The Boussinesq Sequence

The Boussinesq and modified Boussinesq equations may be formulated as Hamiltonian systems [7]. That is,

$$\left(\matrix{U\cr H\cr}\right)_t=\Omega_1\left(\matrix{-U_{xx}/3- U^2\cr \noalign{\vskip 5pt} H\cr}\right)\ ,\eqno (3.1)$$

$$\left(\matrix{\nu\cr \omega\cr}\right)_t=\Omega_2\left( \mat... ...r 2}\nu^2-\textstyle{3\over 2}\omega^2\cr}\right)\ ,\eqno (3.2)$$

where

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...&=\left(\matrix{0&D\cr D&0\cr}\right)\ ,\cr\crcr}}\,\eqno (3.3)$$

$$\Omega_2=\left(\matrix{D&0\cr \noalign{\vskip 5pt} 0&\textstyle{1\over 3}D\cr}\right)\ ,\eqno (3.4)$$

are symplectic operators and

$$\left(\matrix{-U_{xx}/3-U^2\cr \noalign{\vskip 5pt} H\cr}\right)=\nabla H_1\ ,\eqno (3.5)$$

$$\left(\matrix{\omega_x+\nu\omega\cr \noalign{\vskip 5pt} -\n... ...xtstyle{3\over 2}\omega^2\cr}\right)=\nabla H_2\ ,\eqno (3.6)$$

are the functional gradients of the Hamiltonians

$$H_1=\int\left\{{U_x^2\over 6}-{U^3\over 3}+H^2\right\}\ ,\eqno (3.7)$$

$$H_2=\int\left\{{\nu\omega_x-\omega\nu_x\over 2}+{1\over 2}\nu^2 \omega-{1\over 2}\omega^3\right\}\ .\eqno (3.8)$$

By the results of the previous section Eq. (3.2) is invariant under the transformation

$$\left(\matrix{\nu\cr \omega\cr}\right)=A_{\pm}\left(\matrix{\theta\cr z\cr}\right)\ ,\eqno (3.9)$$

where $A_{\pm}$ is defined by (2.23). The three Miura transformations from (3.2) to (3.1) are

(i) $U=-\textstyle{1\over 2}\bigl(\omega^2+\textstyle{4\over 3}(\nu_x+ \textstyle{1\over 4}\nu^2)\bigr)$ ,

$H=-\textstyle{1\over 3}(2\omega_{xx}-\omega^3+\nu_x \omega+3\nu\omega_x+\nu^2\omega)$ ,(3.10)

(ii) $U=z_x-z^2/2+\textstyle{1\over 3}(\theta_x- \theta^2/2)$ ,

$H=\textstyle{1\over 3}(D-2z)\bigl(z_x-z^2/2-(\theta_x -\theta^2/2)\bigr)$ ,(3.11)

with $A_+$ in (3.9); and

(iii) $U=-z_x-z^2/2+\textstyle{1\over 3}(\theta_x- \theta^2/2)$ ,

$H=\textstyle{1\over 3}(D+2z)\bigl[z_x+z^2/2+(\theta_x -\theta^2/2)\bigr]$ ,(3.12)

with $A_-$.

By a theorem of [7], a Miura transformation between two systems with a Hamiltonian structure provides the means for constructing a second Hamiltonian structure for both equations, and, thereby, the recursion operators determining the sequences of higher-order equations. We have from (3.10)-(31.2) the operators

(i) $B_1=-{1\over 3}\left(\matrix{2D+\nu& 3\omega\cr \noalign{\vskip 5pt} D\omega+2(\omega_x+\nu\omega)&2D^2+3D\nu-2\nu_x+\nu^2-3\omega\cr} \right)$ ,(3.13)

(ii) $B_2={1\over 3}\left(\matrix{D-\theta& 3(D-z)\cr \noalign{\vskip 5pt} -(D-2z)(... ...ta)&D^2-3Dz+3z^2\cr &\quad+2(\theta_x-\textstyle{1\over 2}\theta^2)\cr}\right)$ ,(3.14)

