- 1School of Mathematics and Statistics, Anyang Normal University, Anyang, China
- 2School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, China
In this work, we present two finite-dimensional Lie–Poisson Hamiltonian systems associated with the Hirota–Satsuma modified Boussinesq equation by using the nonlinearization method. Moreover, the separation of variables on the common level set of Casimir functions is introduced to study these systems which are associated with a non-hyperelliptic algebraic curve. Finally, in light of the Hamilton–Jacobi theory, the action-angle variables for these systems are constructed, and the Jacobi inversion problem associated with the Hirota–Satsuma modified Boussinesq equation is obtained.
1 Introduction
The Boussinesq-type equations are typical nonlinear integrable equations in mathematical physics and mechanics. We consider the Hirota–Satsuma modified Boussinesq equation
introduced in Hirota and Satsuma [1], which is derived from
by canceling the variable v. Here,
which describes the motion of long waves which are propagated in both directions in shallow water under gravity. Similarity solutions to Eq. 1 are discussed in Quispel et al. [2]; Clarkson [3]. It is shown that this equation has a Lax pair associated with the 3 × 3 matrix spectral problem, from which the Darboux transformation is derived with the help of gage transformation Geng [4]. The corresponding finite-dimensional completely integrable systems in the Liouville sense were derived. As an application, solutions to Eq. 1 are decomposed into solving two compatible Hamiltonian systems of ordinary differential equations Dai and Geng [5]. The explicit Riemann theta function representations of solutions for the Hirota–Satsuma modified Boussinesq hierarchy were studied in He et al. [6].
The separation of variables for finite-dimensional integrable systems is important for constructing action-angle variables. A series of literature studies shows research on finite-dimensional integrable systems associated with hyperelliptic spectral curves (see, e.g., Kuznetsov [7]; Babelon and Talon [8]; Kalnins et al; [9]; Eilbeck et al; [10]; Harnad and Winternitz [11]; Ragnisco [12]; Kulish et al; [13]; Qiao [14]; Zeng [15]; Zhou [16]; Zeng and Lin [17]; Cao et al; [18]; Derkachev [19]; Du and Geng [20]; Du and Yang [21]). However, the study on integrable systems associated with non-hyperelliptic spectral curves is much more complicated (see, e.g., Sklyanin [22]; Adams et al; [23]; Buchstaber et al; [24]; Dickey [25]; Derkachov and Valinevich [26]).
Sklyanin introduced a powerful method of constructing the separated variables for the classical integrable SL (3) magnetic chain, which is associated with a non-hyperelliptic algebraic curve Sklyanin [22]. By this effective way, more general cases are studied Scott [27]; Gekhtman [28]; Dubrovin and Skrypnyk [29]. We follow this method to construct the separable variables for the Lie–Poisson Hamiltonian associated with the Hirota–Satsuma modified Boussinesq Eq. 1 on the common level set of Casimir functions and define action-angle variables with the help of the Hamilton–Jacobi equation. Furthermore, the Jacobi inversion problem for the Hirota–Satsuma modified Boussinesq equation is obtained with action-angle variables.
This paper is organized as follows. In the following section, we will review the Lie–Poisson structure associated with
2 Preliminary
In this section, we introduce some basic notations of Lie–Poisson structures associated with Lie algebra
The Lie algebra
where
are the basis of Lie algebra
Thus, for any functions
with the gradient
The Hamiltonian vector field associated with (3) by a smooth function
The Lie–Poisson structure equations in terms of variables {yij, 1 ≤ i, j ≤ 3} are
The two Casimir functions of the Lie–Poisson structure Eq. 3 are
If we take the direct product of N copies of
and the Hamiltonian vector field associated with a smooth function F is
and the 2N Casimir functions
3 The Lie–Poisson Hamiltonian systems for the Hirota–Satsuma modified Boussinesq equation
According to the Lie–Poisson bracket Eq. 5 on N copies of
and
with Hamiltonians
and
with λ1, … , λN being N distinct parameters and
In fact, the Lie–Poisson Hamiltonian systems Eqs 6, 7 are derived from the 3 × 3 matrix spectral problem
and the auxiliary spectral problem
where u, v are the potentials and λ is a constant spectral parameter. The adjoint representations of the spectral problems Eqs 10, 11 are given by
and
respectively. In order to obtain the Lie–Poisson Hamiltonian systems associated with the Hirota–Satsuma modified Boussinesq Eq. 1, we take N copies of (12)
and N copies of (13)
Now, under the constraint
Eqs 14, 15 are nonlinearized into the Lie–Poisson Hamiltonian systems Eqs 6, 7, respectively.
