Riemann-Hilbert Problems, families of commuting operators and soliton equations

We use Riemann-Hilbert Problems with canonical normalization to develop technique for constructing families of commuting operators. As a result we are able to derive new hierarchies of integrable nonlinear evolution equations.


Introduction
Let Γ be a contour in the complex λ-plane splitting C into two parts C ≡ Γ + ∪ Γ − . By multiplicative Riemann-Hilbert problem (RHP) [4,18] we mean the problem of constructing two functions ξ + (λ) and ξ − (λ) analytic for λ ∈ Γ + and λ ∈ Γ − respectively such that: If the functions ξ ± (λ) are scalar and have no zeroes in their regions of analyticity we can solve the multiplicative RHP by the Plemelj-Sokhotzky formulae [4,18]. We will consider more general and special RHP for functions ξ ± ( x, t, λ) taking values in a simple Lie group G with simple Lie algebra g.
Let us outline the special properties of our RHP. First, as contour Γ we will choose either the real axis R, or a set of straight lines intersecting at the origin of C. Second, we will assume that the solutions and the sewing function depend on two or more auxiliary variables one of which we will call the time t and the others x = (x 1 , . . . , x p ) T will be spatial variables. It is natural that their number p is smaller than the rank of the algebra r = rank g. The third special property is that we will specify explicitly the dependence of the sewing function G( x, t, λ) by: where K and J s belong to the Cartan subalgebra h ∈ g.
In what follows we will start with an RHP whose sewing functions have special dependence on the auxiliary variables t and x s and will demonstrate that their solutions are simultaneous fundamental analytic solutions to a set of commuting operators M , L s . In fact for the simplest nontrivial case when J s , K are real and Γ ≡ R this has been known for long time [21,22,6]. However the RHP was used just for deriving the soliton solutions through the Zakharov-Shabat dressing method [21,22], see also [14,10].
The family of operators M , L s commute provided the first nontrivial coefficient Q 1 ( x, t) in the asymptotic expansion of ξ ± ( x, t, λ) (see eq. (6)) satisfies a certain set of nonlinear evolution equations (NLEE). We will demonstrate on several nontrivial examples that this method allows one to derive new types of integrable interactions. Thus we show that the formal approach of Gel'fand and Dickey [3] can be made more precise.
In Section 2 we outline the main idea of constructing the families of commuting operators [7,8]. The next Section 3 contains several new examples of N -wave type interactions extending the ones found in [12] and their reductions [11,10,9]. The last Section contains discussion and conclusions.
2. RHP with canonical normalization 2.1. RHP and Generalized Zakharov-Shabat operators Consider the simplest nontrivial RHP when Γ ≡ R and the dependence of G( x, t, λ) is provided by eq. (2) with real K and J s ∈ h ⊂ g. Remark 1. With this choice of K and J s it is easy to see that We assume that G( 0, 0, λ) is a smooth bounded function of λ ∈ R and as a consequence it follows that G( x, t, λ) is a smooth bounded function of λ ∈ R for all x and t.
Remark 2. The canonical normalization (4) along with the analyticity properties of the solutions ensure that ξ ± ( x, λ) allow asymptotic expansions of the form: where Q k ( x, t) are smooth functions of x and t vanishing fast enough for x → ∞. The rigorous proof of this fact is out of the scope of the present paper.
It is obvious that belong to the algebra g for any J s and K from g and allow analytic extensions for λ ∈ C ± ; herê ξ ≡ ξ −1 . Since K and J s belong to the Cartan subalgebra h, then Theorem 1 (Zakharov-Shabat, 1974). Let ξ ± (x, t, λ) be solutions to the RHP (3) whose sewing function depends on the auxiliary variables x and t as above. Then ξ ± (x, t, λ) are fundamental analytic solutions of the set of differential operators where Q 1 ( x, λ) is the first nontrivial coefficient in the asymptotic expansion (6) of ξ ± ( x, t, λ).
Proof. Introduce the functions: and using (2) prove that which means that these functions are analytic functions of λ in the whole complex λ-plane. Next we find that for λ → ∞: Then we make use of Liouville theorem to get Lemma 1. The set of operators L s and M have a common FAS, i.e. they all must commute, that is Q 1 ( x, t) satisfies the following NLEE: Proof. Follows naturally from the definitions of the operators L s and M (9).