(iii) $B_3={1\over 3}\left(\matrix{D-\theta& 3(D+z)\cr \noalign{\vskip 5pt} (D+2z)(D-\theta)&D^2+3Dz+3z^2+2(\theta_x-\textstyle{1\over 2}\theta^2)\cr}\right)$ , (3.15)

which determine the first variations of the respective Miura transformations about solutions of (3.2). From [7], the recursion operators (strong symmetries) of (3.1) and (3.2) are

$$M=B\Omega_2 B^{\ast}\Omega_1^{-1}\ ,\eqno (3.16)$$

$$L=\Omega_2 B^{\ast}\Omega_1^{-1} B\ ,\eqno (3.17)$$

where $B$ is (3.13), (3.14), or (3.15), $B^{\ast}$ is the adjoint operator, and

$$\Omega_1^{-1}=\left(\matrix{0&D^{-1}\cr \noalign{\vskip 5pt} D^{-1}&0\cr}\right)\ .\eqno (3.18)$$

The sequences of Boussinesq and modified Boussinesq equations are

$$\left(\matrix{U\cr H\cr}\right)_t=M^n\Omega_1\left(\matrix{-U_{xx}/ 3-U^2\cr \noalign{\vskip 5pt} H\cr}\right)\ ,\eqno (3.19)$$

$$\left(\matrix{\theta\cr z\cr}\right)_t=L^n\Omega_2\left( \ma... ...er 2}\theta^2-\textstyle{3\over 2}z^2\cr}\right)\ ,\eqno (3.20)$$

for $n=0,1,2,\ldots$ .

By direct calculation, using (3.10)-(3.15), we find that

$$M_1=M_2=M_3=M\ ,\qquad L_1=L_2=L_3=L\ ,\eqno (3.21)$$

where the subscript refers to the transformations (3.10), (3.11), (3.12), respectively. This result demonstrates that Eqs. (3.20) are invariant under (3.9), and (3.10) to (3.12) defines Miura transformations from (3.20) to (3.19). For reference,

$$B\Omega_2 B^{\ast}=-{1\over 9}\left(\matrix{4D^3+6UD+3U_x&9HD... ..._{xx}+12U^2)D\cr &-(2U_{xxx}+12UU_x)\cr}\right)\ ,\eqno (3.22)$$

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ... +6zD^{-1}(z_x+\theta z)&\cr}\right)\ .\cr\crcr}}\,\eqno (3.23)$$

At this point, it is convenient to identify the following expressions:

$$s=\theta_x-\textstyle{1\over 2}\theta^2\ , \eqno (3.24)$$

$$\Omega=\left(\matrix{(D-\theta)D(D+\theta)&0\cr \noalign{\vskip 5pt} 0&-\textstyle{1\over 3}D\cr}\right), \eqno (3.25)$$

$$C={1\over 3}\left(\matrix{1&3(D-z)\cr \noalign{\vskip 5pt} -D+2z&D^2-3Dz+3z^2+2s\cr}\right)\ ,\eqno (3.26a)$$

$$R=\left(\matrix{D-\theta&0\cr 0&1\cr}\right)\ ,\eqno (3.26b)$$

$$M_2=C\Omega C^{\ast}\Omega_1^{-1}\ ,\eqno (3.27)$$

$$L_2=\Omega C^{\ast}\Omega_1^{-1} C\ , \eqno (3.28)$$

where

$$U=z_x-\textstyle{1\over 2}z^2+\textstyle{1\over 3}s\ ,\qquad ... ...1\over 3}(D-2z)(z_x- \textstyle{1\over 2}z^2-s)\ .\eqno (3.29)$$

We note the following identifies:

$$M_2=-M\ ,\qquad B=CR\ ,\eqno (3.30)$$

$$RL=-L_2R\ ,\eqno (3.31)$$

$$\left(\matrix{U\cr H\cr}\right)_t=C\left(\matrix{s\cr z\cr}\right)_t =B\left(\matrix{\theta\cr z\cr}\right)_t\ ,\eqno (3.32)$$

$$\Omega_1\left(\matrix{-U_{xx}/3-U^2\cr \noalign{\vskip 5pt} ... ...\vskip 5pt} s+\textstyle{3\over 2}z^2\cr}\right)\ .\eqno (3.33)$$

We now formulate the following theorem.

