Skip to main content

ORIGINAL RESEARCH article

Front. Phys., 22 December 2023
Sec. Mathematical Physics
This article is part of the Research Topic Quasi-Integrability, Nonlinear Evolutions and Physical Applications View all 3 articles

Action-angle variables for the Lie–Poisson Hamiltonian systems associated with the Hirota–Satsuma modified Boussinesq equation

Xue Geng
Xue Geng1*Dianlou DuDianlou Du2Xianguo GengXianguo Geng2
  • 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

utt+13uxxx23u2ux2uxx1utx=0,(1)

introduced in Hirota and Satsuma [1], which is derived from

ut=uxx+23uux+2vx,vt=23uxxx+uuxx+uxvuvx(2)

by canceling the variable v. Here, x1 stands for an inverse operator of = /∂x under conditions x1=x1=1. This equation was initially proposed by Hirota and Satsuma [1] from a Bäcklund transformation of the Boussinesq equation

wtt+13wxx4w2xx=0,

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 sl(3). In Section 3, in the framework of the Lie–Poisson structure on sl(3), two Lie–Poisson Hamiltonian systems associated with the Hirota–Satsuma modified Boussinesq Eq. 1 are presented by using the nonlinearization of the adjoint representations of the 3 × 3 spectral problem and auxiliary spectral one. Moreover, the involution property of conserved integrals is discussed by using the generating function method. In Section 4, on the common level set of Casimir functions, the separated variables are introduced to study these Lie–Poisson Hamiltonian systems. In Section 5, in light of the Hamilton–Jacobi theory, the generating function S for obtaining the canonical transformation from separated variables to action-angle variables is obtained. In Section 6, in terms of the evolution of action-angle variables, the functional independence of conserved integrals is elucidated. Finally, the Jacobi inversion problems for those Lie–Poisson Hamiltonian systems and the Hirota–Satsuma modified Boussinesq Eq. 1 are built.

2 Preliminary

In this section, we introduce some basic notations of Lie–Poisson structures associated with Lie algebra sl(3).

The Lie algebra sl(3) has an invariant nondegenerate symmetric form <A,B>=tr(AB) by means of which we can make an identification sl(3)sl(3)*. For convenience, we choose

sl3=αα=i,j=13αijeij,trα=0,sl3*=yy=i,j=13yijEij,try=0,

where

Eij=δmiδnj,1i,j3,

are the basis of Lie algebra sl(3)*, and the dual bases are given by {eij = Eji, 1 ≤ i, j ≤ 3}. We can confirm that these bases satisfy the commutation relation

Eij,Ekl=δjkEilδliEkj.

Thus, for any functions F(y),G(y)C(sl(3)*), the corresponding Lie–Poisson bracket at the point ysl(3)* is

F,Gy=y,F,G=tryF,G,(3)

with the gradient Fsl(3) defined as

F=k,l=13Fyklekl.

The Hamiltonian vector field associated with (3) by a smooth function F(y)C(sl(3)*) is represented as

XF=F,y.

The Lie–Poisson structure equations in terms of variables {yij, 1 ≤ i, j ≤ 3} are

ylk,ymn=<y,Ekl,Enm>=δlnymkδmkyln,1n,m,l,k3.(4)

The two Casimir functions of the Lie–Poisson structure Eq. 3 are

try2,try3.

If we take the direct product of N copies of sl(3)*, the Lie–Poisson structure becomes

F,Gyj=j=1Nyj,jF,jG,jF=k,l=13Fyjklekl,(5)

and the Hamiltonian vector field associated with a smooth function F is

XjF=jF,yj,j=1,,N,

and the 2N Casimir functions

tryj2,tryj3,j=1,,N.

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 sl(3)*, we discuss the finite-dimensional Lie–Poisson Hamiltonian systems associated with the Hirota–Satsuma modified Boussinesq Eq. 1:

yjx=jH,yj,j=1,,N,(6)

and

yjt=jH1,yj,j=1,,N,(7)

with Hamiltonians

H=r021+r013+r132+3r012r0222r0123,(8)

and

H1=r0112+r022r011+r0222+r023+r012r021+r131+r1122r013r012+r012r132,(9)

with λ1, … , λN being N distinct parameters and rmkl=j=1Nλjmyjkl.

