A Neumann System of the Third Order Differential Operator Associated with the Boussinesq Equation
Abstract: Finite-dimensional integrable Hamiltonian system, obtained through the nonlinearization of the 3 × 3 spectral problem associated with the Boussinesq equation, is investigated. A generating function method starting from the Lax-Moser matrix is used to give an effective way to prove the involutivity of integrals. Finite-parameter solution of the Boussinesq equation is calculated based on the commutative system of ordinary differential equations with these integrals as Hamiltonians. The problem of the third order differential operator associated with the Boussinesq Neumann system put forward by H. Flaschka in 1983 is solved.

1. Introduction

Boussinesq equation

${v}_{tt}=-\frac{1}{3}{\left({v}_{xx}-4{v}^{2}\right)}_{xx}$ (1)

is one of the difficult soliton equations, which has been paid common attention in physical and mathematical fields  - . In 1983, H. Flaschka put forward a problem of the third order differential operator associated with the Boussinesq Neumann system . Some works focus on the decomposition and the structures of the Modified Boussinesq equation    . The decomposition of the Boussinesq Neumann system has not been done thoroughly for a long time. A Neumann system of the Boussinesq equation associated with the third order differential operator is obtained in this paper, which is the extension of the famous KdV Neumann system associated with the second order differential operator. There are many methods to deal with the integrability and involutivity   . A generating function method starting from the Lax-Moser matrix  -  is used to give an effective way to prove the involutivity of integrals.

2. The Generating Function of Integrals

Let ${R}^{6N}$ be the phase space. The canonical coordinates are

${q}^{i}=\left({q}_{1}^{i},\cdots ,{q}_{N}^{i}\right),\text{ }{p}^{i}=\left({p}_{1}^{i},\cdots ,{p}_{N}^{i}\right),\text{ }i=1,2,3.$

Write

$q=\left({q}^{1},{q}^{2},{q}^{3}\right),\text{ }p=\left({p}^{1},{p}^{2},{p}^{3}\right);\text{ }{q}_{k}=\left({q}_{k}^{1},{q}_{k}^{2},{q}_{k}^{3}\right),\text{ }{p}_{k}=\left({p}_{k}^{1},{p}_{k}^{2},{p}_{k}^{3}\right),$

for short. Take $A=diag\left({\alpha }_{1},\cdots ,{\alpha }_{N}\right)$, where ${\alpha }_{1},\cdots ,{\alpha }_{N}$ are distinct spectral constants. We denote:

$〈X,Y〉\triangleq \underset{k=1}{\overset{N}{\sum }}\text{ }\text{ }{X}_{k}{Y}_{k},\text{ }{r}^{ij}=〈{q}^{i},{p}^{j}〉,\text{ }{A}^{ij}=〈A{q}^{i},{p}^{j}〉,$

where X and Y are two N dimensional vectors.

For any matrix $M=\left({M}^{ij}\right)$, the element of its adjoint ${M}^{*}$ is ${M}_{ij}^{*}={\left(-1\right)}^{i+j}{M}_{ji}$, where ${M}_{ji}$ is the cofactor of the element ${M}^{ij}$ in M.

In order to proof the Liouville integrability of the Hamiltonian system, consider the Lax-Moser matrix defined as:

${V}_{\lambda }={V}_{\lambda }\left(p,q\right)={\left({Q}_{\lambda }^{ij}\right)}_{3×3}+{l}_{\lambda },$ (2)

where

${Q}_{\lambda }^{ij}={Q}_{\lambda }\left({q}^{i},{p}^{j}\right)\triangleq \underset{k=1}{\overset{N}{\sum }}\frac{{q}_{k}^{i}{p}_{k}^{j}}{\lambda -{\alpha }_{k}},\text{ }{l}_{\lambda }={l}_{\lambda }\left(p,q\right)=\left(\begin{array}{ccc}0& 0& {r}^{32}\\ {r}^{32}& 0& 2{r}^{12}-{r}^{31}\\ 0& 0& 0\end{array}\right).$