Theorem.    For the Boussinesq sequence

$$\left(\matrix{U\cr H\cr}\right)_t=M^n\Omega_1\left( \matrix{-U_{xx}/3-U^2\cr \noalign{\vskip 5pt} H\cr}\right)\eqno (3.34)$$

and the modified Boussinesq sequence

$$\left(\matrix{\theta\cr z\cr}\right)_t=L^n\Omega_2\left( \ma... ...kip 5pt} -s-\textstyle{3\over 2}z^2\cr} \right)\ ,\eqno (3.35)$$

when $n=0,1,2,3,\ldots$,

$$s=\theta_x-\textstyle{1\over 2}\theta^2\ ,$$

there exists the Bäcklund transformation (BT)

$$U=2{\partial^2\over\partial x^2}\ln\varphi+U_2\ ,\qquad H=2{\partial^2\over\partial x\partial t}\ln\varphi+H_2\ ,\eqno (3.36)$$

$$\theta=-2{\partial\over\partial x}\ln\varphi+\theta_1\ ,\qquad z=z_1\ ,\eqno (3.37)$$

where $(U,H)$, $(U_2,H_2)$ satisfy (3.34); $(\theta,z)$, $(\theta_1,z)$ satisfy (3.35);

$$\theta_1=\varphi_{xx}/\varphi_x\ ,\qquad s=\{\varphi;x\}\ ,\eqno (3.38)$$

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...x}+3z_{1x}\theta_1+ z_1\theta_1^2\}\ ;\cr\crcr}}\,\eqno (3.39)$$

and

$$\left(\matrix{\varphi_t/\varphi_x\cr \noalign{\vskip 5pt} z_t... ...skip 5pt} s+\textstyle{3\over 2}z^2\cr} \right)\ ,\eqno(3.40)$$

$$P=-C^{\ast}\Omega_1^{-1} C\Omega\ .\eqno (3.41)$$

Furthermore, Eqs. (3.35) are invariant under the transformations

$$\left(\matrix{\theta_1\cr \noalign{\vskip 5pt} z_1\cr}\right)... ...theta_2\cr \noalign{\vskip 5pt} z_2\cr}\right)\ ,\eqno (3.42)$$

$$\left(\matrix{\theta_1\cr \noalign{\vskip 5pt} z_1\cr}\right)... ...\theta_3\cr \noalign{\vskip 5pt} z_3\cr}\right)\ ,\eqno (3.43)$$

where

$$A_{\pm}=\left(\matrix{-\textstyle{1\over 2}&\mp\textstyle{3\o... ...style{1\over 2}&-\textstyle{1\over 2}\cr}\right)\ .\eqno (3.44)$$

In addition,

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... .../2-(theta_{2x}-\theta_2^2/2) \bigr)\ ,\cr\crcr}}\,\eqno (3.45)$$

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... .../2-(theta_{3x}-\theta_3^2/2) \bigr)\ ,\cr\crcr}}\,\eqno (3.46)$$

also define solutions $(U_3,H_3)$, $(U_4,H_4)$ of Eqs. (3.34).

Proof.    By (3.21), Eqs. (3.35) are invariant under (3.42), (3.43), and the Miura transformations (3.39), (3.45), and (3.46) from (3.35) to (3.34) are well defined. Now, the identity (when $\theta_1=\varphi_{xx}/\varphi_x$)

$$\left(\matrix{D(D+\theta_1)&0\cr \noalign{\vskip 5pt} 0&1\cr}... ...\theta_1\cr \noalign{\vskip 5pt} z\cr}\right)_t\ ,\eqno (3.47)$$

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ... -s-\textstyle{3\over 2}z^2\cr}\right)\cr\crcr}}\,\eqno (3.48)$$

establishes that $(\theta_1,z)$ is a solution of (3.35), with $\theta_1=\varphi_{xx}/\varphi_x$.

