Action-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system

1.   Introduction

It is well known that Gaudin models describe completely integrable classical and quantum long-range interacting spin chains. Originally it was introduced by M. Gaudin for the simple Lie algebra $\mathfrak{su}(2)$ ^[1] and later generalized to arbitrary semi-simple Lie algebras ^[2,3]. This model attracted considerable interest among theoretical and mathematical physicists, playing a distinguished role in the realm of integrable systems. For example, classical Hamiltonian systems associated with Lax matrices of the Gaudin-type in the context of a general group-theoretic approach ^[4], multi-Hamiltonian formulations (see e.g., ^[5]) and their integrable discretizations through B${\rm \ddot{a}}$cklund transformations (see e.g., ^[6]), separation of variables of the rational Gaudin models (see e.g., ^[7,8,9,10]).

The separation of variables for finite-dimensional integrable systems are important for constructing action-angle variables. Much work on the separation of variables for finite dimensional integrable systems associated with hyperelliptic spectral curves was done(see e.g., ^{[9,10,11,12,13,14,15,16]}). It is worth noting that Sklyanin gave an efficient way to construct separated variables for the classical integrable $SL(3)$ magnetic chain and $\mathfrak{sl}(3)$ Gaudin model related to the non-hyperelliptic spectral curves ^[8]. After this classical work, the general cases ^[17,18,19] and a sufficient condition to guarantee the "separation polynomial" of Sklyanin under the corresponding $r$-matrix were provided in ^[20]. Action-angle variables of the finite-dimensional integrable systems can be constructed by using separated variables from the Liouville theorem and integration of Hamiltonian equations of motion. They were also derived with the help of the scattering data for the first-order matrix differential operator ^[21], or by finding real Darboux coordinates for scattering data ^[22,23], or by using the algebraic-geometric techniques ^[23,24,25]. Action-angle variables of the finite-dimensional integrable systems related to non-hyperelliptic spectral curves can also be obtained in dealing with the stationary equations of soliton hierarchies ^[24,25].

Based on the previous research on Gaudin models ^[8,10], the motivation of this paper is to use the $\mathfrak{gl}_3(\mathbb{C})$ rational Gaudin model governed by the Lax matrix to study the constraint systems associated with the generalized TWRI system: ^[26,27]

where $l, k, m$ are not equal to each other, the real constants $c_{12}, \; c_{21}, \; c_{13}, \; c_{31}, \; c_{23}, \; c_{32}$ are six real constants and $u_{12}, \; u_{21}, \; u_{13}, \; u_{31}, \; u_{23}, \; u_{32}$ are complex functions of two real independent variables $x$ and $t$. Also, as a basic integrable model in mathematical physics, which has important applications in nonlinear optics, plasma physics, acoustics, fluid dynamics, solid-state physics and other fields, many studies have been done on the TWRI system. For example, Zakharov and Manakov ^[28,29] and Kaup ^[30] considered the inverse scattering transformation of the TWRI equations. It is shown that the TWRI equations can be reduced to the generic sixth Painlevé equation ^[31]. Some explicit solutions of the TWRI system are obtained in ^{[32,33,34,35,36,37,38,39]}, including soliton solutions, rational solutions, rogue wave solutions and so on. The finite dimensional Hamiltonian system associated with the TWRI system is proved completely integrable in the Liouville sense ^[26]. Based on the theory of trigonal curves, explicit algebro-geometric solutions of the TWRI system have been achieved in ^[40] by using the asymptotic properties of the Baker-Akhiezer functions and the meromorphic functions ^[41,42].

The outline is as follows. In Section 2, we review some basics about Lie-Poisson structure and coadjoint representaive theory of Lie-algebra $\mathfrak{gl}_3(\mathbb{C})$. In Section 3, Lax matrix $V_\lambda$ is introduced to study the TWRI system (1.1) in the Lie-Poisson structure, the integrability of this restricted system and the relation between the finite dimensional Hamiltonian system and the TWRI system (1.1) are also discussed. In Section 4, the $9N$ dimensional Poisson manifold $(\mathfrak{gl}_3(\mathbb{C})^*)^N$ is reduced to $6N$ dimensional symplectic manifold by making a restriction on $3N$ common level set of Casimir functions, from which $3N$ pairs of separated variables are constructed on the $6N$ dimensional symplectic manifold. In Section 5, based on the Hamilton-Jacobi theory, the generating function to construct the canonical transformation is obtained from separated variables to action-angle variables in implicit form. Further, the functional independence of conserved integrals is proved in terms of the evolution of angle type variables. In addition, the Jacobi inversion problems for the Lie-Poisson Hamiltonian systems related to the TWRI system (1.1) are established.