Let ${L}_{\lambda \xi }=\xi {I}_{3}+{V}_{\lambda }$. Then we have the generating function of integrals:

${F}_{\lambda \xi }=\mathrm{det}{L}_{\lambda \xi }={\xi }^{3}+{F}_{1}\left(\lambda \right){\xi }^{2}+{F}_{2}\left(\lambda \right)\xi +{F}_{3}\left(\lambda \right),$ (3)

where

$\begin{array}{l}{F}_{1}\left(\lambda \right)={Q}_{\lambda }^{11}+{Q}_{\lambda }^{22}+{Q}_{\lambda }^{33},\\ {F}_{2}\left(\lambda \right)=|\begin{array}{cc}{Q}_{\lambda }^{11}& {Q}_{\lambda }^{12}\\ {Q}_{\lambda }^{21}+{r}^{32}& {Q}_{\lambda }^{22}\end{array}|+|\begin{array}{cc}{Q}_{\lambda }^{22}& {Q}_{\lambda }^{23}+2{r}^{12}-{r}^{31}\\ {Q}_{\lambda }^{32}& {Q}_{\lambda }^{33}\end{array}|+|\begin{array}{cc}{Q}_{\lambda }^{11}& {Q}_{\lambda }^{13}+{r}^{32}\\ {Q}_{\lambda }^{31}& {Q}_{\lambda }^{33}\end{array}|,\\ {F}_{3}\left(\lambda \right)=|\begin{array}{ccc}{Q}_{\lambda }^{11}& {Q}_{\lambda }^{12}& {Q}_{\lambda }^{13}+{r}^{32}\\ {Q}_{\lambda }^{21}+{r}^{32}& {Q}_{\lambda }^{22}& {Q}_{\lambda }^{23}+2{r}^{12}-{r}^{31}\\ {Q}_{\lambda }^{31}& {Q}_{\lambda }^{32}& {Q}_{\lambda }^{33}\end{array}|.\end{array}$

A series of polynomials ${F}_{jm}={F}_{jm}\left(q,p\right)$, are defined as the coefficients of the power series expansions as $|\lambda |>\mathrm{max}\left(|{\alpha }_{1}|,\cdots ,|{\alpha }_{N}|\right)$ :

${F}_{j}\left(\lambda \right)=\underset{m=0}{\overset{\infty }{\sum }}\text{ }\text{ }{\lambda }^{-m-1}{F}_{jm},\text{ }j=1,2,3.$

The first few are expressed as $|\lambda |>\mathrm{max}\left(|{\alpha }_{1}|,\cdots ,|{\alpha }_{N}|\right)$ :

$\begin{array}{l}{F}_{10}={r}^{11}+{r}^{22}+{r}^{33},\text{ }{F}_{20}=-3{r}^{12}{r}^{32},\text{ }{F}_{30}={\left({r}^{32}\right)}^{3},\\ {F}_{21}=-{r}^{32}\left({A}^{12}+{A}^{31}\right)-\left(2{r}^{12}-{r}^{31}\right){A}^{32}+\underset{1\le i

By comparing the coefficients of the same power of $\lambda$, general explicit formulas are obtained:

${F}_{1m}=〈{A}^{m}{q}^{1},{p}^{1}〉+〈{A}^{m}{q}^{2},{p}^{2}〉+〈{A}^{m}{q}^{3},{p}^{3}〉,\text{ }m\ge 1;$

$\begin{array}{l}{F}_{2m}=-{r}^{32}\left(〈{A}^{m}{q}^{1},{p}^{2}〉+〈{A}^{m}{q}^{3},{p}^{1}〉\right)-\left(2{r}^{12}-{r}^{31}\right)〈{A}^{m}{q}^{3},{p}^{2}〉\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{\begin{array}{c}k+l=m-1\\ k,l\ge 0\end{array}}{\sum }\text{\hspace{0.17em}}\underset{1\le i