By evaluation of (3.41)

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ... 0&-\textstyle{1\over 3}D\cr}\right)\ ,\cr\crcr}}\,\eqno (3.49)$$

where $s=\{\varphi;x\}$. Thus, by the invariance of the derivative $s$ under the Moebius group and the form of Eqs. (3.40), Eqs. (3.40) are invariant under the transformation

$$\varphi=(a\psi+b)/(c\psi+d)\ , z=z\ .\eqno (3.50)$$

In particular, Eqs. (3.40) are invariant under

$$\varphi=1/\psi\ ,\qquad z=z\ .\eqno (3.51)$$

However,

$$\theta_1={\varphi_{xx}\over\varphi_x}={\psi_{xx}\over\psi_x}-... ...={\psi_{xx}\over\psi_x}+2{\partial\over\partial x}\ln\varphi\ ,$$

which is the BT (3.37) with

$$\theta=\psi_{xx}/\psi_x\ ,\eqno (3.52)$$

and by (3.51) and the previous remarks $(\theta,z)$ is a solution of (3.35). Furthermore, from (3.36), (3.39), (3.51) and (3.52), we find that

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...^3+z\theta_x+3z_x\theta+z\theta^2)\ ,\cr\crcr}}\, \eqno (3.53)$$

demonstrating, by the previous remarks, that $(U,H)$ are solutions of (3.34), completing the proof.

Remark 1.    In certain instances it is preferable to express the equation sequences in terms of the recursion operators of conserved covariants, rather than the ``symmetries". We find for Eqs. (3.34), (3.35), and (3.40) that

$$\left(\matrix{U\cr H\cr}\right)_t=\Omega_1 J^n\left(\matrix{-U_{xx}/ 3-U^2\cr \noalign{\vskip 5pt} H\cr}\right)\ ,\eqno (3.54)$$

$$\left(\matrix{\theta\cr z\cr}\right)_t=\Omega_2 K^n\left( \m... ... 2}\theta^2- \textstyle{3\over 2}z^2\cr}\right)\ ,\eqno (3.55)$$

$$\left(\matrix{\varphi_t/\varphi_x\cr \noalign{\vskip 5pt} z... ..._{xx}/3-U^2\cr \noalign{\vskip 5pt} H\cr}\right)\ ,\eqno(3.56)$$

where

$$\Omega_1^{-1} C\Omega\left(\matrix{z\cr \noalign{\vskip 5pt} ... ...{xx}/3-U^2\cr \noalign{\vskip 5pt} H\cr}\right)\ ,\eqno (3.57)$$

and

$$J=\Omega_1^{-1} B\Omega_2 B^{\ast}\ ,\eqno (3.58)$$

$$K=B^{\ast}\Omega_1^{-1} B\Omega_2\ .\eqno (3.59)$$

Remark 2.    By applying the operator $R$ (3.26b) to the sequence (3.35), using (3.24), (3.25), (3.28), (3.32), (3.33), the sequence of Hamiltonian systems,

$$\left(\matrix{x\cr z\cr}\right)_t=(-L_2)^n\Omega\left( \matr... ...skip 5pt} s+\textstyle{3\over 4}z^2\cr}\right)\ ,\eqno (3.60)$$

is found. From (3.45) and (3.46) we have the Miura transformations

(i) $U=z_x-z^2/2+\textstyle{1\over 3}s\ ,\qquad H=\textstyle{1\over 3}(D-2z)(z_x-z^2/2-s)$ ;(3.61)

(ii) $U=-z_x-z^2/2+\textstyle{1\over 3}s\ ,\qquad H=\textstyle{1\over 3}(D+2z)(z_x+z^2/2+s)$ ;(3.62)

connecting (3.60) to (3.34). From (3.17), (3.23), and (3.35) it is easy to see that (3.35) is invariant under

$$z\rightarrow -z\ ,\eqno (3.63)$$

when $n=2j+1$, $j\ge 0$. By construction the same invariance applied to (3.40) and (3.60). Therefore, when

$$n=2j+1\ ,\qquad j\ge 0\ ,\eqno (3.64)$$

a consistent reduction of (3.35), (3.40), (3.60) is to let

$$z\equiv 0\ .\eqno (3.65)$$

The Miura transformations (3.39), (3.45), and (3.46) are

(i) $U=-\textstyle{2\over 3}(\theta_{1x}+\textstyle{1\over 4} \theta_1^2)\ ,\qquad H=0$ ;(3.66)