2.   Preliminary

We start from a general definition of the Lie-Poisson structure ^[43]. Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Let also $\mathfrak{g}^*$ be the dual of $\mathfrak{g}$ with the natural pairing $\langle A, B\rangle$ between $\mathfrak{g}$ and $\mathfrak{g}^*$. Consider the algebra of smooth function $F\in C^\infty(\mathfrak{g}^*)$, then its gradient $\nabla F\in\mathfrak{g}$ is defined as

For any smooth functions $F$ and $G\in C^\infty (\mathfrak{g}^*)$, Lie-Poisson brakect is defined as

When $\mathfrak{g}$ is the matrix Lie-algebra $\mathfrak{gl}_3(\mathbb{C})$, we take the trace form $\langle A, B\rangle = {\rm tr}(AB)$ as the pairing between $\mathfrak{gl}_3(\mathbb{C})$ and $\mathfrak{gl}_3(\mathbb{C})^*$ and fix the basis of $\mathfrak{gl}_3(\mathbb{C})$ as

where $\delta_{ij}$ are the Kronecker delta functions, which satisfy the commutative relations

Since the trace form $\langle A, B\rangle = {\rm tr}(AB)$ is a non-degenerate pairing, we can make an identification $\mathfrak{gl}_3(\mathbb{C})\cong \mathfrak{gl}_3(\mathbb{C})^{*}$. In the sense of $\langle E_{mn}, e_{kl}\rangle = \delta_{mk}\delta_{nl}$, one can find that the dual basis of $\{E_{kl}, \; 1\leq k, l \leq 3\}$ is $\{e_{kl} = E_{lk}, \; 1\leq m, n \leq 3\}$. Hereafter, we choose $y = \sum\limits_{l, k = 1}^{3}y^{kl}e_{kl}\in \mathfrak{gl}_3(\mathbb{C})^{*}$ and $y^{kl}$ are coordinates on $\mathfrak{gl}_3(\mathbb{C})^{*}$. In these coordinates, the gradient $\nabla F\in \mathfrak{gl}_3(\mathbb{C})$ is determined by

The Lie-Poisson bracket (2.1) become

Specially, the brackets between the coordinates are

Using the cyclicity of the trace, the Hamiltonian vector field associated with (2.5) for a smooth function $F$ is represented as

For any $g\in GL_3(\mathbb{C})$ and $X\in \mathfrak{gl}_3(\mathbb{C})$, the adjoint action of $g\in GL_3(\mathbb{C})$ on its Lie algebra $\mathfrak{gl}_3(\mathbb{C})$ is

Then for any $\xi\in \mathfrak{gl}_3(\mathbb{C})^*$, the coadjoint action ${\rm Ad}_g^*$ of $GL_3(\mathbb{C})$ in the dual space $\mathfrak{gl}_3(\mathbb{C})^*$ is defined as

It follows that ${\rm Ad}_g^*y = gyg^{-1} = {\rm Ad}_gy$. The coadjoint orbit passing through any $y\in \mathfrak{gl}_3(\mathbb{C})^*$ is

According to the representative theory of Lie group, we know that the coadjoint orbits of $GL_3(\mathbb{C})$ are the symplectic leaves of the Lie-Poisson structure on $\mathfrak{gl}_3(\mathbb{C})^*$.

