The Diffiety Institute Preprint Series

Preprint DIPS–5/2004 June 18, 2004

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Decompositions of the loop algebra over so(4) and integrable models of

the chiral equation type

by

O. V. Efimovskaya, V. V. Sokolov

DECOMPOSITIONS OF THE LOOP ALGEBRA OVER so(4) AND INTEGRABLE MODELS OF THE CHIRAL

EQUATION TYPE

O. V. EFIMOVSKAYA AND V. V. SOKOLOV

Abstract. Decompositions of the loop algebra over so(4) are considered, and the exactly integrable nonlinear hyperbolic systems of the principal chiral field equation type are analyzed. New example of such system is found and the Lax representation for this example is constructed.

UDC 517.958+512.77

Introduction. In this article, we consider decompositions of the loop algebra so(4)((λ)) to the direct sum

(1)  so(4)((λ)) = so(4)[[λ]] ⊕ G

of vector spaces, where so(4)[[λ]] denotes the Taylor series Lie subalgebra and G is some Lie subalgebra. This subalgebra G is called a factoring subalgebra. ¿From the results of the paper [2] it follows that the integrable system of the form

(2)  u_{ξ} = [u, Sv + vS^{t}],  v_{η} = [v, S̄u + uS̄^{t}],

is assigned to each factoring subalgebra G; here u, v ∈ so(3), S and S̄ are some constant matrices and the superscript t denotes transposition. The Lax pair for system (2) is expressed in terms of the projector onto the subalgebra G due to decomposition (1).

We will describe integrable models of the form (2) within the framework of [2] under the additional assumption that the matrices S and S̄ are diagonal. In this case system (2) can be rewritten as follows

(3)  u_{ξ} = Λv × u,  v_{η} = Λ̄ u × v,

where

Λ = diag(λ_{1}, λ_{2}, λ_{3}),  Λ̄ = diag(λ̄_{1}, λ̄_{2}, λ̄_{3}).

Here u = (u_{1}, u_{2}, u_{3}) and v = (v_{1}, v_{2}, v_{3}) are vectors in ℝ^{3} and the cross × denotes vector product. Quite obviously, system (3) is compatible with the condition |u| = |v| = 1, which is usually imposed when speaking about applications in physics or geometry.

1991 Mathematics Subject Classification. 35Q58, 37K10, 37K30.

Key words and phrases. Factoring subalgebras, Lax representations, integrable models, loop algebra, Cherednik’s model.

This paper is submitted for publication in the Issue on Geometry of PDE in:

Fundamental’naya i Prikladnaya Matematika / Jour. of Mathematical Sci. (2004).

The work was partially supported by the grants RFBR 02-01-00431 and NSh 1716.2003.1.

The transformations

u_{i} → −u_{i},  v_{i} → −v_{i},  ξ → −ξ,  η → −η

do not spoil the integrability of system (3) for any i. They are equivalent to the change λ_{i} → −λ_{i}, λ̄_{i} → −λ̄_{i}. In addition, compatible transpositions of diagonal elements within the matrices Λ and Λ̄ are also allowed.

Further on, we will show that system (3) admits a Lax representation if the elements of the matrices Λ and Λ̄ satisfy the relations

(4)  λ_{1}λ̄_{1}(λ_{3}^{2} − λ_{2}^{2}) + λ_{2}λ̄_{2}(λ_{1}^{2} − λ_{3}^{2}) + λ_{3}λ̄_{3}(λ_{2}^{2} − λ_{1}^{2}) = 0,
(5)  λ_{1}λ̄_{1}(λ̄_{3}^{2} − λ̄_{2}^{2}) + λ_{2}λ̄_{2}(λ̄_{1}^{2} − λ̄_{3}^{2}) + λ_{3}λ̄_{3}(λ̄_{2}^{2} − λ̄_{1}^{2}) = 0.

If all elements λ_{i} are nonzero, relation (4) implies the equalities

(6) λ̄_{1} = κ_{1}λ_{1} + \frac{κ_{2}}{λ_{1}} ,  λ̄_{2} = κ_{1}λ_{2} + \frac{κ_{2}}{λ_{2}} ,  λ̄_{3} = κ_{1}λ_{3} + \frac{κ_{2}}{λ_{3}} .