(ii) $U=\textstyle{1\over 3}(\theta_{2x}-\theta_2^2/2) \ ,\qquad H=-U_x$ ;(3.67)

(iii) $U=\textstyle{1\over 3}(\theta_{3x}-\theta_3^2/2) \ ,\qquad H=U_x$ .(3.68)

For (3.66) we let

$$\theta_1=-2a\ ,\qquad b=a_x-\textstyle{1\over 2}a^2\ ,\eqno (3.69)$$

$$U=\textstyle{4\over 2}b\ ,\eqno (3.70)$$

and find from Eq. (3.34) that

$$b_t=\textstyle{4\over 27}m_3^j\Omega_3(b_{xx}+4b^2)\ ,\eqno (3.71)$$

for $j=0,1,2,\ldots$, where

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...D^{-1}(D-2a)(D-a)D(D+a)(D+2a)D^{-1}\ .\cr\crcr}}\,\eqno (3.72)$$

For (3.67), (3.68), with

$$s=\theta_{2x}-\theta_2^2/2=\theta_{3x}-\theta_3^2/2$$

or

$$s=\theta_x-\theta^2/2\ ,\eqno (3.73)$$

Equation (3.60) obtains

$$s_t=\textstyle{4\over 27}m_4^j\Omega_4(s_{xx}+\textstyle{1\over 4}s^2)\ ,\eqno (3.74)$$

for $j=0,1,3,\ldots$, where

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...eft(D+{\theta\over 2} \right)D^{-1}\ .\cr\crcr}}\,\eqno (3.75)$$

Equations (3.71) and (3.74) are the sequences of Kuperschmidt/Caudrey-Dobb-Gibbon equations, respectively [4].

To continue the analysis of the Boussinesq sequence, it is necessary to define the discrete symmetries of the modified Boussinesq equations (3.42) and (3.43), as Bäcklund transformations for the singular manifold equation (3.40). That is,

$$\left(\matrix{\varphi_{xx}/\varphi_x\cr \noalign{\vskip 5pt} ... ...}/\psi_x\cr \noalign{\vskip 5pt} z_1\cr}\right)\ .\eqno (3.76)$$

In this way the investigation of the singularities for the Boussinesq and the modified Boussinesq sequences is referred to an investigation of the singularities for the sequence (3.40), which, as in Sec. II, allows a simplified discussion. To begin, for a solution$(\theta,z)$ of (3.35), we define variables $\psi,z)$ by

$$\psi_{xx}/\psi_x=\theta\ ,\qquad z=z\ .\eqno (3.77)$$

Therefore, $\psi$ is determined up to two arbitrary functions of $t$.

On the other hand, with the identification (3.77), $(\psi,z)$ satisfies Eq. (3.40) with the possible inclusions of a term from the null space of the operator,

$$T=\left(\matrix{D(D+\theta)&0\cr 0&1\cr}\right)\ .\eqno (3.78)$$

The general form of a null vector, when $\theta=\psi_{xx}/\psi_x$, is

$$\overline{n}=\left(\matrix{a/\psi_x+b\psi/\psi_x\cr 0\cr}\right)\ , \eqno (3.79)$$

where $(a,b)$ are functions of $t$. Therefore, for an arbitrary $(\psi,z)$ satisfying (3.77),

$$\left(\matrix{\psi_t/\psi_x+a/\psi_x+b(\psi/\psi_x)\cr \noal... ...vskip 5pt} s+\textstyle{3\over 2}z^2\cr}\right)\ ,\eqno (3.80)$$

where $s=\{\psi;x\}$. Now, the right side of (3.80) is expressed entirely in terms of the variables $(s,z)$, which implies that the right side is unchanged in form by the transformation