In fact, the Lie–Poisson Hamiltonian systems Eqs 6, 7 are derived from the 3 × 3 matrix spectral problem

φx=Uφ,φ=φ1φ2φ3,U=010vuλ100,(10)

and the auxiliary spectral problem

φt=Vφ,V=23ux+v13uλλ23uxx+vx+13uv13ux+13u2+v13λu23u10,(11)

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

yx=U,y,(12)

and

yt=V,y,(13)

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)

yjx=Uλj,yj,j=1,,N,(14)

and N copies of (13)

yjt=Vλj,yj,j=1,,N.(15)

Now, under the constraint

u=3r012,v=3r0226r0122,(16)

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

tryj2,tryj3,j=1,,N,

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

ddxVλ=U,Vλ,

and

ddtVλ=V,Vλ,

respectively, where

U=0103r0226r01223r012λ100,V=2r011+r022r012λλ+r0212r013+r1322r022+r011λr0122r01210,

and

Vλ=Vijλ3×3=βλ+j=1Nyjλλj,(17)

with

βλ=00110r012λ1r031+2r012λ11r0320.

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:

F2λ=12trVλ2,F3λ=13trVλ3.(18)

Furthermore, substituting Eqs 17, 18, we have

F2λ=12trVλ2=12trβλ2+j=1Nq1jλλj+j=1Nh2jλλj2l=1FlSλl+1,(19)

where

q1j=trβλyj+kjNtryjykλjλk,h2j=12tryj2,FlS=j=1Nλjlq1j+lj=1Nλjl1h2j,l=1,,

and

F3λ=13trVλ3=13trβλ3+j=1Nq2jλλj+j=1Nq3jλλj2+j=1Nh3jλλj3l=0FlTλl+1,(20)

where

q2j=k=1N1λjλktrβλyjyk+βλykyj+ik,jNtryjykyi+yjyiykλkλi+ik,jNtryjykyi+yjyiykλjλi+trβλ2yj+kjNtryk2yjyj2ykλjλk2,q3j=trβλyj2+kjtryj2ykλjλk,h3j=13tryj3,FlT=j=1Nλjlq2j+lj=1Nλjl1q3j+12ll1j=1Nλjl2h3j,l=0,1,.

From the expressions of F2(λ) and F3(λ) in (19) and 20, we know that for j = 1, , N, q1j, q2j, q3j provide 3N generators of conserved integrals for systems Eqs 6, 7. The Hamiltonian functions Eqs 8, 9 can also be written as

H=F1T(21)

and

H1=F1S,(22)

respectively.

Denoting the variables of F2(λ)-flow and F3(λ)-flow by t2λ and t3λ, respectively, let Vλ2=(vij(λ))3×3; then, the Hamiltonian equations for F2(λ) and F3(λ) are

yjtkλ=jFkλ,yj=1λλjVλk1,yj+Δk1,yj,k=2,3,j=1,,N,(23)

where

Δ1=00λ1V13λV32λ2λ1V13λ0λ1V23λ000Δ2=00λ1v13λv32λ2λ1v13λ0λ1v23λ000.

Taking the sum of Eq. 23 with respect to j from 1 to N, we have

j=1Nyjtkλ=Δk1,j=1NyjVλk1,βλ,

from which we arrive at

βτtkλ=1λτVλk1,βτβλ+Δk1,βτ,k=2,3.(24)

For Casimir functions tr (yj), 1 ≤ jN, it is evident that

tryjtkλ=0,k=2,3.(25)

Proposition 2. The Lax matrix Vτ satisfies the Lax equations along the Fk(λ)-flows:

ddtkλVτ=1λτVλk1+Δk1,Vτ,k=2,3.

Proof. By using (23), (24), and (25), we have

ddtkλVτ=j=N1τλjyjtkλ+βτtkλ=1λτVλk1+Δk1,Vτ+βτtkλ1λτVλk1,βτβλΔk1,βτ=1λμVλk1+k1,Vμ.(26)

Based on Proposition 2, for any λ, τ, it is easy to verify that for l, k = 2, 3,

Flτ,Fkλ=ddtkλFlτ=1ltrddtkλVτl=1ltr1λτVλk1,Vτl=0,

from which we have {qhj, qim} = 0, h, i = 1, 2, 3, j, m = 1, … , N.

Corollary 1. FlS,FlT,l1 are in involution in pairs with respect to the Lie–Poisson bracket Eq. 5.

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

u=3r012,v=3r0226r0122

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

y1,,yj,,yN|tryj2=c2j,tryj3=c3j,j=1,,N(27)

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

detzIVλ=z3F2λzF3λ,(28)

which defines a non-hyperelliptic algebraic curve of genus G=3N2 by introducing variable ζ = a(λ)z:

ζ3+a2λF2λζa3λF3λ=0,

where

aλ=j=1Nλλj.