RHP and Operators of Caudry-Beals-Coifman type
In this Subsection we consider more complicated RHP which is formulated as follows.
• First we introduce the complex valued elements K and J s of the Cartan subalgebra. The conditions [13] Im λα(J) = 0, where ∆ is the root system of g, gives a set of M straight lines, or equivalently, a set of 2M rays l ν starting from the origin. We then define the contour as Γ ≡ 2M ∪ ν=1 l ν . • To each ray l ν (15) one can associate the subset of roots δ ν ⊂ ∆ and the corresponding subalgebra g ν ⊂ g. Then the corresponding sewing function takes values in the corresponding subgroup G ν ( x, t, λ) ∈ G ν and are bounded functions for λ ∈ l ν , see remark 1.
The rest of the details are the same as in the previous Subsection. Quite similarly is formulated the generalization of the Zakharov-Shabat theorem. The family of commuting operators formally coincides with the ones in eqs. (9); the difference is that now the Cartan subalgebra elements are complex. As a result however, the operator L in (9) takes the form of a CBC system [2, 1, 13].

Jets of order k
Another natural generalization consists in formulating the RHP on the complex plane of λ.
with the canonical normalization (4). The Zaharov-Shabat method can easily be generalized also for the RHP (16). The result is formulated as Theorem 2. Let ξ ± (x, t, λ) be solutions to the RHP (16) whose sewing function depends on the auxiliary variables x and t as follows: where K and J s belong to the Cartan subalgebra h ∈ g. Then ξ ± (x, t, λ) are fundamental analytic solutions of the set of differential operators Here U s ( x, t, λ) and V ( x, t, λ) are the jets of order k of J s (x, λ) and K(x, λ), i.e.: where the subscript + means that we retain only the nonnegative powers of λ.
Proof. Introduce the functions: and using (16) and (17) prove that Thus we find that these functions are analytic functions of λ in the whole complex λ k -plane. Next we find that: and make use of Liouville theorem to get Let us now express U s (x) ∈ g in terms of the asymptotic coefficients Q s in eq. (6).
Thus for the first three coefficients of U s ( x, t, λ) we get: and similar expressions for V ( x, t, λ) with J s replaced by K.
Lemma 2. The set of operators L s and M (18) have a common FAS, i.e. they all must commute, that is, the set of functions Q 1 ( x, t), . . . , Q k ( x, t) satisfy the following NLEE: Proof. Follows naturally from the definitions of the operators L s and M (18).
Remark 3. Obviously one can use families of operators with different maximal powers of λ.
Using them one can derive not only the relevant N -wave type equations, but also the higher order NLEE of the hierarchy.

Remark 4.
Considering RHP of the form (16) one must take special care of the behavior of the solutions when λ → 0. We are not going to discuss here the conditions that are necessary to impose so that the RHP will be properly defined leving it for the future.