Relation (5) is equivalent to the equation

κ_{1}κ_{2}(λ_{3}^{2} − λ_{2}^{2})(λ_{1}^{2} − λ_{3}^{2})(λ_{2}^{2} − λ_{1}^{2}) = 0.

The case κ_{2} = 0 corresponds to the known integrable model by I. Cherednik ([3]), while the case κ_{1} = 0 was discovered by I. Golubchik and V. Sokolov, see [1, 2]. If κ_{i} ≠ 0, then without loss of generality we put λ_{1} = λ_{2}. Then we have λ̄_{1} = λ̄_{2}, while λ_{3}, λ̄_{3} are arbitrary. Possibly, this case is new.

We conjecture that Eq. (4-5) are not only sufficient, but also necessary for exact integrability of the model given in Eq. (3). In principle, one can use the symmetry approach or the Painlevé test to prove this.

1. Factoring subalgebras for the loop algebras over so(3) and so(4)

In the sequel, we represent the elements of so(4) as block–diagonal matrices which consist of two blocks, each of them belonging to so(3). By e_{1}, e_{2}, and e_{3} we denote the standard basis in so(3); namely, we put

e_{1} = ( \begin{matrix}0 & 1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , e_{2} = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ −1 & 0 & 0 \end{matrix} ) , e_{3} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & −1 & 0 \end{matrix} ) .

DECOMPOSITIONS OF LOOP ALGEBRA AND INTEGRABLE MODELS 3

Obviously, any factoring subalgebra G in so(4)((λ)) contains the series

(7)

E_{1} = \frac{1}{λ} ( \begin{matrix}e_{1} & 0 \\ 0 & 0 \end{matrix} ) + ( \begin{matrix}∗ & 0 \\ 0 & c̄ \end{matrix} ) + O(λ),
E_{2} = \frac{1}{λ} ( \begin{matrix}e_{2} & 0 \\ 0 & 0 \end{matrix} ) + ( \begin{matrix}∗ & 0 \\ 0 & ā \end{matrix} ) + O(λ),
E_{3} = \frac{1}{λ} ( \begin{matrix}e_{3} & 0 \\ 0 & 0 \end{matrix} ) + ( \begin{matrix}∗ & 0 \\ 0 & b̄ \end{matrix} ) + O(λ),

(8)

Ē_{1} = \frac{1}{λ} ( \begin{matrix}0 & 0 \\ 0 & e_{1} \end{matrix} ) + ( \begin{matrix}c & 0 \\ 0 & ∗ \end{matrix} ) + O(λ),
Ē_{2} = \frac{1}{λ} ( \begin{matrix}0 & 0 \\ 0 & e_{2} \end{matrix} ) + ( \begin{matrix}a & 0 \\ 0 & ∗ \end{matrix} ) + O(λ),
Ē_{3} = \frac{1}{λ} ( \begin{matrix}0 & 0 \\ 0 & e_{3} \end{matrix} ) + ( \begin{matrix}b & 0 \\ 0 & ∗ \end{matrix} ) + O(λ),

where

(9)  a = ∑ a_{i}e_{i},  b = ∑ b_{i}e_{i},  c = ∑ c_{i}e_{i},
(10)  ā = ∑ ā_{i}e_{i},  b̄ = ∑ b̄_{i}e_{i},  c̄ = ∑ c̄_{i}e_{i}

are some elements of so(3).

It is clear that the sum of the Lie subalgebra G, which is generated by series (7-8), and the Taylor series subalgebra coincides with whole algebra so(4)((λ)). But for generic elements of the form (7), (8) this sum is not direct.