A smooth function $h$ on $\mathfrak{gl}_3(\mathbb{C})^*$ is a Casimir if an only if it is invariant under the coadjoint action

Choose $\tilde{h}_k(y) = {\rm tr}(y^k)$, where $y\in\mathfrak{gl}_3(\mathbb{C})^*, \; k = 1, 2, 3, \cdots,$ we have

Thus, we choose three Casimir functions as ${\rm tr}(y), \; \frac{1}{2}{\rm tr}(y^2)$ and $\frac{1}{3}{\rm tr}(y^3).$

To study the Lie-Poisson Hamiltonian systems, we will use the direct product of $N$ copies of $\mathfrak{gl}_3(\mathbb{C})^*$ which is denoted by $(\mathfrak{gl}_3(\mathbb{C})^*)^N$. The standard Lie-Poisson structure on $(\mathfrak{gl}_3(\mathbb{C})^*)^N$ is given by

where $y_j\in\mathfrak{gl}_3(\mathbb{C})^*, \; j = 1, 2, \cdots, N$.

The Hamiltonian vector field associated with Lie-Poisson bracket (2.4) for a smooth function $F$ is

and the $3N$ Casimir functions are ${\rm tr}(y_j), \; \frac{1}{2}{\rm tr}\big(y_j^2\big), \; \frac{1}{3}{\rm tr}\big(y_j^3\big), j = 1, \ldots, N.$

3.   The Lie-Poisson Hamiltonian systems

In this section, we shall use the $\mathfrak{gl}_3(\mathbb{C})$ rational Gaudin model governed by the Lax matrix

to study the TWRI system ($\beta_l$ are different constants).

To obtain the Lie-Poisson Hamiltonian systems from the Lax matrix (3.1), we choose

as the Hamiltonians. Let $t_{k\lambda}$ be the variables of $\mathcal{F}_k(\lambda)$, according to Hamiltonian vector field (2.5), the Lie-Poisson Hamiltonian systems for $\mathcal{F}_k(\lambda), k = 1, 2, 3$ are

Proposition 3.1. The Lax matrix $V_\tau$ satisfies the Lax equations along the $\mathcal{F}_k(\lambda)$-flows:

with $\lambda$, $\tau$ are two different constant spectral parameters.

Proof. By making use of (3.3), one infers

Based on Proposition 3.1, for any $\lambda, \, \tau$, a direct calculation shows that

Therefore, $\mathcal{F}_k(\lambda), k = 1, 2, 3$ can be regarded as the generating function of integrals of the Hamiltonian systems generated from it. Using (3.1), we arrive at

where

Similarly, we have

where

which implies $\{E_{lj}, E_{km}\} = 0, \ l, k = 1, 2, 3, \; j, m = 1, \ldots, N$ and the following fact.

Corollary 3.1. Functions $F_{1, l}, \; F_{2, l}, \; F_{3, l}\; (l\geq 1)$ are in involution in pairs with respect to the Lie-Poisson bracket (2.4).

Now, we consider two Lie-Poisson Hamiltonian systems generated by the Hamiltonians

where

and

respectively, where $\alpha_1, \; \alpha_2, \; \alpha_3$ are different constants and $c_{lk} = \frac{\beta_l-\beta_k}{\alpha_l-\alpha_k}, \; 1\leq l, k\leq 3, \; l\neq k$.

The equations of motion for $H$ and $H_1$ are

and

which are exactly the Lie-Poisson Hamiltonian systems associated with TWRI system (1.1).

In fact, the Lie-Poisson Hamiltonian systems (3.8) and (3.9) are generated respectively by the $N$ copies of adjoint representations of the spectral problems

and

under the Bargmann constraint

Proposition 3.2. The Lie-Poisson Hamiltonian systems (3.8) and (3.9) admit the Lax representations

and

respectively, where

with

It follows that the integrals of motion of the Lie-Poisson Hamiltonian systems (3.8) and (3.9) are provided by $\mathcal{F}_k(\lambda), k = 1, 2, 3$.

Now we know that there are $3N$ Casimir functions of the Lie-Poisson structure (2.4): ${\rm tr}(y_j), \; \frac{1}{2}{\rm tr}\big(y_j^2\big), \; \frac{1}{3}{\rm tr}\big(y_j^3\big), j = 1, \ldots, N$ and $3N$ involutive first integrals $E_{1, j}, \; E_{2, j}, \; E_{3, j}, j = 1, \cdots, N$. Hence, they are integrable.