$$\psi\rightarrow e^{-\int^t b}\left\{\psi_1-\int^t ae^{\int^t b} ds \right\}\ ,\eqno (3.81)$$

where $(\psi_1,z)$ satisfies (3.40). Thus for an appropriate choice of the time-dependent ``constants" of integration, there exists a solution of (3.77) [for ``arbitrary" $(\theta,z)$] so that $(\psi,z)$ satisfies (3.40). From (3.81),

$$\psi_{xx}/\psi_x=\psi_{1xx}/\psi_{1x}=\theta\ .\eqno (3.82)$$

Furthermore, $(\psi_1,z)$ is uniquely determined up to transformations of the form

$$\psi_1=a\psi+b\ ,\eqno (3.83)$$

where $(a,b)$ are (time-independent) constants, and [modulo (3.83)] the transformation $(\theta,z)\leftrightarrow(\psi_1,z)$ is one to one. Therefore, the Bäcklund transformation (3.76) is well defined for Eqs. (3.40). Alternatively, let $(\psi,z)$ be a known solution of (3.40) and, applying (3.76), substitute for $(\varphi_{xx}/\varphi_x, z)$ in the right side of (3.40). By the invariance of (3.35), the equation for $z$ is satisfied identically, while $\varphi_t/\varphi_x$ is a known function of $(x,t)$, as is $\varphi_{xx}/\varphi_x$, which determined $\varphi$ uniquely up to the equivalence (3.83). In a similar way it can be shown that

$$s=\{\varphi;x\}\ ,\qquad z=z\eqno (3.84)$$

define a transformation from (3.60) to (3.40) which determines a unique $\varphi$, [modulo (3.50)], as a solution of(3.40).

We next propose to classify the singularities of (3.40) according to their ``leading-order" behavior and observe the effect of the transformations (3.50) and (3.76) on these singularities.

Recall from Sec. II that Eqs. (3.40) have, when $n=0$, two types of singularities, (2.34) and (2.35). With the notation

$$\varphi_{xx}/\varphi_x\simeq k\epsilon^{-1}+\cdots~,\qquad z\simeq \beta\epsilon^{-1}+\cdots~,\eqno (3.85)$$

these are represented, to the leading order, by Table I, where $\alpha=k+1$, $\alpha_+=-\alpha_-$. To the leading order the symmetry (3.76) is represented by the transformation

$$k^{\prime}=-\textstyle{1\over 2}k\mp\textstyle{3\over 2}\beta... ...m\textstyle{1\over 2}k-\textstyle{1\over 2}\beta\ ,\eqno (3.86)$$

and the inversion, $\varphi\rightarrow 1/\varphi^{\prime}$ by

$$\alpha^{\prime}=-\alpha\ .\eqno (3.87)$$

In the expansion of $\varphi$ in (3.40) we have

$$\varphi=\varphi_0\epsilon^{\alpha}+\cdots~,\eqno (3.88)$$

hence, (3.87). Note (3.87) does not apply to singularities of the form

$$\varphi=\varphi_0+\varphi_1\epsilon^{\alpha}\ ,\eqno (3.89)$$

when real $(\alpha)>0$. [See (2.34)]. Thus (3.87) does not apply to the last line of Table I. The entries in the left and right side of Table I are, however, separately closed under (3.86). The above remarks will apply to the entire Boussinesq sequence.

Now by a leading-order analysis it is possible to establish that ail singularities of the sequence (3.40) are of the form (3.85), where $k$ or $\beta$ might vanish separately. Thus, it is required to find the values of $(k,\beta)$ that are consistent with (3.85) for each equation in the sequence (3.40). With (3.85),

$$\widehat{V}_0=\left(\matrix{z\cr \noalign{\vskip 5pt} \{\var... ...r 2}((k+1)^2-2)\bigr\}\epsilon^{-2}\cr} \right)\ ,\eqno (3.90)$$

where $\epsilon=x+\epsilon(t)$. And, using (3.41),

$$p^j\widehat{V}_0\simeq\widehat{P}_j\left(\matrix{\epsilon^{-m... ... \noalign{\vskip 5pt} \epsilon^{-m-1}\cr}\right)\ ,\eqno (3.91)$$

