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ORIGINAL RESEARCH article

Front. Phys., 22 December 2023

Sec. Mathematical Physics

Volume 11 - 2023 | https://doi.org/10.3389/fphy.2023.1285301

This article is part of the Research TopicQuasi-Integrability, Nonlinear Evolutions and Physical ApplicationsView 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+13(uxxx23u2ux2ux1xut)x=0,(1)

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

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

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

wtt+13(wxx4w2)xx=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

sl(3)={αα=3i,j=1αijeij,tr(α)=0},sl(3)*={yy=3i,j=1yijEij,tr(y)=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,G}(y)=y,[F,G]=tr(y[F,G]),(3)

with the gradient Fsl(3) defined as

F=3k,l=1Fyklekl.

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

tr(y2),tr(y3).

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

{F,G}(yj)=Nj=1yj,[jF,jG],jF=3k,l=1Fykljekl,(5)

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

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

and the 2N Casimir functions

tr(y2j),tr(y3j),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=r210+r130+r321+3r120r2202(r120)3,(8)

and

H1=(r110)2+r220r110+(r220)2+r230+r120r210+r311+r1212r130r120+r120r321,(9)

with λ1, … , λN being N distinct parameters and rklm=Nj=1λmjyklj.

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=[U(λj),yj],j=1,,N,(14)

and N copies of (13)

yjt=[V(λj),yj],j=1,,N.(15)

Now, under the constraint

u=3r120,v=3r2206(r120)2,(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

tr(y2j),tr(y3j),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=(0103r2206(r120)23r120λ100),V=(2r110+r220r120λλ+r2102r130+r3212r220+r110λr1202r12010),

and

Vλ=(Vij(λ))3×3=β(λ)+Nj=1yjλλj,(17)

with

β(λ)=(00110r120λ1(r310+2r120)λ1(1r320)0).

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:

F2(λ)=12tr(V2λ),F3(λ)=13tr(V3λ).(18)

Furthermore, substituting Eqs 17, 18, we have

F2(λ)=12tr(V2λ)=12tr(β(λ)2)+Nj=1q1jλλj+Nj=1h2j(λλj)2l=1FSlλl+1,(19)

where

q1j=tr(β(λ)yj)+Nkjtr(yjyk)λjλk,h2j=12tr(y2j),FSl=Nj=1λljq1j+lNj=1λl1jh2j,l=1,,

and

F3(λ)=13tr(V3λ)=13tr(β(λ)3)+Nj=1q2jλλj+Nj=1q3j(λλj)2+Nj=1h3j(λλj)3l=0FTlλl+1,(20)

where

q2j=Nk=11λjλk[tr(β(λ)yjyk+β(λ)ykyj)+Nik,jtr(yjykyi+yjyiyk)λkλi+Nik,jtr(yjykyi+yjyiyk)λjλi+tr(β(λ)2yj)+Nkjtr(y2kyjy2jyk)(λjλk)2,q3j=tr(β(λ)y2j)+kjtr(y2jyk)λjλk,h3j=13tr(y3j),FTl=Nj=1λljq2j+lNj=1λl1jq3j+12l(l1)Nj=1λl2jh3j,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=FT1(21)

and

H1=FS1,(22)

respectively.

Denoting the variables of F2(λ)-flow and F3(λ)-flow by t2λ and t3λ, respectively, let V2λ=(vij(λ))3×3; then, the Hamiltonian equations for F2(λ) and F3(λ) are

yjtkλ=[jFk(λ),yj]=1λλj[Vk1λ,yj]+[Δk1,yj],k=2,3,j=1,,N,(23)

where

Δ1=(00λ1V13(λ)V32(λ)2λ1V13(λ)0λ1V23(λ)000)Δ2=(00λ1v13(λ)v32(λ)2λ1v13(λ)0λ1v23(λ)000).

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

Nj=1yjtkλ=[Δk1,Nj=1yj][Vk1λ,β(λ)],

from which we arrive at

β(τ)tkλ=1λτ[Vk1λ,β(τ)β(λ)]+[Δk1,β(τ)],k=2,3.(24)

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

tr(yj)tkλ=0,k=2,3.(25)

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

ddtkλVτ=[1λτVk1λ+Δk1,Vτ],k=2,3.