Proposition 1. For any factoring subalgebra G, the following commutator relations are satisfied

(11) ( \begin{matrix}[E_{2}, Ē_{2}] \\ [E_{2}, Ē_{3}] \\ [E_{2}, Ē_{1}] \end{matrix} ) = ( \begin{matrix}0 & a_{1} & −a_{3} \\ 0 & b_{1} & −b_{3} \\ 0 & c_{1} & −c_{3} \end{matrix} ) ( \begin{matrix}E_{2} \\ E_{3} \\ E_{1} \end{matrix} ) +

( \begin{matrix}0 & −ā_{1} & ā_{3} \\ ā_{1} & 0 & −ā_{2} \\ −ā_{3} & ā_{2} & 0 \end{matrix} ) ( Ē_{2} Ē_{3} Ē_{1} ) ,

(12) ( \begin{matrix}[E_{3}, Ē_{2}] \\ [E_{3}, Ē_{3}] \\ [E_{3}, Ē_{1}] \end{matrix} ) = ( \begin{matrix}−a_{1} & 0 & a_{2} \\ −b_{1} & 0 & b_{2} \\ −c_{1} & 0 & c_{2} \end{matrix} ) ( \begin{matrix}E_{2} \\ E_{3} \\ E_{1} \end{matrix} ) +

( \begin{matrix}0 & −b̄_{1} & b̄_{3} \\ b̄_{1} & 0 & −b̄_{2} \\ −b̄_{3} & b̄_{2} & 0 \end{matrix} ) ( Ē_{2} Ē_{3} Ē_{1} ) ,

(13) ( \begin{matrix}[E_{1}, Ē_{2}] \\ [E_{1}, Ē_{3}] \\ [E_{1}, Ē_{1}] \end{matrix} ) = ( \begin{matrix}a_{3} & −a_{2} & 0 \\ b_{3} & −b_{2} & 0 \\ c_{3} & −c_{2} & 0 \end{matrix} ) ( \begin{matrix}E_{2} \\ E_{3} \\ E_{1} \end{matrix} ) +

( \begin{matrix}0 & −c̄_{1} & c̄_{3} \\ c̄_{1} & 0 & −c̄_{2} \\ −c̄_{3} & c̄_{2} & 0 \end{matrix} ) ( Ē_{2} Ē_{3} Ē_{1} ) .

Proof. Since the sum of so(4)[[λ]] and G is supposed to be direct, the dimension of the vector space generated by elements (7), (8) and their pairwise commutators must be equal to 12. In other words, for any i and j commutators [E_{i}, Ē_{j}] are some linear combinations of generators (7), (8). For instance,

[E_{1}, Ē_{2}] = ∑_{i=1}^{3} m_{i}E_{i} + n_{i}Ē_{i}

for some constants m_{i} and n_{i}. Substituting the corresponding decompositions from (7), (8) for E_{1}, Ē_{2} into the above identity and equating the coefficients of λ^{−1}, we obtain explicit expressions for m_{i} and n_{i} in terms of the coefficients from formula (9). These expressions are given by the first row of matrix formula (13). In this manner, we obtain all identities (11-13). □

It is easy to see that for any factoring subalgebra G its projections onto the first and second blocks are factoring subalgebras for the algebra so(3)((λ)). Relations (11-13) describe the “interaction” between these two factoring subalgebras. If all constants a_{i}, b_{i}, c_{i}, ā_{i}, b̄_{i}, c̄_{i} are equal to zero simultaneously, then the algebra G is the direct sum of two factoring subalgebras for so(3)((λ)).

Another necessary condition for the sum of so(4)[[λ]] and G to be direct is that the dimension of the vector space spanned over the vectors E_{i}, [E_{i}, E_{j}], and [E_{i}, [E_{j}, E_{k}]] be equal to 9. The same condition should be fulfilled for the generators Ē_{1}, Ē_{2}, Ē_{3}. The corresponding commutator relations coincide with the identities for the loop algebra so(3)((λ)) obtained in [4]. Namely, the following statement is valid.