where $m=3j+1$, $j=0,1,2,3,\ldots$ and $\widehat{P}_j=\widehat{P}_j(k,\beta,m)$, $\widehat{P}_0=\widehat{V}_{0j}$. Also,

$$\widehat{P}_j=A_{j-1}\widehat{P}_{j-1}\ ,\eqno (3.92)$$

where, from (3.41),

$$A_j=C_j^{\ast}\Omega_j^{-1} C_j\Omega_j\ ,\eqno (3.93)$$

Table I.    $n=0$.

$\alpha_+$		$k$		$\beta$			$\alpha_-$		$k$		$\beta$
$-$1		$-$2		0			1		0		0
2		1		$\pm$1

$$C_j^{\ast}=\left(\matrix{1&2\beta-m-2\cr \noalign{\vskip 5pt}... ...(m+2)\beta\cr &+(m+2)(m+3)+1-(k+1)^2\cr}\right)\ ,\eqno (3.94)$$

$$\Omega_{1j}^{-1}\left(\matrix{0&1/(m+4)\cr \noalign{\vskip 5pt} 1/(m+3)&0\cr}\right)\ ,\eqno (3.95)$$

$$C_j=\left(\matrix{1&-3(\beta+m+2)\cr \noalign{\vskip 5pt} 2\... ...(m+3)\beta+(m+2)(m+3)\cr &+1-(k+1)^2\cr}\right)\ ,\eqno (3.96)$$

$$\Omega_j=\left(\matrix{(k+m+2)(k-m)&0\cr \noalign{\vskip 5pt} 0&\textstyle{1\over 3}\cr}\right)\ ,\eqno (3.97)$$

and $m=3j+1$. Consider the $(j+1)$-th equation in sequence (3.40). We require that (i) the leading-order term

$$P^{j+1}\widehat{V}_0\simeq\widehat{P}_{j+1}\left(\matrix{\eps... ...noalign{\vskip 5pt} \epsilon^{-m-4}\cr}\right)\ .\eqno (3.98)$$

$$m=3j+1$$

vanishes. Or, when

$$\varphi=\varphi_0+\varphi_{k+1}\epsilon^{k+1}+\cdots\ ,\eqno (3.99)$$

with $\varphi_0=\varphi_0(t)\ne 0$

$${\varphi_t\over\varphi_x}\simeq {\varphi_{0t}\over(k+1)\varphi_{k+1}} \epsilon^{-k}\ ,\eqno (3.100)$$

that

(ii) $${\left(\matrix{\bigl[\varphi_{0t}/(k+1) \varphi_{k+1}\bigr]\epsil... ...\matrix{\epsilon^{-m-3}\cr \noalign{\vskip 5pt} \epsilon^{-m-4}\cr}\right)\ .}$$(3.101)

In case (i), we have

$$A_j\widehat{P}_j\equiv 0\ ,\eqno (3.102)$$

which, by (3.92), includes the leading-order conditions of this type for all the preceding equations in the sequence. Therefore, it is sufficient (by recursion) to evaluate (3.102) when

$$\widehat{P}_j\ne 0\ ,\qquad\det\vert A_j\vert =0\ .\eqno (3.103)$$

In case (ii) it can be shown that

$$\varphi_{0t}\simeq (k+1)\varphi_{k+1}\eqno (3.104)$$

and (3.101) becomes

$$\widehat{P}_{j+1}=\left(\matrix{c\cr 0\cr}\right)\ ,\eqno (3.105)$$

where

$$k=m+3\ .\eqno (3.106)$$

In both cases (3.102) and (3.105) are polynomials in $(k,\beta,m)$ that determine the allowed values of $(k,\beta)$ in (3.85). The zeroth-order equation is evaluated in Table I. The first-order equation is

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...xx}+10s_xz_x+5z^3s+5zs^2\cr}\right)\ .\cr\crcr}}\,\eqno (3.107)$$

For this equation we find the results in Table II, which is found by solving (3.102), (3.103), and (3.105) with $j=0$. The complete list of singularities for this equation is found by striking the last line from Table I and adjoining it to Table II. The upper block of singularities (type 1) corresponds to solving (3.102) and (3.103) with

$$\det(C_0^{\ast}\Omega_{10}^{-1} C_0)=0\ .\eqno (3.108)$$

The middle block (type 2) corresponds to

$$\det(\Omega_0)=0\eqno (3.109)$$

and the lower block (type 3) to (3.105) with $j=0$.