$\begin{array}{l}{F}_{3m}={\left({r}^{32}\right)}^{2}〈{A}^{m}{q}^{3},{p}^{2}〉\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{r}^{32}\underset{\begin{array}{c}k+l=m-1\\ k,l\ge 0\end{array}}{\sum }\left(|\begin{array}{cc}〈{A}^{k}{q}^{2},{p}^{1}〉& 〈{A}^{l}{q}^{2},{p}^{2}〉\\ 〈{A}^{k}{q}^{3},{p}^{1}〉& 〈{A}^{l}{q}^{3},{p}^{2}〉\end{array}|-|\begin{array}{cc}〈{A}^{k}{q}^{1},{p}^{2}〉& 〈{A}^{l}{q}^{1},{p}^{3}〉\\ 〈{A}^{k}{q}^{3},{p}^{2}〉& 〈{A}^{l}{q}^{3},{p}^{3}〉\end{array}|\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(2{r}^{12}-{r}^{31}\right)\underset{\begin{array}{c}k+l=m-1\\ k,l\ge 0\end{array}}{\sum }|\begin{array}{cc}〈{A}^{k}{q}^{1},{p}^{1}〉& 〈{A}^{l}{q}^{1},{p}^{2}〉\\ 〈{A}^{k}{q}^{3},{p}^{1}〉& 〈{A}^{l}{q}^{3},{p}^{2}〉\end{array}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\underset{\begin{array}{c}j+k+l=m-2\\ j,k,l\ge 0\end{array}}{\sum }|\begin{array}{ccc}〈{A}^{j}{q}^{1},{p}^{1}〉& 〈{A}^{k}{q}^{1},{p}^{2}〉& 〈{A}^{l}{q}^{1},{p}^{3}〉\\ 〈{A}^{j}{q}^{2},{p}^{1}〉& 〈{A}^{k}{q}^{2},{p}^{2}〉& 〈{A}^{l}{q}^{2},{p}^{3}〉\\ 〈{A}^{j}{q}^{3},{p}^{1}〉& 〈{A}^{k}{q}^{3},{p}^{2}〉& 〈{A}^{l}{q}^{3},{p}^{3}〉\end{array}|,\text{ }m\ge 2.\end{array}$

Expand ${F}_{j}\left(\lambda \right)$ in non-negative power of $\lambda$ :

${F}_{j}\left(\lambda \right)=\underset{m=1}{\overset{\infty }{\sum }}\text{ }\text{ }{\lambda }^{m-1}{F}_{j,-m},\text{ }j=1,2,3$

as $|\lambda |<\mathrm{max}\left(|{\alpha }_{1}|,\cdots ,|{\alpha }_{N}|\right)$. By comparing the coefficients of the same power of $\lambda$, we have:

${F}_{1,-m}=-\left(〈{A}^{-m}{q}^{1},{p}^{1}〉+〈{A}^{-m}{q}^{2},{p}^{2}〉+〈{A}^{-m}{q}^{3},{p}^{3}〉\right),\text{ }m\ge 1;$

$\begin{array}{l}{F}_{2,-m}={r}^{32}\left(〈{A}^{-m}{q}^{1},{p}^{2}〉+〈{A}^{-m}{q}^{3},{p}^{1}〉\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(2{r}^{12}-{r}^{31}\right)〈{A}^{-m}{q}^{3},{p}^{2}〉\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{\begin{array}{c}k+l=m+1\\ k,l\ge 1\end{array}}{\sum }\text{\hspace{0.17em}}\underset{1\le i