Proposition 2. Suppose G is a factoring subalgebra; then the following relations are satisfied

(14) ( \begin{matrix}[E_{1}, [E_{2}, E_{3}]] \\ [E_{3}, [E_{1}, E_{2}]] \\ [E_{2}, [E_{3}, E_{1}]] \end{matrix} ) = A ( \begin{matrix}[E_{3}, E_{1}] \\ [E_{1}, E_{2}] \\ [E_{2}, E_{3}] \end{matrix} ) + B ( \begin{matrix}E_{2} \\ E_{3} \\ E_{1} \end{matrix} )

and

(15) ( \begin{matrix}[E_{3}, [E_{2}, E_{3}]] + [E_{1}, [E_{1}, E_{2}]] \\ [E_{1}, [E_{3}, E_{1}]] + [E_{2}, [E_{2}, E_{3}]] \\ [E_{2}, [E_{1}, E_{2}]] + [E_{3}, [E_{3}, E_{1}]] \end{matrix} ) =

C ( \begin{matrix}[E_{3}, E_{1}] \\ [E_{1}, E_{2}] \\ [E_{2}, E_{3}] \end{matrix} ) + D ( \begin{matrix}E_{2} \\ E_{3} \\ E_{1} \end{matrix} ) ,

where A, B, C, and D are some matrices of the form

(16)  A = ( \begin{matrix}−u & w & 0 \\ u & 0 & −v \\ 0 & −w & v \end{matrix} ) ,  B = ( \begin{matrix}−α & β & 0 \\ α & 0 & −γ \\ 0 & −β & γ \end{matrix} ) ,

(17)  C = ( \begin{matrix}x & v & −w \\ −v & y & u \\ w & −u & z \end{matrix} ) ,  D = ( \begin{matrix}ε & γ & −β \\ −γ & τ & α \\ β & −α & δ \end{matrix} )

and u, v, w, x, y, z, α, β, γ, δ, τ, and ε are some numbers such that the conditions tr C = tr D = 0 are satisfied. Moreover, there exist the constants k_{1} and k_{2} such that

(18)  k_{1}A + k_{2}B = 0,  k_{1}C + k_{2}D = 0.

The analogous commutator relations link the generators Ē_{1}, Ē_{2}, Ē_{3}: one should put bars over all symbols in Eq. (14-18).

Now we recall a general construction from the paper [2]. Let

(19)  L = \frac{d}{dη} + ∑ u_{i}E_{i},  M = \frac{d}{dξ} + ∑ v_{i}Ē_{i}.

Then the relation [L, M] = 0 is equivalent to the system

(20) u_{ξ} = [u, v_{1}c + v_{2}a + v_{3}b],  v_{η} = [v, u_{1}c̄ + u_{2}ā + u_{3}b̄],

where u = ∑ u_{i}e_{i}, v = ∑ v_{i}e_{i}. Indeed, the commutator [L, M] lies in G, and its asymptotic behaviour is Z/λ + O(1), while system (20) is equivalent to the condition that the residue Z be equal to zero. The algebra G does not contain non-zero Taylor series, therefore the equation Z = 0 is equivalent to the condition [L, M] = 0.

One easily checks that Eq. (20) can be rewritten in the form (2), where

(21) S = ( \begin{matrix}\frac{1}{2}(a_{2} − b_{3} + c_{1}) & a_{3} & −c_{3} \\ b_{2} & \frac{1}{2}(−a_{2} + b_{3} + c_{1}) & c_{2} \\ −b_{1} & a_{1} & \frac{1}{2}(a_{2} + b_{3} − c_{1}) \end{matrix} )

and S̄ is obtained by putting the bar over each symbol in the formula above.

Within this paper, we assume that the matrices S and S̄ are diagonal, which implies

a_{1} = ā_{1} = a_{3} = ā_{3} = b_{1} = b̄_{1} = b_{2} = b̄_{2} = c_{2} = c̄_{2} = c_{3} = c̄_{3} = 0.

Then the matrices Λ and Λ̄ in Eq. (3) are

(22)  Λ = diag(c_{1}, a_{2}, b_{3}),  Λ̄ = diag(c̄_{1} , ā_{2}, b̄_{3}).

We also assume that none of these matrices is zero, otherwise system (3) becomes trivial.

