_es) + sf.24E±(e2-64)) ,
T± =exp (T±i5E±{ei_es) + T±24E±(e2_£4)) .
Note that the reduction condition (3,7) on the FAS reflects also on their asvmptoties for x ^ as follows:
C'(S±(x, t, X)) =S±(x,t,Xuu), C(S±(x,t,X)) =S±+1(x,t,Xuu),
C(T±(x, t, X)) =T± (x, t, Xuv), C(T±(x, t, X)) =T±v+l(x,t,Xuv),
(or u = 0,1..., 19. Thus, we have recovered all S' (X, t) and T' (X, t).
ii) It remains to recover D+(A) and D-(A) (or d^>a(A)) using the fact that they are analytic functions of A in the sector Qv and Qv-i respectively. In addition, it follows from from equation (3,10) that (see Table 4)
d+a - d-;a ^ (1 - Su,aSu,-a) , A e^v, a e,
d+;a -d-;a =ln (1 - r+aT-a) , A el v, a e ^
for v = 0,1..., 19, which follow from equations (3,10), In particular, for k = 0,1 we have:
d+a d-;a =ln (1 - 4««--a^ A e ^ a e{ei - e4],
d+a - d-a =ln (1 - s-J, A e lu a e {ei - e5, e2 - e4],
and similar expressions in terms of r+a and T--a, k = 0,1,
iii) Comparing asvmptotics (3,9) of the FAS for x — we easily find that the sewing functions Gv0 in (3,8) are
GM(A, t) = S-(A, t)S+(X, t) = D-(A)T+(A, t)T-(A, t)D+(A), A e lk, k = 0,1. Thus we know the left hand side in the relation
D-(A)Gk,o(A, t)D +(A, t) = TT+(A, t)T-(A, t), k = 0,1, (4.2)
Tk0 (A, )
Gauss factors, which has a unique solution. This means that knowing 7! we can recover 72. Quite analogously one can prove th at knowing 72 we can uniquely re cover 7i.
iv) The RHP has unique regular solution. Suppose we have constructed the solution (x, t, A) in the sector Qv, Then we recover the potential from the well known relation
Q(x, t) = lim A (J - CvJC-i(x,t, A)) .
C(Q(x, t)) = Q(x, t).
□
5. Discussion and conclusions
We specified in [13] the choice od the corresponding Kae-Moodv algebras and formulated
the specific Lax operators and the corresponding direct and scattering problems. In each of
(2)
the cases one needs to take into account specific peculiarities. For example, in the case of A5 , after taking the average on the Coxeter automorphism, the elements B[2k - 1, 4] belong to the center of the algebra instead to its Cartan subalgebra.
The constructions that we outlined allow one to apply the dressing Zakharov-Shabat method and derive the soliton solutions of the corresponding mKdV and 2-dimensional Toda field theories. One may expect additional difficulties in this, due to the fact that the Coxeter symmetries require that even the simplest dressing factors must contain at leat 2h simple poles (that is, 1^d 20 poles) whose residues Pk must be related by the Coxeter automorphism. Therefore, it is important that deriving the projectors we must strictly stick to the construction of the FAS in each of the sectors of analytieitv.
The main ideas in this and many previous publications of the author, see e.g. [8], [9], [10]) are based on the notion of fundamental analytic solution introduced by A.B, Shabat [32], [33], Another important trend started by A.B, Shabat and his collaborators concerns the classification of the integrable NLEE, see [28], [34], [36], [1], [35], [29], [30] and the numerous references therein. The idea is based on the theorem that if a given nonlinear evolution equation possesses a master symmetry, then it has an infinite number of integrals of motion and therefore, it should be integrable.
a)
о—о—о—о—о
а1 а2 а4
а5
а1 а2 а4 а5
)
ßl ß2
Рис, 2: Dynkin diagrams (DD) of A5 and related Кае-Moody algebras: a) DD of A5 ~ s 1(6); b) extended DD of A^; c) DD of Af.
The final remark here concerns the fact that the one-to-one correspondence between the minimal sets of scattering data and the potential Q(x, t) follows also from the expansions over the squared solutions of L, see [8], [10], [9], [22], [21], These ideas will be published elsewhere.
Acknowledgements I am grateful to Dr. Alexander Stefanov and Dr. Stanislav Varbev for useful discussions,
A. Basis and grading of A^1
The rank of the algebra A^ ~ s 1(6) is 5, the Coxeter number is h = 6 and its exponents are 1, 2, 3 4, 5. The root system and the set of simple roots aj of A^1 ~ s 1(6) are
A =A+ u A-, A± = {±(ej — e k), 1 k^ 6}, aj =j — ej+u j = 1,---,5-The Cartan-Wevl basis of A^ in the typical representation is as follows: Hej-ek = Ejj - Ekk, Eej -ek = Ejk, E-a = El,
[Ha ,Eß ] = (a,ß )Eß, [Ea,Eß ] = Na,ßEa+ß.
The numbers Na,ß = — Nß,a are non-vanishing if and only if a + ß E A.