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

νi3F2μiνiF3μi=0.(29)

It follows from (28) that νi should be an eigenvalue of the matrix Vμi. Therefore, there must exist such a similarity transformation

VμiṼμi=KiVμiKi1

for each i that the matrix Ṽμi is block-triangular

Ṽ21μi=Ṽ31μi=0,(30)

and νi is the eigenvalue of Vμi split from the upper block

νi=Ṽ11μi.(31)

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:

Kk=100k10001.

Note that the matrix

ṼλKkVλK1k=V11λkV12λV12λV13λV21λ+kV11λkkV12λ+V22λV22λ+kV12λV23+kV13λV31λkV32λV32λV33λ

depends on two parameters λ and k. Hence, we can consider condition Eq. 30 as the set of two algebraic equations

Ṽ21λ=V21λ+kV11λkkV12λ+V22λ=0,Ṽ31λ=V31λkV32λ=0(32)

for two variables λ and k. By eliminating k from (32) yields the polynomial equation for λ:

V32λV31λV11λV22λ+V32λ2V21λV31λ2V12λ.=0.(33)

Based on (33), we can define the polynomial B(λ) of degree 3N as

Bλ=V32λV31λV11λV22λ+V32λ2V21λV31λ2V12λ.nλaλ3,(34)

where

nλ=i=13N2λμi,aλ=j=1Nλλjj=0NajλNja0=1.(35)

Expressing k from Ṽ31(λ)=0 as k = V31(λ)/V32(λ) and substituting it into the definition Eq. 31 of νi yields

νi=Ṽ11μi=V11μiV12μiV31μiV32μi,i=1,,3N2,(36)

thereby giving rise to 3N pairs of variables μi, νi. Let

Aλ=V11λV12λV31λV32λ,(37)

with the help of (4) and (17), it is easy to see that

Vlkτ,Vmnλ=1λτVmkτVmkλδlnVlnτVlnλδmk,

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

Aτ,Aλ=0,Bτ,Bλ=0,Aτ,Bλ=1λτBλV322λV322τBτ.(38)

Proposition 4. {μi, νi, 1 ≤ i ≤ 3N} are canonical coordinates, that is,

μi,μj=0,νi,νj=0,νi,μj=δij.

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

0=F,Bμj=F,Bλ|λ=μj+BμjF,μj

or

F,μj=F,Bλ|λ=μjBμj,(39)

for any function F, in the same way, we have

νi,F=Aμi,F=Aμ,F|μ=μi+Aμiμi,F.

Now, we turn to prove {νi, μj} = δij. Starting with

νi,μj=Aμ,μj|μ=μi+Aμiμi,μj=Aμ,μj|μ=μi,

using (39) and the third equation of (38), we arrive at

νi,μj=Aμ,Bλ|λ=μjμ=μiBμj=1μiμjV322μjV322μiBμiBμj.

The last expression vanishes for μiμj 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

12trβλ2+j=1Nq1jλλjb2λaλl=1flSλl+1,13trβλ3+j=1NI2jλλj+j=1NI3jλλj2b3λa2λl=0flTλl+1,

where

b2λ=I1λN2+I2λN3+IN3λ2+IN2λ+IN1,b3λ=λ2N1+INλ2N2++I3N3λ+I3N2,(40)

from which we can rewrite the generating functions F2(λ),F3(λ) as

F2λ=b2λaλ+j=1NC2jλλj2=l=1flSλl+1+j=1NC2jλλj2R2λa2λ,(41)
F3λ=b3λa2λ+j=1NC3jλλj3=l=0flTλl+1+j=1NC3jλλj3,(42)

with R2(λ)=a(λ)b2(λ)+a2(λ)j=1NC2j(λλj)2,C2j=12c2j,C3j=13c3j.

The comparison of the coefficients of λl (l = 0, … , N − 1) in equation

b2λ=aλl=1flSλl+1

and the comparison of the coefficients of λl (l = 0, 1, … , 2N − 1) in equation

b3λ=a2λl=0flTλl+1,

respectively, yield

Ij=i=1jaifjiS,j=1,,N1,IN+k=l=0k+1i,j0i+j=laiajfk+1lT,k=0,,2N2.