$\begin{array}{l}{F}_{3,-m}=-{\left({r}^{32}\right)}^{2}〈{A}^{m}{q}^{3},{p}^{2}〉\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{r}^{32}\underset{\begin{array}{c}k+l=m+1\\ k,l\ge 1\end{array}}{\sum }\left(|\begin{array}{cc}〈{A}^{-k}{q}^{2},{p}^{1}〉& 〈{A}^{-l}{q}^{2},{p}^{2}〉\\ 〈{A}^{-k}{q}^{3},{p}^{1}〉& 〈{A}^{-l}{q}^{3},{p}^{2}〉\end{array}|-|\begin{array}{cc}〈{A}^{-k}{q}^{1},{p}^{2}〉& 〈{A}^{-l}{q}^{1},{p}^{3}〉\\ 〈{A}^{-k}{q}^{3},{p}^{2}〉& 〈{A}^{-l}{q}^{3},{p}^{3}〉\end{array}|\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-\left(2{r}^{12}-{r}^{31}\right)\underset{\begin{array}{c}k+l=m+1\\ k,l\ge 1\end{array}}{\sum }|\begin{array}{cc}〈{A}^{-k}{q}^{1},{p}^{1}〉& 〈{A}^{-l}{q}^{1},{p}^{2}〉\\ 〈{A}^{-k}{q}^{3},{p}^{1}〉& 〈{A}^{-l}{q}^{3},{p}^{2}〉\end{array}|\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{\begin{array}{c}j+k+l=m+2\\ j,k,l\ge 1\end{array}}{\sum }|\begin{array}{ccc}〈{A}^{-j}{q}^{1},{p}^{1}〉& 〈{A}^{-k}{q}^{1},{p}^{2}〉& 〈{A}^{-l}{q}^{1},{p}^{3}〉\\ 〈{A}^{-j}{q}^{2},{p}^{1}〉& 〈{A}^{-k}{q}^{2},{p}^{2}〉& 〈{A}^{-l}{q}^{2},{p}^{3}〉\\ 〈{A}^{-j}{q}^{3},{p}^{1}〉& 〈{A}^{-k}{q}^{3},{p}^{2}〉& 〈{A}^{-l}{q}^{3},{p}^{3}〉\end{array}|,\text{ }m\ge 1.\end{array}$

3. The Involutivity of Integrals

The involutivity is critical to the integrability of the Hamiltonian system, which is defined as the Poisson bracket of two conserved integrals being zero. A direct calculation gives:

Proposition 1. The Hamiltonian system for the ${F}_{\lambda \xi }$ -flow is expressed as:

$\frac{\text{d}{q}_{k}}{\text{d}{t}_{\lambda \xi }}={W}_{\lambda \xi \mu }{q}_{k},\text{ }\frac{\text{d}{p}_{k}}{\text{d}{t}_{\lambda \xi }}=-{W}_{\lambda \xi \mu }^{\text{T}}{p}_{k},$ (4)

where

${W}_{\lambda \xi \mu }=\frac{1}{\lambda -{\alpha }_{k}}{L}_{\lambda \xi }^{*}+{L}_{0\lambda \xi },{L}_{0\lambda \xi }=\left(\begin{array}{ccc}0& 0& {L}_{23}\\ -2{L}_{23}& 0& {L}_{13}-{L}_{21}\\ 0& 0& 0\end{array}\right).$

Proof. From the Equation (3) and the property of Hamiltonian system, we calculate the partial derivatives of ${q}_{k}$ and ${p}_{k}$ with respect to ${t}_{\lambda \xi }$, then the results are obtained. $\square$

Lemma 2. ${\epsilon }_{k}={q}_{k}{p}_{k}^{\text{T}}$ satisfies the Lax equation along the ${t}_{\lambda \xi }$ -flow:

$\frac{\text{d}{\epsilon }_{k}}{\text{d}{t}_{\lambda \xi }}=\left[{W}_{\lambda \xi \mu },{\epsilon }_{k}\right]=\frac{1}{\lambda -{\alpha }_{k}}\left[{L}_{\lambda \xi }^{*},{\epsilon }_{k}\right].$

Proof.