The Dvnkin diagram of A5 algebra and the extended Dvnkin diagrams of A^ and A^2 are shown in Figure 2,
Let us now briefly outline how to define Kae-Moodv algebra starting from a simple Lie algebra g which in our case is chosen to be A5 ~ s 1(6). First we use a Coxeter automorphism to introduce a grading in the Lie algebra A5:
g = e g(k\ g = e Us,
k=0 s=0
where the linear subspaces are such that
C1XC-1 = u-kX, X E g(k), C1YC-1 = u-sY) Y E Is, where u1 = e^r, Each of the gradings satisfies
[g(k), g(m)] e g(k+m), g,] e 0S+P, (A.l)
where (k + m^d (s +p) are understood modulo 6. The indices for A^1 are everywhere taken modulo 6. Using this grading, we can now construct polynomials in A and X-1 such that
N
X(A) = ^ ASXS, Xs E g(s), (A.2)
s= — 00
which are the elements of the Kac-Moody algebra [25], [4], Here the upper index of the subspace s is evaluated modulo 6. Obviously the commutator of two such polynomials in A and A- due to the properties of grading (A.l) will again be of form (A,2), Of course, the rigorous definition of Kae-Moodv algebra requires additional structures, which we do not mention now,
A5
produces two Kac-Moodv algebras A5i) with height 1 and A52) with height 2,
Ci Ci
0 ~ A5, This is Z6 automorphism. With this automorphism we effectively work with Kac-Moodv algebra A5i). Indeed, each of these choices satisfies Cf = 1, Cf = 1 and each of these automorphisms induces a grading in g.
In what follows, the choice of the automorphisms is specified by
Ci
/0 1 0 0 0 0 1 0 0 0 0 0
0 0 1 0 0 0 0 w1 0 0 0 0
0 0 0 0 0 0 1 0 0 1 0 0 , C1 = 0 0 0 0 w2 0 0 w3 0 0 0 0
0 0 0 0 0 1 0 0 0 0 w4 0
—1 0 0 0 0 0 0 0 0 0 0
(A.3)
J10) and Ci = JO1) along
C6 = C6 = 1 Ci
with the more general ones J(k\ which provide a convenient basis in A5i) which satisfies the above gradings, see [3], [24], [4], [25]:
J (k)
/ j,j+Sw1 Ej,j+S-, tj,j + S
{
1
- 1
if if
j + S ^ 6 j + k > 6.
Here 6 x 6 matrices Ekm are defined as (Ekm)sp = 5ksSmp. The elements of this basis satisfy the commutation relations
J(k), J(m)
(, ,-ms , ,-kl\ j(k+m) [W1 - W1 ) Js+l ■
C-1Jifc)Ci
w.
v(fc)
It is also easy to confirm that
C1-iJik)Ci = u-kJ(k\
and
T(k) T(m) = . ,-sm j(k+m) Js Jp = Mi Js+p ,
Using this, the bases in each of the linear subspaces can be specified as follows
(Jf)-1 = (JfT
0(fc) =l.c. {J{sk), s = 1,..., 6}, The basis that we constructed for A^ is
= l.c. { J(k\ k=1,..., 6}.
g(0) :l.c■{ J10),
j (0) T (0) T (0) T (0)-.
J2 , J3 , J4 , J5 },
, rj(2) j(2) j(2) j(2) j(2) j(2) -i
. i.C, { J1 , J2 , J3 , J4 , J5 , J6 }
(4) (4) T(4) T(4) T(4) T(4) T(4) 1
. i.C, { J1 , J2 , J3 , J4 , J5 , J6 }
(2) (2)
g(1) tc .{ J11),J21),J31),J41),JÎ1),J61)}, g(3) tc .{ j13),j23),j31),j43),j53),j63)}, g(5) :i.c.{ j15),j25),j35),j45),jî5),j65)}.
B. Basis and grading of a52)
(2)
A5
C2 = Ci o V, which is a composition of Ci with the external automorphism V of A5, and V is
generated by the symmetry of its Dvnkin diagram. In the five-dimensional space of roots, the
mapping V acts as V : ek — - e7-k, k = 1,..., 6. On any of the root vectors X, V act as
V (X) = -s2xts2-\
S2 = Ei 6 — E2.5 + E3 4 — E4 3 + E5 2 — E,
^6,1-
Note that S
—S2. Obviouslv, V splits the Lie algebra g ~ A5 into two: g bases, corresponding to the positive roots, are given as follows:
(0) • {£+, £+, £+-, 1 ^ i<j^ 3},
It], n 1 11 ' ^ ^ } '
0V 0
(1).
St, ij '
S -, j
£± = Eij T (—IT Here we can identify the root vectors:
1 ^ i< j ^ 3} ±
go U g1, whose
(B.l)
S± = Eß ± (-1T~JEß,
£t
jj
E.
jj-
E.
±
±
°ij
E
±
6 i + ^ 7
S ±, j
E+ 7
St..