Let

νi=Sμi,i=1,,3N2,

with the help of Eq. 29, we have the completely separable Hamilton–Jacobi equations:

Sμi3b2μiaμi+j=1NC2jμiλj2Sμib3μia2μi+j=1NC3jμiλj3=0,

for i = 1, … , 3N − 2, from which we can obtain an implicit complete integral of Hamilton–Jacobi equations for the generating functions F2(λ) and F3(λ):

S=j=13N2Sjμj=Sμ1,,μ3N2;I1,,I3N2=j=13N20μjzdλ,(43)

where z satisfies Eq. 28.

Now, let us consider a canonical transformation from (μ, ν) to (ϕ, I) generated by the generating function S:

i=13N2νidμi+i=13N2ϕidIi=dS,

which satisfies

νi=Sμi,ϕi=SIi.(44)

From Eqs 28, 4144, we have

ϕi=SIi=j=13N20μjzIidλ=j=13N20μjaλzλNi1Rλdλ,i=1,,N1,j=13N20μjλ3Ni2Rλdλ,i=N,,3N2,(45)

where R(λ) = 3a2(λ)z2R2(λ). Thus, by using (40), (41), and (42), the generating functions of integrals can be rewritten as

F2λ=j=1NC2jλλj2+I1λN2++IN1aλK2I1,,IN1,λ,F3λ=j=1NC3jλλj3+λ2N1+INλ2N2++I3N2a2λK3IN,,I3N2,λ.

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 F2(λ),F3(λ) in terms of action-angle variables Ij, ϕj, j = 1, … , 3N − 2 are

ϕjt2λ=K2λIj=λNj1aλ,1jN1K2λIj=0,Nj3N2,Ijt2λ=K2λϕj=0,1j3N2,(46)
ϕjt3λ=K3λIj=0,1jN1K3λIj=λ3Nj2a2λ,Nj3N2,Ijt3λ=K3λϕj=0,1j3N2.(47)

Proposition 5. Let t2,l and t3,l be the variables of FlS-flow and FlT-flow, respectively; then, we have

dϕdt2,1,,dϕdt2,N1,dϕdt3,1,,dϕdt3,2N1=Q1100Q22,(48)

where

Q11=1A1A2AN21A1AN31A11,Q22=1B1B2B2N21B1B2N31B11

with Aks being the coefficients in the expansion

λNaλ=k=0Akλk,

which could be represented through the power sums of λl, δk=l=1Nλlk,

A0=1,A1=δ1,A2=12δ2+δ12,

with the recursive formula

Ak=1kδk+i,j1i+j=kδiAj,

and Brs are the comparison of the coefficients of λr, r = 0, 1, … in

λ2Na2λ=k=0Akλk2=r=0Brλr,

which can be written as B0=A02=1,B1=2A1,,Br=i,j0i+j=rAiAj with the supplementary definition Ak = Bk = 0, k = 1, 2, ….

Proof. According to the definition of the Lie–Poisson bracket,

Ijt2λ=l=01λl+1Ij,FlS=l=01λl+1dIjdt2,l,Ijt3λ=l=01λl+1Ij,FlT=l=01λl+1dIjdt3,l,ϕjt2λ=l=01λl+1ϕj,FlS=l=01λl+1dϕjdt2,l,ϕjt3λ=l=01λl+1ϕj,FlT=l=01λl+1dϕjdt3,l,(49)

for j = 1, … , 3N − 2. By using Eqs (46), (47), and 49, it is easy to see that

l=01λl+1Ij,FlS=l=01λl+1Ij,FlT=0,j=1,,3N2,l=01λl+1ϕj,FlS=λNj1aλ=k=0Akλk+j+1,j=1,,N1,l=01λl+1ϕj,FlT=0,j=1,,N1,l=01λl+1ϕj,FlS=0,j=N,,3N2,l=01λl+1ϕj,FlT=λ3Nj2a2λ=k=0Bkλk+j+2N,j=N,3N2.(50)

By comparing the coefficients of λl−1 in (50), we get the Lie–Poisson brackets

Ij,FlS=0,Ij,FlT=0,j=1,,3N2,ϕj,FlS=Alj,ϕj,FlT=0,j=1,,N1,ϕj,FlS=0,ϕj,FlT=Bl+Nj1,j=N,,3N2,(51)

thereby providing the nondegeneracy matrix Eq. 48.

Proposition 6. F1S,,FN1S,F1T,,F2N1T given in Eqs 19, 20 are functionally independent.

Proof. We only need to prove the linear independence of the gradients:

F1S,,FN1S,F1T,F2N1T.