${\stackrel{˙}{\epsilon }}_{k}={\stackrel{˙}{q}}_{k}{p}_{k}^{\text{T}}+{q}_{k}{\stackrel{˙}{p}}_{k}^{\text{T}}=\left(W{q}_{k}\right){p}_{k}^{\text{T}}+{q}_{k}{\left(-{W}^{\text{T}}{q}_{k}\right)}^{\text{T}}=W{\epsilon }_{k}-{\epsilon }_{k}W=\left[W,{\epsilon }_{k}\right].$ $\square$

Proposition 3. ${L}_{\mu \eta }$ satisfies the Lax equation along the ${t}_{\lambda \xi }$ -flow:

$\frac{\text{d}{L}_{\mu \eta }}{\text{d}{t}_{\lambda \xi }}=\left[{W}_{\lambda \xi \mu },{L}_{\mu \eta }\right].$ (5)

Proof. ${Q}_{\mu }$ can be expressed by ${\epsilon }_{k}$ as:

${Q}_{\mu }=\underset{k=1}{\overset{N}{\sum }}\frac{{\epsilon }_{k}}{\mu -{\alpha }_{k}}.$

Resorting to Lemma 2, we have:

$\frac{\text{d}{Q}_{\mu }}{\text{d}{t}_{\lambda \xi }}=\left[{W}_{\lambda \xi \mu },{L}_{\mu \eta }\right]-\left[{L}_{0\lambda \xi },{l}_{\mu }\right],$

$\frac{\text{d}{L}_{\mu \eta }}{\text{d}{t}_{\lambda \xi }}-\left[{W}_{\lambda \xi \mu },{L}_{\mu \eta }\right]=\frac{\text{d}{l}_{\mu }}{\text{d}{t}_{\lambda \xi }}-\left[{L}_{0\lambda \xi },{l}_{\mu }\right]=0.$ $\square$

Lemma 4. The two determinants are true as $a,b,c,d=1,2,3$ :

$|\begin{array}{cc}{Q}_{\lambda }\left({q}^{a},{p}^{b}\right)& {Q}_{\lambda }\left({q}^{a},{p}^{d}\right)\\ {Q}_{\lambda }\left({q}^{c},{p}^{b}\right)& {Q}_{\lambda }\left({q}^{c},{p}^{d}\right)\end{array}|=\frac{1}{2}\underset{k=1}{\overset{N}{\sum }}\underset{l=1}{\overset{N}{\sum }}\frac{1}{\left(\lambda -{\alpha }_{k}\right)\left(\lambda -{\alpha }_{l}\right)}|\begin{array}{cc}{q}_{k}^{a}& {q}_{l}^{a}\\ {q}_{k}^{c}& {q}_{l}^{c}\end{array}||\begin{array}{cc}{p}_{k}^{b}& {p}_{k}^{d}\\ {p}_{l}^{b}& {p}_{l}^{d}\end{array}|.$ (6)

$\begin{array}{l}det{\left({Q}_{\lambda }\left({q}^{i},{p}^{j}\right)\right)}_{3×3}\\ =\frac{1}{6}\underset{j=1}{\overset{N}{\sum }}\underset{k=1}{\overset{N}{\sum }}\underset{l=1}{\overset{N}{\sum }}\frac{1}{\left(\lambda -{\alpha }_{j}\right)\left(\lambda -{\alpha }_{k}\right)\left(\lambda -{\alpha }_{l}\right)}|\begin{array}{ccc}{q}_{j}^{1}& {q}_{k}^{1}& {q}_{l}^{1}\\ {q}_{j}^{2}& {q}_{k}^{2}& {q}_{l}^{2}\\ {q}_{j}^{3}& {q}_{k}^{3}& {q}_{l}^{3}\end{array}||\begin{array}{ccc}{p}_{j}^{1}& {p}_{k}^{2}& {p}_{l}^{3}\\ {p}_{j}^{1}& {p}_{k}^{2}& {p}_{l}^{3}\\ {p}_{j}^{1}& {p}_{k}^{2}& {p}_{l}^{3}\end{array}|.\end{array}$ (7)

From (6) and (7), we have:

${F}_{1}\left(\lambda \right)={F}_{10}\frac{{\beta }_{1}\left(\lambda \right)}{\alpha \left(\lambda \right)}=\left({r}^{11}+{r}^{22}+{r}^{33}\right)\frac{{\beta }_{1}\left(\lambda \right)}{\alpha \left(\lambda \right)}=\underset{k=1}{\overset{N}{\sum }}\frac{{E}_{1k}}{\lambda -{\alpha }_{k}},$ (8)