The Lax representation and the involution property of conserved integrals are also given by using the generating function method.
Since the Lie–Poisson structure Eq. 5 has 2N Casimir functions
thus to prove the integrability of the Lie–Poisson Hamiltonian systems Eqs 6, 7, it is necessary to find 3N functionally independent Poisson commuting integrals. By using the constraint Eq. 16, after a direct calculation, we can get the following proposition.
Proposition 1. The Lie–Poisson Hamiltonian systems Eqs 6, 7 admit the Lax representations
and
respectively, where
and
with
It follows that the integrals of motion for the Lie–Poisson Hamiltonian systems Eqs 6, 7 are provided by the spectral invariants of Lax matrix Vλ. Therefore, one has the generating function of integrals for systems Eqs 6, 7:
Furthermore, substituting Eqs 17, 18, we have
where
and
where
From the expressions of
and
respectively.
Denoting the variables of
where
Taking the sum of Eq. 23 with respect to j from 1 to N, we have
from which we arrive at
For Casimir functions tr (yj), 1 ≤ j ≤ N, it is evident that
Proposition 2. The Lax matrix Vτ satisfies the Lax equations along the
Proof. By using (23), (24), and (25), we have
Based on Proposition 2, for any λ, τ, it is easy to verify that for l, k = 2, 3,
from which we have {qhj, qim} = 0, h, i = 1, 2, 3, j, m = 1, … , N.
Corollary 1.
By observing Eqs 21, 22, we know that {H, H1} = 0. Thus, some solutions of the Hirota–Satsuma modified Boussinesq Eq. 1 can be obtained by solving two compatible Hamiltonian systems of ordinary differential equations.
Proposition 3. Let yj be a compatible solution of the Lie–Poisson Hamiltonian systems Eqs 6, 7, then
solves the Hirota–Satsuma modified Boussinesq Eq. (1).
4 Separation of variables
In this section, we construct the separable variables on the common level set of the Casimir functions
to deal with the Lie–Poisson Hamiltonian systems. The characteristic polynomial of Lax matrix Vλ for the Hirota–Satsuma modified Boussinesq Eq. 1 is an independent constant with variables x and t in the expansion
which defines a non-hyperelliptic algebraic curve of genus
where
With the application of Sklyanin’s method given in Sklyanin [22], a half of the variables of separation μi (i = 1, … , 3N − 2) should be defined as zeros of some polynomial B(λ) with degree 3N − 2, and the corresponding conjugate variables νi (i = 1, … , 3N − 2) are related to μi by the secular equation
It follows from (28) that νi should be an eigenvalue of the matrix
for each i that the matrix
and νi is the eigenvalue of
Therefore, the problem is reduced to a determination of the matrix Ki and polynomial B(λ). Let us consider K(k) to be as follows:
Note that the matrix
depends on two parameters λ and k. Hence, we can consider condition Eq. 30 as the set of two algebraic equations
for two variables λ and k. By eliminating k from (32) yields the polynomial equation for λ:
Based on (33), we can define the polynomial B(λ) of degree 3N as
where
Expressing k from
thereby giving rise to 3N pairs of variables μi, νi. Let
with the help of (4) and (17), it is easy to see that
from which, together with the definitions of B by (34) and A by (37), the Lie–Poisson brackets for B(λ) and A(τ) satisfy
Proposition 4. {μi, νi, 1 ≤ i ≤ 3N} are canonical coordinates, that is,
Proof. The commutativity of Bs Eq. 38 obviously entrains the commutativity of μj (zeros of B(λ)). The Poisson brackets including νj can be calculated by using the implicit definition of μj. From B (μj) = 0, for j = 1, … , 3N, it follows that
or
for any function F, in the same way, we have
Now, we turn to prove {νi, μj} = δij. Starting with
using (39) and the third equation of (38), we arrive at
The last expression vanishes for μi ≠ μj due to B (μi) = B (μj) = 0 and is evaluated via L’Hôpital’s rule for μi = μj to produce the proclaimed result. The commutativity of νs can be shown in the same way, starting from the first equation of (38).
5 Action-angle variables and Jacobi inversion problems
Let us start with
where
from which we can rewrite the generating functions
with
The comparison of the coefficients of λl (l = 0, … , N − 1) in equation
and the comparison of the coefficients of λl (l = 0, 1, … , 2N − 1) in equation
respectively, yield
Let
with the help of Eq. 29, we have the completely separable Hamilton–Jacobi equations:
for i = 1, … , 3N − 2, from which we can obtain an implicit complete integral of Hamilton–Jacobi equations for the generating functions
where z satisfies Eq. 28.