Suppose

k=1N1ckFkS+m=12N1cN+m1FmT=0,

we have

0=k=1N1ckϕj,FkS+m=12N1cN+m1ϕj,FmT=k=1N1ckdϕjdt2,k+m=12N1cN+m1dϕjdt3,m.

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 F1S,,FN1S, F1T,,F2N1T, which are involutive in pairs and functionally independent.

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

Ijx=Ij0,ϕjx=ϕj0,j=1,,N1,ϕj0+BNjx,j=N,,3N2.(52)

Thus, combining Eq. 45 with (52) yields the Jacobi inversion problem for the Lie–Poisson Hamiltonian system Eq. 6

ϕj0=k=13N20μkaλzλNj1Rλdλ,j=1,,N1,ϕj0+BNjx=k=13N20μkλ3Nj2Rλdλ,j=N,,3N2.

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

Ijt=Ij0,ϕjt=ϕj0+A1jt,j=1,,N1,ϕj0,j=N,,3N2.(53)

According to Eqs 45, 53, we have the Jacobi inversion problem for the Lie–Poisson Hamiltonian system Eq. 7

ϕj0+A1jt=k=13N20μkaλzλNj1Rλdλ,j=1,,N1,ϕj0=k=13N20μkλ3Nj2Rλdλ,j=N,,3N2.

The compatible solution of systems Eqs 6, 7 in terms of action-angle variables Ij, ϕj is

Ijx,t=Ij0,0,ϕjx,t=ϕj0,0+A1jt,j=1,,N1,ϕj0,0+BNjx,j=N,,3N2.(54)

From (45) and (54), we finally obtain the Jacobi inversion problem for the Hirota–Satsuma modified Boussinesq Eq. 1:

ϕj0,0+A1jt=k=13N20μkaλzλNj1Rλdλ,j=1,,N1,ϕj0,0+BNjx=k=13N20μkλ3Nj2Rλdλ,j=N,,3N2.

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.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

References

1. Hirota R, Satsuma J. Nonlinear evolution equations generated from the backlund transformation for the Boussinesq equation. Prog Theor Phys (1977) 57:797–807. doi:10.1143/ptp.57.797

CrossRef Full Text | Google Scholar

2. Quispel GRW, Nijhoff FW, Capel HW. Linearization of the Boussinesq equation and the modified Boussinesq equation. Phys Lett A (1982) 91:143–5. doi:10.1016/0375-9601(82)90817-9

CrossRef Full Text | Google Scholar

3. Clarkson PA. New similarity solutions for the modified Boussinesq equation. J Phys A: Math Gen (1989) 22:2355–67. doi:10.1088/0305-4470/22/13/029

CrossRef Full Text | Google Scholar

4. Geng XG. Lax pair and Darboux transformation solutions of the modified Boussinesq equation. Acta Mathematicae Applicatae Sinica (1988) 11:324–8.

Google Scholar

5. Dai HH, Geng XG. Finite-dimensional integrable systems through the decomposition of a modified Boussinesq equation. Phys Lett A (2003) 317:389–400. doi:10.1016/j.physleta.2003.08.049

CrossRef Full Text | Google Scholar

6. He GL, Geng XG, Wu LH. The trigonal curve and the integration of the Hirota-Satsuma hierarchy. Math Methods Appl Sci (2017) 40:6581–601. doi:10.1002/mma.4476

CrossRef Full Text | Google Scholar

7. Kuznetsov VB. Quadrics on real Riemannian spaces of constant curvature: separation of variables and connection with Gaudin magnet. Theor Math Phys (1992) 33:3240–54. doi:10.1063/1.529542

CrossRef Full Text | Google Scholar

8. Babelon O, Talon M. Separation of variables for the classical and quantum Neumann model. Nucl Phys B (1992) 379:321–39. doi:10.1016/0550-3213(92)90599-7

CrossRef Full Text | Google Scholar

9. Kalnins EG, Kuznetsov VB, Miller W. Quadrics on complex Riemannian spaces of constant curvature, separation of variables, and the Gaudin magnet. J Math Phys (1994) 35:1710–31. doi:10.1063/1.530566

CrossRef Full Text | Google Scholar

10. Eilbeck JC, Enol’skii V, Kuznetsov V, Tsiganov A. Linear r-matrix algebra for classical separable systems. J Phys A: Math Gen (1994) 27:567–78. doi:10.1088/0305-4470/27/2/038

CrossRef Full Text | Google Scholar

11. Harnad J, Winternitz P. Classical and quantum integrable systems in 263-1263-1263-1and separation of variables. Commun Math Phys (1995) 172:263–85. doi:10.1007/BF02099428