One can check that in this case the structure of the generators (7),(8) is of the form

(23)  E_{i} = ( \begin{matrix}q_{i} e_{i} & 0 \\ 0 & p̄_{i} e_{i} \end{matrix} ) ,  Ē_{i} = ( \begin{matrix}p_{i} e_{i} & 0 \\ 0 & q̄_{i} e_{i} \end{matrix} ) ,

where q_{i} and q̄_{i} are some scalar Laurent series with the asymtotics 1/λ + O(1) and p_{i}, p̄_{i} are Taylor series.

Now we describe all factoring subalgebras with the generators given by Eq. (23). Substituting Eq. (23) into commutation relations (11-15), we get an overdetermined system of algebraic equations for q_{i}, p_{i}, q̄_{i}, p̄_{i}. Quite obviously, the relations

u = ū = v = v̄ = w = w̄ = α = ᾱ = β = β̄ = γ = γ̄ = 0

are its compatibility conditions.

The system consists of the equations

(24)  \begin{aligned}c_{1}q_{2} − q_{3}p_{1} + b̄_{3}p_{2} & = 0, \\ a_{2}q_{1} − q_{3}p_{2} + b̄_{3}p_{1} & = 0, \\ b_{3}q_{2} − q_{1}p_{3} + c̄_{1}p_{2} & = 0, \end{aligned}  \begin{aligned}c_{1}q_{3} − q_{2}p_{1} + ā_{2}p_{3} & = 0, \\ a_{2}q_{3} − q_{1}p_{2} + c̄_{1}p_{3} & = 0, \\ b_{3}q_{1} − q_{2}p_{3} + ā_{2}p_{1} & = 0, \end{aligned}

(25)  \begin{aligned}y q_{1}q_{2} − τ q_{3} + q_{1}^{2}q_{3} − q_{2}^{2}q_{3} & = 0, \\ z q_{2}q_{3} − δ q_{1} + q_{1}q_{2}^{2} − q_{1}q_{3}^{2} & = 0, \\ x q_{1}q_{3} − ε q_{2} + q_{2}q_{3}^{2} − q_{1}^{2}q_{2} & = 0, \end{aligned}

(26)  \begin{aligned}ȳ p_{1}p_{2} − τ̄ p_{3} + p_{1}^{2}p_{3} − p_{2}^{2}p_{3} & = 0, \\ z̄ p_{2}p_{3} − δ̄ p_{1} + p_{1}p_{2}^{2} − p_{1}p_{3}^{2} & = 0, \\ x̄ p_{1}p_{3} − ε̄ p_{2} + p_{2}p_{3}^{2} − p_{1}^{2}p_{2} & = 0. \end{aligned}

There is also the system symmetric to (24-26) with bars over all symbols.

¿From Eq. (24) and (25) it follows that

(27)  p_{1} = c_{1} + O(λ),  p_{2} = a_{2} + O(λ),  p_{3} = b_{3} + O(λ),
q_{2}^{2} − q_{1}^{2} = \frac{y}{λ} + O(1),  q_{3}^{2} − q_{2}^{2} = \frac{z}{λ} + O(1),  q_{1}^{2} − q_{3}^{2} = \frac{x}{λ} + O(1).

Excluding the unknowns q_{1}, q_{2}, q_{3} from Eq. (24), we obtain the equation

a_{2}ā_{2}(p_{1}^{2} − p_{3}^{2}) + b_{3}b̄_{3}(p_{2}^{2} − p_{1}^{2}) + c_{1}c̄_{1}(p_{3}^{2} − p_{2}^{2}) = 0,

whence, by using Eq. (27),(22), we deduce that condition (4) is necessary. Condition (5) is obtained if we start from the system symmetric to Eq. (24). The equation

(28)  p_{1}^{2}(q_{2}^{2} − q_{3}^{2}) + p_{2}^{2}(q_{3}^{2} − q_{1}^{2}) + p_{3}^{2}(q_{1}^{2} − q_{2}^{2}) = 0

also follows from Eq. (24). Also, formulae (28) and (27) imply the equation

c_{1}^{2}z + a_{2}^{2}x + b_{3}^{2}y = 0.