Now, let us consider a canonical transformation from (μ, ν) to (ϕ, I) generated by the generating function S:
which satisfies
where R(λ) = 3a2(λ)z2 − R2(λ). Thus, by using (40), (41), and (42), the generating functions of integrals can be rewritten as
The variables I1, … , I3N−2 will be variables of action type, and the conjugate variables ϕ1, … , ϕ3N−2 will be the corresponding angles.
The Hamiltonian canonical equations for the generating functions
Proposition 5. Let t2,l and t3,l be the variables of
where
with Aks being the coefficients in the expansion
which could be represented through the power sums of λl,
with the recursive formula
and Brs are the comparison of the coefficients of λr, r = 0, 1, … in
which can be written as
Proof. According to the definition of the Lie–Poisson bracket,
for j = 1, … , 3N − 2. By using Eqs (46), (47), and 49, it is easy to see that
By comparing the coefficients of λ−l−1 in (50), we get the Lie–Poisson brackets
thereby providing the nondegeneracy matrix Eq. 48.
Proposition 6.
Proof. We only need to prove the linear independence of the gradients:
Suppose
we have
Hence, c1 = c2 = ⋯ = c3N−2 = 0 since the coefficient determinant is equal to 1 by matrix Eq. 48.Remark. Corollary 1 and the present Proposition completely prove the Liouville integrability of the Lie–Poisson Hamiltonian systems Eqs 6, 7 with the Hamiltonians Eqs 21, 22, and 3N − 2 integrals
After fixing the values of the 2N Casimir functions in (27), based on (51), using (21), the solution of system Eq. 6 in terms of action-angle variables ϕj, Ij is
Thus, combining Eq. 45 with (52) yields the Jacobi inversion problem for the Lie–Poisson Hamiltonian system Eq. 6
For the Lie–Poisson Hamiltonian system Eq. 7 with respect to Lie–Poisson bracket Eq. 51, using (22), we obtain the solution of system Eq. 7 in terms of action-angle variables ϕj, Ij
According to Eqs 45, 53, we have the Jacobi inversion problem for the Lie–Poisson Hamiltonian system Eq. 7
The compatible solution of systems Eqs 6, 7 in terms of action-angle variables Ij, ϕj is
From (45) and (54), we finally obtain the Jacobi inversion problem for the Hirota–Satsuma modified Boussinesq Eq. 1:
6 Conclusion
In this paper, two finite-dimensional Lie–Poisson Hamiltonian systems associated with a 3 × 3 spectral problem related to the Hirota–Satsuma modified Boussinesq equation are presented. Separation of variables for the integrable systems with non-hyperelliptic spectral curves is constructed by using the method proposed by Sklyanin. Then, 3N-2 pairs of action-angle variables are introduced with the help of Hamilton–Jacobi theory. The Jacobi inversion problems for these Lie–Poisson Hamiltonian systems and the Hirota–Satsuma modified Boussinesq equation are discussed. Furthermore, based on the Jacobi inversion problems, we may use the algebro-geometric method to obtain the multi-variable sigma-function solutions, which will be left to future research. The methods in this paper can be applied to other systems of soliton hierarchies with 3 × 3 matrix spectral problems, even 4 × 4 matrix spectral problems.
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.
Author contributions
XuG: writing–original draft and writing–review and editing. DD: writing–review and editing. XiG: writing–review and editing.
Funding
The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This work was supported by the National Natural Science Foundation of China (Nos. 12001013 and 11271337) and the Key Scientific Research Projects of the Universities in Henan Province (Project No. 22A110005).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Keywords: Hirota–Satsuma modified Boussinesq equation, non-hyperelliptic algebraic curve, separation of variables, action-angle variables, Jacobi inversion problem
Citation: Geng X, Du D and Geng X (2023) Action-angle variables for the Lie–Poisson Hamiltonian systems associated with the Hirota–Satsuma modified Boussinesq equation. Front. Phys. 11:1285301. doi: 10.3389/fphy.2023.1285301
Received: 29 August 2023; Accepted: 01 December 2023;
Published: 22 December 2023.
Edited by:
Stefani Mancas, Embry–Riddle Aeronautical University, United StatesReviewed by:
Haret Rosu, Instituto Potosino de Investigación Científica y Tecnológica (IPICYT), MexicoYunqing Yang, Zhejiang Ocean University, China
Copyright © 2023 Geng, Du and Geng. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Xue Geng, Z2VuZ3h1ZTE5ODVAMTYzLmNvbQ==