${F}_{2}\left(\lambda \right)={F}_{20}\frac{{\beta }_{2}\left(\lambda \right)}{\alpha \left(\lambda \right)}=-3{r}^{12}{r}^{32}\frac{{\beta }_{2}\left(\lambda \right)}{\alpha \left(\lambda \right)}=\underset{k=1}{\overset{N}{\sum }}\frac{{E}_{2k}}{\lambda -{\alpha }_{k}},$ (9)

${F}_{3}\left(\lambda \right)={F}_{30}\frac{{\beta }_{3}\left(\lambda \right)}{\alpha \left(\lambda \right)}={\left({r}^{32}\right)}^{3}\frac{{\beta }_{3}\left(\lambda \right)}{\alpha \left(\lambda \right)}=\underset{k=1}{\overset{N}{\sum }}\frac{{E}_{3k}}{\lambda -{\alpha }_{k}},$ (10)

where $\alpha \left(\lambda \right)={\prod }_{k=1}^{N}\left(\lambda -{\alpha }_{k}\right)$, and ${\beta }_{1}\left(\lambda \right),{\beta }_{2}\left(\lambda \right),{\beta }_{3}\left(\lambda \right)$ are polynomials of degree $N-1$. From (8)-(10), another group of conserved integrals ${E}_{1k},{E}_{2k},{E}_{3k}$ is obtained:

${E}_{1k}=\underset{i=1}{\overset{3}{\sum }}\text{ }\text{ }{q}_{k}^{i}{p}_{k}^{i},\text{ }1\le k\le N;$

$\begin{array}{l}{E}_{2k}=-{r}^{32}\left({q}_{k}^{1}{p}_{k}^{2}+{q}_{k}^{3}{p}_{k}^{1}\right)-\left(2{r}^{12}-{r}^{31}\right){q}_{k}^{3}{p}_{k}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{\begin{array}{c}1\le l\le N\\ l\ne k\end{array}}{\sum }\frac{1}{{\alpha }_{k}-{\alpha }_{l}}\underset{1\le i

$\begin{array}{l}{E}_{3k}={\left({r}^{32}\right)}^{2}{q}_{k}^{3}{p}_{k}^{2}+{r}^{32}\underset{\begin{array}{c}1\le l\le N\\ l\ne k\end{array}}{\sum }\frac{1}{{\alpha }_{k}-{\alpha }_{l}}\left(|\begin{array}{cc}{q}_{k}^{2}& {q}_{l}^{2}\\ {q}_{k}^{3}& {q}_{l}^{3}\end{array}||\begin{array}{cc}{p}_{k}^{1}& {p}_{k}^{2}\\ {p}_{l}^{1}& {p}_{l}^{2}\end{array}|-|\begin{array}{cc}{q}_{k}^{1}& {q}_{l}^{1}\\ {q}_{k}^{3}& {q}_{l}^{3}\end{array}||\begin{array}{cc}{p}_{k}^{2}& {p}_{k}^{3}\\ {p}_{l}^{2}& {p}_{l}^{3}\end{array}|\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(2{r}^{12}-{r}^{31}\right)\underset{\begin{array}{c}1\le l\le N\\ l\ne k\end{array}}{\sum }\frac{1}{{\alpha }_{k}-{\alpha }_{l}}|\begin{array}{cc}{q}_{k}^{1}& {q}_{l}^{1}\\ {q}_{k}^{3}& {q}_{l}^{3}\end{array}||\begin{array}{cc}{p}_{k}^{1}& {p}_{k}^{2}\\ {p}_{l}^{1}& {p}_{l}^{2}\end{array}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{\begin{array}{c}1\le j\le N\\ j\ne k\end{array}}{\sum }\text{\hspace{0.17em}}\underset{\begin{array}{c}1\le l\le N\\ l\ne j,k\end{array}}{\sum }\frac{1}{2\left({\alpha }_{j}-{\alpha }_{k}\right)\left({\alpha }_{l}-{\alpha }_{k}\right)}|\begin{array}{ccc}{q}_{j}^{1}& {q}_{k}^{1}& {q}_{l}^{1}\\ {q}_{j}^{2}& {q}_{k}^{2}& {q}_{l}^{2}\\ {q}_{j}^{3}& {q}_{k}^{3}& {q}_{l}^{3}\end{array}||\begin{array}{ccc}{p}_{j}^{1}& {p}_{k}^{2}& {p}_{l}^{3}\\ {p}_{j}^{1}& {p}_{k}^{2}& {p}_{l}^{3}\\ {p}_{j}^{1}& {p}_{k}^{2}& {p}_{l}^{3}\end{array}|,\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\le k\le N.\end{array}$