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

ddtkλVτ=Nj=1τλjyjtkλ+β(τ)tkλ=[1λτVk1λ+Δk1,Vτ]+β(τ)tkλ1λτ[Vk1λ,β(τ)β(λ)][Δk1,β(τ)]=[1λμVk1λ+k1,Vμ].(26)

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

{Fl(τ),Fk(λ)}=ddtkλFl(τ)=1ltr(ddtkλVlτ)=1ltr([1λτVk1λ,Vlτ])=0,

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

Corollary 1. FSl,FTl,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=3r120,v=3r2206(r120)2

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|tr(y2j)=c2j,tr(y3j)=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

det(zIVλ)=z3F2(λ)zF3(λ),(28)

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

ζ3+a2(λ)F2(λ)ζa3(λ)F3(λ)=0,

where

a(λ)=Nj=1(λλj).

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

ν3iF2(μi)νiF3(μi)=0.(29)

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

Vμi˜Vμi=KiVμiK1i

for each i that the matrix ˜Vμi is block-triangular

˜V21(μi)=˜V31(μi)=0,(30)

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

νi=˜V11(μi).(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:

K(k)=(100k10001).

Note that the matrix

˜VλK(k)VλK1(k)=(V11(λ)kV12(λ)V12(λ)V13(λ)V21(λ)+kV11(λ)k(kV12(λ)+V22(λ))V22(λ)+kV12(λ)V23+kV13(λ)V31(λ)kV32(λ)V32(λ)V33(λ))

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

{˜V21(λ)=V21(λ)+kV11(λ)k(kV12(λ)+V22(λ))=0,˜V31(λ)=V31(λ)kV32(λ)=0(32)

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

V32(λ)V31(λ)[V11(λ)V22(λ)]+V32(λ)2V21(λ)V31(λ)2V12(λ).=0.(33)

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

B(λ)=V32(λ)V31(λ)[V11(λ)V22(λ)]+V32(λ)2V21(λ)V31(λ)2V12(λ).n(λ)a(λ)3,(34)

where

n(λ)=3N2i=1(λμi),a(λ)=Nj=1(λλj)Nj=0ajλNj(a0=1).(35)

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

νi=˜V11(μi)=V11(μi)V12(μi)V31(μi)V32(μi),i=1,,3N2,(36)

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

A(λ)=V11(λ)V12(λ)V31(λ)V32(λ),(37)

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

{Vlk(τ),Vmn(λ)}=1λτ[(Vmk(τ)Vmk(λ))δln(Vln(τ)Vln(λ))δmk],

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

{{A(τ),A(λ)}=0,{B(τ),B(λ)}=0,{A(τ),B(λ)}=1λτ(B(λ)V232(λ)V232(τ)B(τ)).(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,B(μj)}={F,B(λ)}|λ=μj+B(μj){F,μj}

or

{F,μj}={F,B(λ)}|λ=μjB(μj),(39)

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

{νi,F}={A(μi),F}={A(μ),F}|μ=μi+A(μi){μi,F}.

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

{νi,μj}={A(μ),μj}|μ=μi+A(μi){μi,μj}={A(μ),μj}|μ=μi,

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

{νi,μj}={A(μ),B(λ)}|μ=μiλ=μjB(μj)=1μiμj(V232(μj)V232(μi)B(μi)B(μj)).

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

12tr(β(λ)2)+Nj=1q1jλλjb2(λ)a(λ)l=1fSlλl+1,13tr(β(λ)3)+Nj=1I2jλλj+Nj=1I3j(λλj)2b3(λ)a2(λ)l=0fTlλl+1,

where

b2(λ)=I1λN2+I2λN3+IN3λ2+IN2λ+IN1,b3(λ)=λ2N1+INλ2N2++I3N3λ+I3N2,(40)

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

F2(λ)=b2(λ)a(λ)+Nj=1C2j(λλj)2=l=1fSlλl+1+Nj=1C2j(λλj)2R2(λ)a2(λ),(41)
F3(λ)=b3(λ)a2(λ)+Nj=1C3j(λλj)3=l=0fTlλl+1+Nj=1C3j(λλj)3,(42)

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

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

b2(λ)=a(λ)(l=1fSlλl+1)

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

b3(λ)=a2(λ)(l=0fTlλl+1),

respectively, yield

Ij=ji=1aifSji,j=1,,N1,IN+k=k+1l=0(i,j0i+j=laiaj)fTk+1l,k=0,,2N2.