Taking into account the condition x + y + z = 0, we conclude that either the relation

(29) z = κ(a_{2}^{2} − b_{3}^{2}),  x = κ(b_{3}^{2} − c_{1}^{2}),  y = κ(c_{1}^{2} − a_{2}^{2})

holds for some κ, or the relation c_{1}^{2} = a_{2}^{2} = b_{3}^{2} is valid. The second case corresponds to the unit matrix Λ (up to the transformations mentioned in the introduction).

The unknows p_{1}, p_{2}, p_{3} are also found from system (24): they are

(30)  \begin{aligned}p_{1} & = \frac{c_{1}q_{1}q_{2}^{2} + a_{2}b̄_{3}q_{2}q_{3} + c̄_{1}b_{3}b̄_{3}q_{1}}{q_{1}q_{2}q_{3} − c̄_{1}ā_{2}b̄_{3}} , \\ p_{2} & = \frac{a_{2}q_{2}q_{3}^{2} + b_{3}c̄_{1}q_{1}q_{3} + ā_{2}c_{1}c̄_{1}q_{2}}{q_{1}q_{2}q_{3} − c̄_{1}ā_{2}b̄_{3}} , \\ p_{3} & = \frac{b_{3}q_{3}q_{1}^{2} + c_{1}ā_{2}q_{1}q_{2} + b̄_{3}a_{2}ā_{2}q_{3}}{q_{1}q_{2}q_{3} − c̄_{1}ā_{2}b̄_{3}} . \end{aligned}

The denominators in these expressions are nonzero due to Eq. (27). One can check that the other three equations in system (24) are equivalent to the relations

c_{1}z = a_{2}b̄_{3} − ā_{2}b_{3},  a_{2}x = b_{3}c̄_{1} − b̄_{3}c_{1},  b_{3}y = c_{1}ā_{2} − c̄_{1}a_{2}, (31)
c_{1}δ = c̄_{1}(a_{2}ā_{2} − b_{3}b̄_{3}), a_{2}ε = ā_{2}(b_{3}b̄_{3} − c_{1}c̄_{1}), b_{3}τ = b̄_{3}(c_{1}c̄_{1} − a_{2}ā_{2}), (32)

since equations (25) hold.

Now we consider system (25). Taking into account the condition x + y + z = ε + δ + τ = 0, from Eq. (25) we get

δq_{1}^{2} + εq_{2}^{2} + τq_{3}^{2} = 0, zq_{2}^{2}q_{3}^{2} + xq_{1}^{2}q_{3}^{2} + yq_{1}^{2}q_{2}^{2} = 0.

Three cases follow from these relations and the second condition in Eq. (18). Namely, either we obtain the equality x = y = z = 0, or the equation ε = δ = τ = 0 holds, or, finally, two series among q_{1}, q_{2}, and q_{3} coincide. In the last case, we put q_{2} = q_{1} without loss of generality; then we also conclude that either τ = y = 0 or q_{3} = q_{1}.

Case 1. Consider the case x = y = z = 0. In addition, suppose that q_{i} ≠ q_{j} for any i ≠ j (unless we get Case 3). System (25) acquires the form

q_{2}^{2} − q_{3}^{2} = δ,  q_{3}^{2} − q_{1}^{2} = ε,  q_{1}^{2} − q_{2}^{2} = τ.

¿From Eq. (31) and (32) it follows that the relations

c̄_{1} = kc_{1},  ā_{2} = ka_{2},  b̄_{3} = kb_{3},
δ = k^{2}(a_{2}^{2} − b_{3}^{2}),  ε = k^{2}(b_{3}^{2} − c_{1}^{2}),  τ = k^{2}(c_{1}^{2} − a_{2}^{2})

hold. Here k is a nonzero parameter which can be eliminated from system (3) by a scaling of the independent variable η. The parameter λ in the factoring subalgebra problem is defined up to the change