The first two typical members of TWRI vector fields $\{X_m\}$ are ^[26]

The zero-curvature equation $U_t = W_x- [U, W]$ leads to the TWRI system (1.1)

From Corollary 3.1, it is not difficult to verify the involutivity $\{H, H_1\} = 0$, which implies the commutativity of Hamiltonian vector fields. The importance of Hamiltonians $H, \; H_1$ are that the differential $f_*$ maps $[\nabla_jH, y_j]$, $[\nabla_jH_1, y_j]$ exactly into the TWRI vector fields $X_0, \; X_1$

respectively. Thus solutions of TWRI system (1.1) can be obtained by solving two compatible Lie-Poisson Hamiltonian systems with ordinary differential equations:

Proposition 3.3. Let $y_j$ be a compatible solution of the Lie-Poisson Hamiltonian systems (3.8) and (3.9). Then

solves TWRI system (1.1).

4.   Separation of variables

In this section, we will give the separated canonical coordinates on the common level set of the Casimir functions to deal with the Lie-Poisson Hamiltonian systems.

Remark 1. Let $(C_{1j}, \; C_{2j}, \; C_{3j})$ be regular values of the map defined by the Casimirs: $y_j\rightarrow ({\rm tr}(y_j), \frac{1}{2}{\rm tr}\big(y_j^2\big), \frac{1}{3}{\rm tr}\big(y_j^3\big)).$ Then restricted on the common level set of Casimir functions

the $9N$ dimensional Poisson manifold $(\mathfrak{gl}_3(\mathbb{C})^*)^N$ is naturally reduced to a $6N$ dimensional symplectic manifold, by which $3N$ pairs of canonical variables can be introduced.

In the following, we give the first $3N-2$ pairs of separated variables $\mu_i, \nu_i$ by Sklyanin's method ^[8].

In fact, the characteristic polynomial of Lax matrix $V_\lambda$ for the TWRI system (1.1) is a constant independent of variables $x$ and $t$ with the expansion

which defines a non-hyperelliptic algebraic curve by introducing variable $\zeta = a(\lambda)z$:

with

Follow the method in ^[8], the canonical separated variables $\mu_i\; (i = 1, \ldots, 3N-2)$ are defined by zeros of some polynomial $B(\lambda)$ and the corresponding conjugate coordinates $\nu_i\; (i = 1, \ldots, 3N-2)$ related to $\mu_i$ by the equation

It follows from (2.3) that $\nu_i$ are eigenvalues of the matrix $V_{\mu_i}$. Therefore, there must exists a similarity transformation $V_{\mu_i}\rightarrow \widetilde{V}_{\mu_i} = PV_{\mu_i} P^{-1}$ for each $i$ such that the matrix $\widetilde{V}_{\mu_i}$ is block-triangular

and $\nu_i$ is the eigenvalue of $V_{\mu_i}$ splitted from the upper block,

Thus, the problem is reduced to determining the matrix $P$ and polynomial $B(\lambda)$. Let $P$ be of the form:

Note that the matrix

depends on two parameters $\lambda$ and $p$. Hence, we can consider the condition (4.4) as the set of two algebraic equations

for two variable $\lambda$ and $p$. Eliminating $p$ from (4.6), one obtains the polynomial equation for $\lambda$:

Based on (3.1) and (4.7), the polynomial $B(\lambda)$ of degree $3N-2$ can be denote as

where

Expressing $p$ from $\widetilde{V}_{31}(\lambda) = 0$ as $p = -V_{31}(\lambda)/V_{32}(\lambda)$ and substituting it into Eq (4.5) of $\nu_i$, we arrive at

which produces $3N-2$ pairs of variables $\mu_i, \nu_i.$ It is easy to see from (2.3) and the above expressions that

with $\lambda, \; \tau$ are two different constant spectral parameters. We have from (4.11) that

which, together with $B(\lambda)$ defined by (4.7), $B(\lambda)$ and $A(\tau)$ satisfy

Proposition 4.1. The canonical separated variables $\mu_j$ and $\nu_j$ constructed from $B(\lambda)$ in (4.8) and $A(\mu_j)$ in (4.10) satisfy the relations