CrossRef Full Text | Google Scholar

12. Ragnisco O. Dynamical r-matrices for integrable maps. Phys Lett A (1995) 198:295–305. doi:10.1016/0375-9601(95)00056-9

CrossRef Full Text | Google Scholar

13. Kulish PP, Rauch-Wojciechowski S, Tsiganov AV. Stationary problems for equation of the KdV type and dynamical r-matrices. J Math Phys (1996) 37:3463–82. doi:10.1063/1.531575

CrossRef Full Text | Google Scholar

14. Qiao ZJ. Modified r-matrix and separation of variables for the modified Korteweg-de Vries (MKdV) hierarchy. Physica A (1997) 243:129–40. doi:10.1016/S0378-4371(97)00260-4

CrossRef Full Text | Google Scholar

15. Zeng YB. Separation of variables for constrained flows. J Math Phys (1997) 38:321–9. doi:10.1063/1.531851

CrossRef Full Text | Google Scholar

16. Zhou RG. Lax representation, r -matrix method, and separation of variables for the Neumann-type restricted flow. J Math Phys (1998) 39:2848–58. doi:10.1063/1.532424

CrossRef Full Text | Google Scholar

17. Zeng YB, Lin RL. Families of dynamical r-matrices and Jacobi inversion problem for nonlinear evolution equations. J Math Phys (1998) 39:5964–83. doi:10.1063/1.532608

CrossRef Full Text | Google Scholar

18. Cao CW, Wu YT, Geng XG. Relation between the Kadometsev-Petviashvili equation and the confocal involutive system. J Math Phys (1999) 40:3948–70. doi:10.1063/1.532936

CrossRef Full Text | Google Scholar

19. Derkachev SE. The r-matrix factorization, q-operator, and variable separation in the case of the xxx spin chain with the SL(2, C) symmetry group. Theor Math Phys (2011) 169:1539–50. doi:10.1007/s11232-011-0131-x

CrossRef Full Text | Google Scholar

20. Du DL, Geng X. On the relationship between the classical Dicke-Jaynes-Cummings-Gaudin model and the nonlinear Schrödinger equation. J Math Phys (2013) 54:053510. doi:10.1063/1.4804943

CrossRef Full Text | Google Scholar

21. Du DL, Yang X. An alternative approach to solve the mixed AKNS equations. J Math Anal Appl (2014) 414:850–70. doi:10.1016/j.jmaa.2014.01.041

CrossRef Full Text | Google Scholar

22. Sklyanin EK. Separation of variables in the classical integrable sl(3) magnetic chain. Commun Math Phys (1992) 150:181–91. doi:10.1007/BF02096572

CrossRef Full Text | Google Scholar

23. Adams MR, Harnad J, Hurtubise J. Darboux coordinates and Liouville-Arnold integration in loop algebras. Commun Math Phys (1993) 155:385–413. doi:10.1007/BF02097398

CrossRef Full Text | Google Scholar

24. Buchstaber VM, Leykin DV, Ènol’skii VZ. Uniformization of Jacobi varieties of trigonal curves and nonlinear differential equations. Funktsional Anal I Prilozhen (2000) 34:159–71. doi:10.1007/bf02482405

CrossRef Full Text | Google Scholar

25. Dickey LA. Soliton equations and Hamiltonian systems. World Scientific (2003).

Google Scholar

26. Derkachov SE, Valinevich PA. Separation of variables for the quantum SL(3,ℂ) spin magnet: eigenfunctions of Sklyanin b-operator. J Math Sci (2019) 242:658–82. doi:10.1007/s10958-019-04505-5

CrossRef Full Text | Google Scholar

27. Scott DRD. Classical functional bethe ansatz for sl(n): separation of variables for the magnetic chain. J Math Phys (1994) 35:5831–43. doi:10.1063/1.530712

CrossRef Full Text | Google Scholar

28. Gekhtman MI. Separation of variables in the classical sl(n) magnetic chain. Commun Math Phys (1995) 167:593–605. doi:10.1007/BF02101537

CrossRef Full Text | Google Scholar

29. Dubrovin B, Skrypnyk T. Separation of variables for linear lax algebras and classical r-matrices. J Math Phys (2018) 59:091405. doi:10.1063/1.5031769

CrossRef Full Text | Google Scholar

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 States

Reviewed by:

Haret Rosu, Instituto Potosino de Investigación Científica y Tecnológica (IPICYT), Mexico
Yunqing 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==

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.