(33)  λ ↦ λ + k_{2}λ^{2} + k_{3}λ^{3} + ⋯,

therefore, without loss of generality, we assume that the conditions

q_{1}^{2} − k^{2}c_{1}^{2} = q_{2}^{2} − k^{2}a_{2}^{2} = q_{3}^{2} − k^{2}b_{3}^{2} = \frac{1}{λ^{2}}

are fulfilled. We finally have

q_{1} = \frac{\sqrt{1 + k^{2}c_{1}^{2}λ^{2}}}{λ} ,  q_{2} = \frac{\sqrt{1 + k^{2}a_{2}^{2}λ^{2}}}{λ} ,  q_{3} = \frac{\sqrt{1 + k^{2}b_{3}^{2}λ^{2}}}{λ} ,
p_{1} = c_{1}\sqrt{1 + k^{2}a_{2}^{2}λ^{2}}\sqrt{1 + k^{2}b_{3}^{2}λ^{2}} + ka_{2}b_{3}λ\sqrt{1 + k^{2}c_{1}^{2}λ^{2}},
p_{2} = a_{2}\sqrt{1 + k^{2}c_{1}^{2}λ^{2}}\sqrt{1 + k^{2}b_{3}^{2}λ^{2}} + kc_{1}b_{3}λ\sqrt{1 + k^{2}a_{2}^{2}λ^{2}},
p_{3} = b_{3}\sqrt{1 + k^{2}c_{1}^{2}λ^{2}}\sqrt{1 + k^{2}a_{2}^{2}λ^{2}} + kc_{1}a_{2}λ\sqrt{1 + k^{2}b_{3}^{2}λ^{2}}.

The formulae for the functions q̄_{i}, p̄_{i} are similar.

Now we consider the upper block in the equation [L, M] = 0 for the operators L and M, which are given in Eq. (19), and obtain the Lax operators

(34)  L = \frac{d}{dη} + ∑_{i=1}^{3} u_{i}q_{i}e_{i},  M = \frac{d}{dξ} + ∑_{i=1}^{3} v_{i}p_{i}e_{i}

for Cherednik’s model. Here the coefficients belong to so(3). The lower block defines another Lax pair which is completely analogous.

Case 2. Consider the case ε = δ = τ = 0; assume that q_{i} ≠ q_{j} for any i ≠ j. Taking into account Eq. (29), we rewrite system (25) in the form

\frac{q_{1}q_{2}}{q_{3}} − κb_{3}^{2} = \frac{q_{1}q_{3}}{q_{2}} − κa_{2}^{2} = \frac{q_{2}q_{3}}{q_{1}} − κc_{1}^{2} = \frac{1}{λ},

whence we get

q_{1} = \frac{\sqrt{1 + κa_{2}^{2}λ}\sqrt{1 + κb_{3}^{2}λ}}{λ}, q_{2} = \frac{\sqrt{1 + κc_{1}^{2}λ}\sqrt{1 + κb_{3}^{2}λ}}{λ},

q_{3} = \frac{\sqrt{1 + κc_{1}^{2}λ}\sqrt{1 + κa_{2}^{2}λ}}{λ},

p_{1} = c_{1}λq_{1},  p_{2} = a_{2}λq_{2},  p_{3} = b_{3}λq_{3},

c_{1}c̄_{1} = a_{2}ā_{2} = b_{3}b̄_{3} = κc_{1}a_{2}b_{3}.

One can easily obtain analogous expressions that define the functions q̄_{i}, p̄_{i}.

Case 3a. Consider the case q_{1} = q_{2} and assume τ = y = 0. ¿From Eq. (28) it follows that p_{1}^{2} = p_{2}^{2}, and Eq. (27) implies the condition c_{1}^{2} = a_{2}^{2}. Also, from Eq. (24) we easily get c̄_{1}^{2} = ā_{2}^{2}. Consider the case p_{2} = p_{1}, c_{1} = a_{2}, c̄_{1} = ā_{2}. By definition, put c̄_{1} = tc_{1} and k = b̄_{3} + b_{3}t, where k and t are arbitrary constants; further on, we define λ from the equation

q_{1}^{2} − c̄_{1}^{2} = \frac{1}{λ^{2}} .