Let

νi=Sμi,i=1,,3N2,

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

(Sμi)3(b2(μi)a(μi)+Nj=1C2j(μiλj)2)Sμi(b3(μi)a2(μi)+Nj=1C3j(μiλj)3)=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=3N2j=1Sj(μj)=S(μ1,,μ3N2;I1,,I3N2)=3N2j=1μj0zdλ,(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:

3N2i=1νidμi+3N2i=1ϕidIi=dS,

which satisfies

νi=Sμi,ϕi=SIi.(44)

From Eqs 28, 4144, we have

ϕi=SIi=3N2j=1μj0zIidλ={3N2j=1μj0a(λ)zλNi1R(λ)dλ,i=1,,N1,3N2j=1μj0λ3Ni2R(λ)dλ,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

F2(λ)=Nj=1C2j(λλj)2+I1λN2++IN1a(λ)K2(I1,,IN1,λ),F3(λ)=Nj=1C3j(λλj)3+λ2N1+INλ2N2++I3N2a2(λ)K3(IN,,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={K2(λ)Ij=λNj1a(λ),1jN1K2(λ)Ij=0,Nj3N2,Ijt2λ=K2(λ)ϕj=0,1j3N2,(46)
ϕjt3λ={K3(λ)Ij=0,1jN1K3(λ)Ij=λ3Nj2a2(λ),Nj3N2,Ijt3λ=K3(λ)ϕj=0,1j3N2.(47)

Proposition 5. Let t2,l and t3,l be the variables of FSl-flow and FTl-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

λNa(λ)=k=0Akλk,

which could be represented through the power sums of λl, δk=Nl=1λkl,

A0=1,A1=δ1,A2=12(δ2+δ21),

with the recursive formula

Ak=1k(δk+i,j1i+j=kδiAj),

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

λ2Na2(λ)=(k=0Akλk)2=r=0Brλr,

which can be written as B0=A20=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+1{Ij,FSl}=l=01λl+1dIjdt2,l,Ijt3λ=l=01λl+1{Ij,FTl}=l=01λl+1dIjdt3,l,ϕjt2λ=l=01λl+1{ϕj,FSl}=l=01λl+1dϕjdt2,l,ϕjt3λ=l=01λl+1{ϕj,FTl}=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+1{Ij,FSl}=l=01λl+1{Ij,FTl}=0,j=1,,3N2,l=01λl+1{ϕj,FSl}=λNj1a(λ)=k=0Akλk+j+1,j=1,,N1,l=01λl+1{ϕj,FTl}=0,j=1,,N1,l=01λl+1{ϕj,FSl}=0,j=N,,3N2,l=01λl+1{ϕj,FTl}=λ3Nj2a2(λ)=k=0Bkλk+j+2N,j=N,3N2.(50)

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

{Ij,FSl}=0,{Ij,FTl}=0,j=1,,3N2,{ϕj,FSl}=Alj,{ϕj,FTl}=0,j=1,,N1,{ϕj,FSl}=0,{ϕj,FTl}=Bl+Nj1,j=N,,3N2,(51)

thereby providing the nondegeneracy matrix Eq. 48.

Proposition 6. FS1,,FSN1,FT1,,FT2N1 given in Eqs 19, 20 are functionally independent.

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

FS1,,FSN1,FT1,FT2N1.

Suppose

N1k=1ckFSk+2N1m=1cN+m1FTm=0,

we have

0=N1k=1ck{ϕj,FSk}+2N1m=1cN+m1{ϕj,FTm}=N1k=1ckdϕjdt2,k+2N1m=1cN+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 FS1,,FSN1, FT1,,FT2N1, 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

Ij(x)=Ij(0),ϕj(x)={ϕj(0),j=1,,N1,ϕj(0)+BNjx,j=N,,3N2.(52)

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

{ϕj(0)=3N2k=1μk0a(λ)zλNj1R(λ)dλ,j=1,,N1,ϕj(0)+BNjx=3N2k=1μk0λ3Nj2R(λ)dλ,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

Ij(t)=Ij(0),ϕj(t)={ϕj(0)+A1jt,j=1,,N1,ϕj(0),j=N,,3N2.(53)

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

{ϕj(0)+A1jt=3N2k=1μk0a(λ)zλNj1R(λ)dλ,j=1,,N1,ϕj(0)=3N2k=1μk0λ3Nj2R(λ)dλ,j=N,,3N2.

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

Ij(x,t)=Ij(0,0),ϕj(x,t)={ϕj(0,0)+A1jt,j=1,,N1,ϕj(0,0)+BNjx,j=N,,3N2.(54)

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

{ϕj(0,0)+A1jt=3N2k=1μk0a(λ)zλNj1R(λ)dλ,j=1,,N1,ϕj(0,0)+BNjx=3N2k=1μk0λ3Nj2R(λ)dλ,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.

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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 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, gengxue1985@163.com

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.