Obviously, E+i-ep E+ +e. and E+e. are the generators of sp(6) corresponding to its positive roots; E—-e^d E-i+e. provide the positive roots of g^ It is easy to confirm that they satisfy standard commutation relations, taking into account the Z2-grading such as
[E±, E±a] = Ha, [H, E±] = a(H)E±, [E-,E-] = n-jE^, [E-,E+] = n+jE-^,
etc. Let us now take into account the Coxeter automorphism which is given by
C2(X) = CiV (X )C-1 = —CiS2XTS-1C-1.
One can check that C2,0 = 1, so the Coxeter number is h2 = 10. This automorphism C2 splits the roots of A5 into three orbits each containing 10 roots. The grading condition is
[g(k),g(0] c g(k+l), k,l = 1,..., 10,
where k + l is taken modulo 1^, We assume that the orbits start from the root vectors E12, E34
and E13. We consider also the action of C2 also on the Cartan generators. The basis for each g(k) C2
sf = (Eij), n{k) = (Ell),
s=0
s=0
U2
2 t i
e io .
It is easy to check that C2(S(k)) = uks(k), C2{U(f)) = <4H^/i.e. £(k) and n(k) belong to g(k) We will provide this basis explicitly:
(k) s12
£
(k) 34
£
(k) 13
n[k)
( 0 1 0 0 U-3k - U2 0
U-5k - U2 0 U-2k U2 0 0 0
0 U-7k - U2 0 0 0 f -4k - U2
0 0 0 0 0 0
U-8k U2 0 0 0 0 -U-k
\ 0 0 U-9k U2 0 U -6k U2 0
( 0 0 0 U2- 6k 0 0
0 0
-U-k 0
V 0
0 0
f -5k
0 0
, -6k \ U2
0 0
0 0
-3k 2
0 0
U2 U.
0 0 0
0
-8k 2
-7k
-U2 0 0
1 0 0
-5k
-U. 1 0
-8k 2
U0
0 0
-9k
0 0
— U0
7k
-9k 2
0
=c#,kdiag (u2 5k 7k, U 9k
U-4k - U2 U-2k U2 0 0 0 0
0 0 U2- k \
0 3k 2k
-U- -U-
0 U-4k - U2 0
0 0 0
0 0 0
0 0 0 /
-9k 2 0,U-3k, U2- k ),
where ch, k = - 1. Since wf = -1 it is easy to see that cH,k = 0 for k = 13 5 7 and 9, Thus, the subspaee g(p) has a nontrivial section with the Cartan subalgebra if and only if p is an exponent of A52). It is easy to confirm that ¿+ provides a basis for the sub algebra sp(6) of a52) . Then the basis in each of the subspaees g(k) is as follows
g( ) =1.C■ {'l1, <22, ^-33}, g( ) =1-C■ {<21, ¿32, ¿43, <15},
0( ) =1.C, {<?3+, ¿¿¡, ¿-4}
0(4) =l.c.{¿5+1, ¿1-2, ¿-3}
0(6) =1.C.{¿+5, ¿3-2}
0( ) =l.c, {^+3, ¿+4, ¿41}
g( ) =1.C-{<14, ¿25, ¿-3, ¿24}
g() =l,e,^+;, ¿61, ¿33 ¿11, ¿33 ¿22}, (B-2)
0 =1.C^¿4l, ¿52, ¿31, ¿42}, g( ) =1.C-{<+!, ¿23, ¿34, ¿51}-
As a result, the rank of A^ is 3 h = 10 and its exponents are 1 3 5 7 9, see [5], [4].
(2) A5
automorphism as an element of the Cartan subgroup. More precisely, one can use the
C2
2iri
C12(X) = —S2XJS2-1, S2 = diag (1, —w2,w2, — w3>4, —w§), w = e^, (B.3)
and where the transposition is taken with respect to the second diagonal of the matrix. With choice for the Coxeter automorphism, the set of admissible roots of A52) acquires the form
¿l30 = 2(E1,5 + E2,6 ), ¿-/0 =2(E5,1 + E6,2) CT =%1 +
¿Pi =C(Ei+1,i + E7-i ,6-i), ¿-l =(Ei,i+1 + E6-i ,7-i )( 1, M/i =Mi+1 — , ¿/3 =CE4,3, ¿-/3 =E3,4 C 1, = — M3 + M■4,
where i = 1, 2, (Ekm)ab = and = Ei,i — E7-,7-,
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Vladimir Stefanov Gerdjikov, Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. Georgi Bonehev Str,, Block 8, 1113, Sofia, Bulgaria
Sankt-Petersburg State University of Aerospace Instrumentation
B. Morskava, 67A,
190000, St-Petersburg, Russia
Institute for Advanced Physical Studies,
111 Tsarigradsko chaussee,
1784, Sofia, Bulgaria
Institute for Nuclear Research and Nuclear Energy
Bulgarian Academy of Sciences,
72 Tsarigradsko Chaussee, Blvd.,
1784, Sofia, Bulgaria
E-mail: vgerdj ikovSmath. bas. bg