Then, the solution is

q_{1} = q_{2} = \frac{\sqrt{1 + t^{2}c_{1}^{2}λ^{2}}}{λ} , q_{3} = \frac{\sqrt{4 + k^{2}λ^{2}}}{2λ} + \frac{k}{2} − b_{3}t,

p_{1} = p_{2} = \frac{1}{2}c_{1} (kλ + \sqrt{4 + k^{2}λ^{2}})\sqrt{1 + t^{2}c_{1}^{2}λ^{2}},

p_{3} = b_{3} + \frac{1}{2}tc_{1}^{2}λ (kλ + \sqrt{4 + k^{2}λ^{2}}).

The Lax pair (19), (23) assigned to this solution is one of the main results of the present paper. The constants t and k look a bit artificial; they are introduced in order to absorb the degenerate case which is considered below. The case p_{2} = −p_{1} is treated quite analogously and results in the same system (3).

Case 3b. Let the relation q_{1} = q_{2} = q_{3} hold. ¿From Eq. (25) we obviously get x = y = z = δ = ε = τ = 0. Also, from Eq. (24) we obtain

(35)  (p_{1}, p_{2}, p_{3}) = t (c_{1}, a_{2}, b_{3}),

where t is a Taylor series in λ.

The terms a_{2}, b_{3}, c_{1} do not equal zero simultaneously; consequently, the constant term t_{0} of the series t is also nontrivial and the relation

(36)  (c̄_{1}, ā_{2}, b̄_{3}) = t_{0} (c_{1}, a_{2}, b_{3})

holds. By using Eq. (35), from Eq. (24) we obtain the equations

(c_{1}^{2} − a_{2}^{2})(1 − t) = 0,  (b_{3}^{2} − a_{2}^{2})(1 − t) = 0,  (c_{1}^{2} − b_{3}^{2})(1 − t) = 0.

Clearly enough, either all p_{i} are constants or a_{2}^{2} = b_{3}^{2} = c_{1}^{2}. In the first case, without loss of generality we assume that a_{2} = c_{1} = ā_{2} = c̄_{1} = p_{1} = p_{2} = 0. The final result is

q_{1} = q_{2} = q_{3} = \frac{1}{λ}, p_{1} = p_{2} = 0, p_{3} = b_{3}.

We finally note that these formulae are a particular subcase within Case 3a if we put c_{1} = 0.

In the second case, suppose that c_{1} = a_{2} = b_{3}. Then Eq. (36) implies c̄_{1} = ā_{2} = b̄_{3}, and therefore p_{1} = p_{2} = p_{3}. The solution

q_{i} = \frac{1}{λ} + c̄_{1}, p_{i} = c_{1}(1 + c̄_{1}λ)

is absorbed in Case 2 if κ = 1/c̄_{1} and c_{1} = b_{3} = a_{2}.

Acknowledgements. The authors thank M. V. Pavlov for stimulating discussions.

References

[1] I. Z. Golubchik, V. V. Sokolov, One more instance of the classical Yang–Baxter equation, in: Funkts. analiz i ego pilozh., 34 (2000) 4, 75–78

[2] I. Z. Golubchik, V. V. Sokolov, Compatible Lie brackets and integrable equations of the type of the principal chiral field model, in: Funkts. analiz i ego pilozh., 36 (2002) 3, 9-19 (in Russian).

[3] I. V. Cherednik, On the integrability of the two–dimensional asymmetric chiral O(3)-field and its quantum analog, in: Yadernaya fizika, 33 (1981), 278–282 (in Russian).

[4] V. V. Sokolov, On the decompositions of the loop algebra over so(3) to a sum of two subalgebras, in: Doklady RAN(to appear).

Translated by A. V. KISELEV

119992 Russia, Moscow, GSP-2, Vorobjevy gory, Lomonosov MSU, Faculty of Mathematics and Mechanics, Chair of Higher Algebra.

  E-mail address: efvitaly@mail.ru

119334 Russia, Moscow, Kosygin str., 2, Landau Institute for Theoretical Physics.

  E-mail address: sokolov@landau.ac.ru