Examples of new types of N -wave interactions
The integrability of the well known N -wave equations in two-dimensional space-time was discovered by Zakharov and Manakov [19]. To this end they used the Lax pair 1 , a 2 , . . . , a n ), where the potential Q(x, t∆) is a n × n matrix with Q kk = 0. The compatibility of this pair is which is a system of n(n − 1) equations for the off-diagonal elements of Q(x, t). This system admits the natural reduction C 0 QC 0 = q † where C 0 = diag (1, 1 , . . . n−1 ) and k = ±1. After the N -wave system (28) reduces to n(n − 1)/2 equations for Q km (x, t), k < m.
The N -wave are easily generalized to any other simple Lie algebra, see [5]. Indeed, let us consider the simple Lie algebra g of rank r with root system ∆ and Cartan-Weyl basis H s , E α , α ∈ ∆; assume also that the Cartan generators H s satisfy H s , H k = δ sk . Then consider the Lax pair where ∆ + ⊂ ∆ is the subset of positive roots. The N -wave equations have as Hamiltonian where X, Y is the Killing form between the elements X, Y ∈ g. So typically one may say that a given set of NLEE are of N -wave type if: i) they are contain first order derivatives with respect to x and t; ii) the coefficients a k and b k are real which physically means that the group velocity of each of these waves is real; and iii) the nonlinearities in the equations are quadratic in the fields q α . These types of N -wave equations were known for a long time. A number of their inequivalent Z 2 -reductions for the low-rank Lie algebras were described in [11]. Obviously, using the ISM one can show, that in terms of the scattering data of L these equations become linear evolution equations. Note that the scattering data of L are determined through the scattering matrix T (λ, t) for λ ∈ R -the continuous spectrum of L, and some additional data characterizing the discrete spectrum of L. The inverse scattering problem for the operator L is best of all reduced to a RHP (3) on the real line.
Our aim in this Section is to demonstrate new examples of qualitatively different N -wave equations which are also integrable.
3.1. CBC systems and 4-wave equations with complex group velocities, k = 1. The 4-wave equations below can be solved by the ISM applied to two operators of CBC type related to the g so(5) algebra. So we consider the Lax pair (29) with Here a 1 is a complex number that can be assumed to be a 1 = e iϕ 0 and ∆ 1,+ ≡ {e 1 ± e 2 , e 1 , e 2 } is the set of positive roots of the algebra so (5).
The important difference between (31) and the Lax pair (27) for g so (5) is that now J and K have complex eigenvalues. As a consequence of this the continuous spectrum of L 1 and M 1 fills up the real axis R and two lines Re ±iϕ 0 closing angles ±φ 0 with R. Again the ISP for the operator L 1 reduces to a RHP, whose solution consists of 6 fundamental analytic solutions χ ν (x, t, λ), ν = 1, . . . , 6 -one for each of the 6 sectors Ω ν into which the lines R and Re ±iϕ 0 split the complex λ-plane.
We need in addition a Z 2 -reduction which must relate q j and p j ; several types of such reductions have been described in [11]. However the reduction we will use below where C 1 = E 12 + E 21 + E 33 − E 45 − E 54 corresponds to the Weyl reflection with respect to the root e 1 − e 2 . This involution leads to a set of algebraic constraints on q j (x, t) and p j (x, t). For convenience we introduce where w 0 (x, t) and The Hamiltonian is given by: Let us briefly compare the new 4-wave system (34) with the well known 4-wave system (see Chapter 3 of [20]) which are obtained with the generalized Zakharov-Shabat system with real valued J = a 1 H 1 + a 2 H 2 and K = b 1 H 1 + b 2 H 2 and with the standard reduction p k = q * k : The obvious differences are: i) the new 4-wave equation is a system of equations for 3 complex fields w 2 , w 3 , w 4 and 2 real fields q 1 /a 1 , a 1 p 1 (instead of 4 complex fields q α ); ii) the group velocities are now complex (instead of real); iii) the interaction 'strength' between the different waves is different (instead of being equal to κ).
Of course there will be differences between the soliton solutions, but these will be given elsewhere. (3) and

New types of
Fix up k = 2. Then the Lax pair becomes where Impose a Z 2 -reduction of type a) with A = diag (1, , 1), 2 = 1. Thus Q 1 and Q 2 get reduced into: New type of integrable 3-wave equations: where the interaction constant κ and u 3 are given by: The diagonal terms in the Lax representation are λ-independent. Two of them read: These relations are satisfied identically as a consequence of the NLEE.
3.3. New types of 3-wave interactions, k = 3. Our last example of 3-wave interactions is similar to the one, reported in [12] and involves Lax pair, which is cubic in λ: With this choice U and V automatically satisfy the reduction condition [17] with ω = exp(2πi/3). We can also impose the involution with = ±1, 1 = ±1 which gives: and analogous relations for v ij . The compatibility condition now gives: Introduce also the coefficients Q 1,2 which automatically satisfy the above reductions: Thus we get the following parametrization for U k and V k or in components: The expressions for v jk differ from (52) only by replacing a k by b k . The corresponding NLEE take the form: Inserting here the notations from eq. (52) we obtain the four NLEE for the four independent functions w 1 , w 2 , R 1 and R 2 .

Conclusions and open questions
We proposed a method for constructing new integrable NLEE based on the use of the RHP with canonical normalization combined with the Mikhailov's reduction group [16,17]. Obviously we can derive many new classes of such equations making appropriate choices of the: i) the order k of the jets, ii) the simple Lie algebra g and iii) the reduction group and its realization as a subgroup of the group of automorphisms of g.
Since the method is based on the RHP one can apply Zakharov-Shabat dressing method for constructing their explicit (N -soliton) solutions.
The new NLEE are expected to be Hamiltonian. So one must show that the jets U ( x, t, λ) can be viewed as elements of more complicated co-adjoint orbits of the relevant Kac-Moody algebra, generated by the chosen grading of f. The corresponding Poisson brackets can be derived using the results of Kulish and Reyman [15] and imposing the reduction conditions as constraints.
The last but not least important problem concerns the possible physical applications of these equations.