Proof. The commutativity of $B$ in (4.13) implies the commutativity of $\mu_j$ (zeros of $B(\lambda)$). The Poisson brackets of $\nu_j$ can be calculated by using the definition of $\mu_j.$ From $B(\mu_j) = 0$ for $j = 1, \ldots, 3N-2$, it follows that

that is

for any function $F$. In the same way we have

Now we turn to prove $\{\nu_i, \mu_j\} = \delta_{ij}.$ Noting that

we obtain by using (4.14) and the third equation in (4.13) that

which vanishes for $\mu_i\neq \mu_j$ due to $B(\mu_i) = B(\mu_j) = 0$ and is evaluated via L'Hôpital rule for $\mu_i = \mu_j$ to produce the proclaimed result. Similarly, the commutativity of $\nu_i\; (1\leq i\leq 3N-2)$ can be shown by the first equation of (4.13).

Apart from the $3N-2$ pairs of separated variables above, we should add 2 pairs of conjugate variables to prove the canonical structure on the common level set of Casimir functions (4.1).

The last $2$ pairs of conjugate variables can be defined as follows by direct calculation.

Proposition 4.2. Assume that the 2 pairs of additional canonical separated variables are defined by

Then on the common level set of Casimir functions (4.1), we have

It is shown that $\mu_j, \; \nu_j, \; j = 1, \cdots, N$ are $3N$ pairs of conjugate variables.

5.   Action-angle variables and Jacobi inversion problems

In this section, the action-angle variables will be introduced by resorting to Hamilton-Jacobi theory. As a by-product, the functional independence of conserved integrals for the Liouville integrability of Lie-Poisson Hamiltonian systems (3.8) and (3.9) will be proved. Further, the Jacobi inversion problems for systems (3.8), (3.9) and TWRI Eq (1.1) will be built by using the canonical transformation from the separated variables to the action-angle variables

Let

where

from which we can rewrite the generating functions $\mathcal{F}_1(\lambda), \; \mathcal{F}_2(\lambda), \; \mathcal{F}_3(\lambda)$ as

with

The comparison of the coefficients of $\lambda^l, \; l = 0, \ldots, N-1$ in equation

and the comparison of the coefficients of $\lambda^l, \; l = 0, 1, \ldots, 2N-1$ in equation

respectively, yield

Let

We obtain from (4.3) that the completely separated Hamilton-Jacobi equations:

with $i = 1, \ldots, 3N-2$, from which we get an implicit complete integral of Hamilton-Jacobi equations for the generating functions $\mathcal{F}_2(\lambda)$ and $\mathcal{F}_3(\lambda)$:

where $z$ satisfies (4.2).

Now let us consider a canonical transformation from the variables $\mu_i, \nu_i, (i = 1, \dots, 3N-2)$ to variables $\phi_i$ and $I_i, (i = 1, \ldots, 3N-2)$, generated by the generating function $S$:

that satisfies

Resorting to Eqs (5.3), (5.4), (4.2) and (5.2), we arrive at

where $R(\lambda) = 3a^2(\lambda)z^2-2a(\lambda)R_1(\lambda)z+\frac{1}{2}R_1^2(\lambda)-R_2(\lambda)$. From Eqs (5.1) and (5.2), the generating functions of integrals can be rewritten as

Functions $I_1, \cdots, I_{3N-2}$ and $\phi_1, \ldots, \phi_{3N-2}$ are variables of action type and the corresponding variables of angles, respectively. In the following, we will use these action-angle variables to discuss the equations of motion for the Lie-Poisson Hamiltonian systems generated by the Lax matrix (3.1). The Hamiltonian canonical equations for the generating functions $\mathcal{F}_2(\lambda)$ and $\mathcal{F}_3(\lambda)$ in terms of action-angle variables $I_j$ and $\phi_j$, $\; j = 1, \ldots, 3N-2$, are as follows

Let $t_{2, l}$ and $t_{3, l}$ represent the variables of $F_{2, l}$-flow and $F_{3, l}$-flow, respectively. According to the definition of the Lie-Poisson bracket, one infers

Proposition 5.1.

with

where $A_k$'s are the coefficients in the expansion

which could be represented through the power sums of $\lambda_j$, that is

with the recursive formula

and $B_r$'s are obtained by comparing the coefficients of $\lambda^r, \; r = 0, 1, \ldots,$ in equation

where $B_0 = A_0^2 = 1, \; B_1 = 2A_1, \cdots, B_r = \sum\limits_{m, n\geq 0 \atop m+n = r}A_mA_n$ with the supplementary definition $A_{-k}$ = $B_{-k} = 0$, $k = 1, 2, \ldots$.