From the definition of involutivity of two conserved integrals and direct calcultion, we have:

Proposition 5. The integrals

$\left\{{F}_{1m},{F}_{2m},{F}_{3m},m\in Z;{E}_{1k},{E}_{2k},{E}_{3k},1\le k\le N\right\}$

are involutive in pairs:

$\left({F}_{im},{F}_{jn}\right)=0,\text{ }\forall i,j=1,2,3;\text{ }\forall m,n=0,1,2,\cdots .$ (11)

$\left({F}_{i,-m},{F}_{j,-n}\right)=0,\text{ }\forall i,j=1,2,3;\text{ }\forall m,n=,1,2,\cdots .$ (12)

$\left({F}_{im},{F}_{j,-n}\right)=0,\text{ }\forall i,j=1,2,3;\text{ }\forall m=0,1,2,\cdots ;n=1,2,\cdots .$ (13)

$\left({E}_{ik},{E}_{jl}\right)=0,\text{ }\forall i,j=1,2,3;\text{ }\forall 1\le k,l\le N.$ (14)

$\left({F}_{im},{E}_{jl}\right)=0,\text{ }\forall i,j=1,2,3;\text{ }\forall m\in Z;1\le l\le N.$ (15)

4. Hamiltonian Systems

By direct calculations, the canonical equations of the Hamiltonian systems $\left({F}_{31}\right)$ and $\left({F}_{21}\right)$ can be expressed as:

$\begin{array}{l}{q}_{x}=\frac{\partial {F}_{31}}{\partial p}=Uq,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}_{x}=-\frac{\partial {F}_{31}}{\partial q}=-{U}^{\text{T}}p;\\ {q}_{t}=\frac{\partial {F}_{21}}{\partial p}=Vq,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}_{t}=-\frac{\partial {F}_{21}}{\partial q}=-{V}^{\text{T}}p;\end{array}$ (16)

where $U,V$ are $3N×3N$ matrices:

$U=\left(\begin{array}{ccc}{r}^{31}& 1& {r}^{11}-{r}^{22}\\ -\left[2{r}^{11}+{r}^{33}+{\left({r}^{31}\right)}^{2}\right]& -{r}^{31}& A+2{A}^{32}+2{r}^{13}+2{r}^{21}+\left({r}^{11}-{r}^{22}\right){r}^{31}\\ 1& 0& 0\end{array}\right);$

$V=\left(\begin{array}{ccc}{r}^{22}+{r}^{33}& 0& -A+{A}^{32}-{r}^{13}\\ -\left(A+2{A}^{32}+{r}^{21}\right)& {r}^{11}+{r}^{33}& {r}^{31}A-{A}^{12}-{A}^{31}-{r}^{23}\\ -{r}^{31}& -1& {r}^{11}+{r}^{22}\end{array}\right).$

Proposition 6. Let $q\left(x,t\right),p\left(x,t\right)$ be compatible solution of the $\left({F}_{31}\right)$ and $\left({F}_{21}\right)$ flow on the level set $\left\{\left(p,q\right)\in {ℝ}^{6N}:{r}^{32}=1,{r}^{12}=0\right\}$. Then the Boussinesq Equation (1) has a finite-parameter solution $\left(u,v\right)=B\left(q,p\right)$ given as:

$u=-\frac{3}{2}{A}^{32}+\frac{1}{2}{F}_{31},\text{ }v=\frac{3}{2}{r}^{11}-\frac{1}{2}{F}_{10}.$ (17)

Proof. A direct calculation gives:

$\begin{array}{l}{r}_{x}^{11}=-{r}^{13}+{r}^{21}+\left({r}^{11}-{r}^{22}\right){r}^{31},\\ {r}_{x}^{22}=3{A}^{32}+2{r}^{13}+{r}^{21}+\left({r}^{11}-{r}^{22}\right){r}^{31},\\ {r}_{x}^{33}=-3{A}^{32}-{r}^{13}-2{r}^{21}-2\left({r}^{11}-{r}^{22}\right){r}^{31},\\ {r}_{x}^{13}=-{A}^{12}+{r}^{23}+{r}^{13}{r}^{31}+\left({r}^{11}-{r}^{22}\right)\left({r}^{33}-{r}^{11}\right),\\ {r}_{x}^{21}={A}^{31}+2{r}^{31}{A}^{32}-{r}^{23}+2{r}^{13}{r}^{31}-\left({r}^{11}-{r}^{22}\right)\left(2{r}^{11}+{r}^{33}\right),\end{array}$

$\begin{array}{l}{r}_{x}^{23}={A}^{33}-{A}^{22}+2\left({r}^{33}-{r}^{22}\right){A}^{32}-{r}^{23}{r}^{31}+\left(-2{r}^{11}-2{r}^{22}+{r}^{33}\right){r}^{13},\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(-{r}^{11}-{r}^{22}+2{r}^{33}\right){r}^{21}-{r}^{13}{\left({r}^{31}\right)}^{2}+\left({r}^{11}-{r}^{22}\right)\left({r}^{33}-{r}^{22}\right){r}^{31},\end{array}$

$\begin{array}{l}{r}_{x}^{31}=3{r}^{11},\text{ }{A}_{x}^{12}=-{A}^{11}+{A}^{22}+2{r}^{31}{A}^{21}+\left({r}^{11}-{r}^{22}\right){A}^{32},\\ {A}_{x}^{31}={A}^{11}-{A}^{33}-{r}^{31}{A}^{31}+\left[2{r}^{11}+{r}^{33}+{\left({r}^{31}\right)}^{2}\right]{A}^{32},\\ {A}_{x}^{32}={A}^{12}-{A}^{31}+{r}^{31}{A}^{32}.\end{array}$

So from the above calculations and (16), we have:

${v}_{t}=-{u}_{x},\text{ }{u}_{t}=\frac{1}{3}{\left({v}_{xx}-4{v}^{2}\right)}_{x},$

and

${v}_{tt}={\left(-{u}_{x}\right)}_{t}=-{\left({u}_{t}\right)}_{x}=-\frac{1}{3}{\left({v}_{xx}-4{v}^{2}\right)}_{xx}$

which is exactly the “good” Boussinesq Equation (1). $\square$

5. Conclusion

The third order differential operator associated with the Boussinesq Neumann system is a critical point in researching problems of integrable system. In this paper, we obtain a Neumann system of the Boussinesq equation associated with the third order differential operator, which is the extension of the famous KdV Neumann system associated with the second order differential operator. By means of generating function from the Lax-Moser matrix, we prove the involutivity of integrals successfully. Meanwhile, a finite-parameter solution to the Boussinesq equation is obtained naturally.

Acknowledgements

This work is supported by the Foundation (Grant No. 11601123 and 202010463050).

Cite this paper: Cao, J. and Han, J. (2020) A Neumann System of the Third Order Differential Operator Associated with the Boussinesq Equation. Journal of Applied Mathematics and Physics, 8, 2861-2868. doi: 10.4236/jamp.2020.812211.
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