Proof. Using (5.6), (5.8) and (5.10), it is easy to see that

By comparing the coefficients of $\lambda^{-l-1}$ in (5.12), we deduce the Poisson brackets

Thus, the non-degenerate matrix takes on the form (5.11).

Proposition 5.2. $F_{2, 1}, \cdots, F_{2, N-1}, \; F_{3, 1}, \cdots, F_{3, 2N-1}$ given in (3.4) and (3.5) are functionally independent.

Proof. We need only prove the linear independence of the gradients: $\nabla F_{2, 1}, \cdots, \nabla F_{2, N-1},$ $\nabla F_{3, 1}, \cdots\nabla F_{3, 2N-1}.$ Suppose

It is easy to calculate that

which implies that $c_1 = c_2 = \cdots = c_{3N-2} = 0$ because the coefficient determinant is equal to 1 by (5.11).

Remark 2. From Corollary 3.1 and the above Proposition, it is proved that the Lie-Poisson Hmailtonian systems (3.8) and (3.9) with the Hamiltonians (3.6) and (3.7) are complete integrable in the Liouville sense because their $3N-2$ integrals $F_{2, 1}, \cdots, F_{2, N-1}$, $F_{3, 1}, \cdots, F_{3, 2N-1}$ are involutive in pairs and functionally independent.

For given values of the $3N$ Casimir functions in (4.1), $F_{0, l} = \sum_{j = 1}^N\lambda_j^lC_{1j}$ are constants, which means that $\{\phi_j, F_{0, l}\} = 0$. Based on (5.13) and (3.6), the solution of the Lie-Poisson Hmailtonian system (3.8) in terms of action-angle variables $\phi_j$ and $I_j$ is

Thus, combining (5.5) with (5.14) give rise to the Jacobi inversion problem

For the Lie-Poisson Hamiltonian system (3.9) with respect to Lie-Poisson bracket (5.13), we obtain by using (3.7) that

With the help of (5.5) and (5.15), we deduce the Jacobi inversion problem

The compatible solution of Lie-Poisson Hamiltonian systems (3.8) and (3.9) in terms of action-angle variables $\phi_j$ and $I_j$ is

Thus, by making use of (5.5) and (5.16), we arrive at the Jacobi inversion problem for the TWRI Eq (1.1)

6.   Conclusions

In this paper, two finite dimensional Lie-Poisson Hamiltonian systems associated with a $3\times3$ spectral problem related to three-wave resonant interaction system are presented with the help of nonlinearization method. In the framework of Lie-Poisson structure, it is easier to prove the integrability for these finite-dimensional Lie-Poisson Hamiltonian systems in Liouville sense. $3N$ pairs of separation of variables for these integrable systems with non-hyperelliptic spectral curves are constructed and $3N-2$ pairs of them are proposed by using Sklyanin's method. In addition, apart from the variables $\mu_k, \; \nu_k(k = 1, \cdots, 3N-2)$, we add two pairs of conjugate variables. $3N-2$ pairs of action-angle variables are introduced with the help of Hamilton-Jacobi theory. The Jacobi inversion problems for the these Lie-Poisson Hamiltonian systems and three-wave resonant interaction system are discussed. Furthermore, based upon the Jacobi inversion problems, we may use the algebro-geometric method to get the multi-variable sigma-function solutions, which will be left to a future publication. The methods in this paper can be applied to other systems of soliton hierarchies with $3\times3$ matrix spectral problems.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos.12001013, 11626140 and 11271337).

Conflict of interest

The authors declare that they have no competing interests.