We now claim that the solution of (3.102), (3.103), and (3.105) for the $(j+1)$-th equation is shown in Tables III and IV. In Tables III and IV the type 1, 2, and 3 blocks of singularities are identified as before. The following observations are straight forward to verify [using (3.86)]. Identifying blocks of singularities in Table III or IV as left (L) or right (R) and type 1, 2, or 3; then within a fixed table, we have the following.

(1) The values of $(k,\beta)$ in the sets (i) (type 3, type 1L) and (ii) (type 1R, type 2R) are invariant under (3.86).

(2) (i) Any singularity of type 3 can be mapped into a singularity of type IL by (3.86). (ii) Any singularity of type 1R can be mapped into a singularity of type 2R by (3.86).

(3) Under the transformation, $\varphi\rightarrow 1/\varphi$, (i) type 1L $\leftrightarrow$ type 1R and (ii) type 2L $\leftrightarrow$ type 2R.

(4) Since the singularities of type 2L correspond (with $m=3j+1$) identically to what would be the type 3 with $m=3(j-1)+1$, every singularity of type 2L$(j)$ can (by observation 2) be mapped into a singularity of type 1L$(j-1)$. Recall that to obtain all the singularities of the $(j+1)$-th equation, it is required to adjoin the types obtained from Tables III or IV with $m\rightarrow m-3,m-6$, etc., deleting in each instance the type 3 block.

(5) By a recursive application of observations (2)-(4), all the singularities described in Tables III and IV can be mapped into the first line of Table I.

Now it is easy to show that any singularity of Eq. (3.40) with $k=-2$, $\beta=0$ is (1) meromorphic and (2) depends on the maximum number of arbitrary ``constants" allowed for by the differential equation. (See Sec. II and [4]). By the obvious reconstructions, all the singularities mapped by (3.86) and (3.87) into the one with $(k=2,\ \beta=0)$ will be meromorphic. Therefore, if the claim that Tables III and IV represent the general forms of allowed singularities is valid, the above remarks demonstrate that the sequence (3.40), and, by implication, the Boussinesq sequence, identically posses the Painlevé property.

Table II.    First-order equation.

$\alpha_+$		$k$		$\beta$			$\alpha_-$		$k$		$\beta$
$-$7		$-$8		0			7		6		0
$-$4		$-$5		$\pm$1			4		3		$\mp$1
$-$1		$-$2		$\pm$2			1		0		$\mp$2
2		1		$\pm$3			$-$2		$-$3		$\mp$3
2		1		$\pm$1			$-$2		$-$3		$\pm$1
5		4		0
5		4		$\pm$2
5		4		$\pm$4

Table III.    $(j+1)$-th equation, $m=3j+1$ even.

$\alpha_+$		$k$		$\beta$			$\alpha_-$		$k$		$\beta$
$-2m-5$		$-2m-6$		0			$2m+5$		$2m+4$		0
$-2m-2$		$-2m-3$		$\pm 1$			$2m+2$		$2m+1$		$\pm 1$
$-2m+1$		$-2m$		$\pm 2$			$2m-1$		$2m-2$		$\mp 2$
$\vdots$		$\vdots$		$\vdots$			$\vdots$		$\vdots$		$\vdots$
$m+1$		$m$		$\pm(m+2)$			$-m-1$		$-m-2$		$\pm(m+2)$
$m+1$		$m$		0			$-m-1$		$-m-2$		0
$m+1$		$m$		$\pm 2$			$-m-1$		$-m-2$		$\mp 2$
$\vdots$		$\vdots$		$\vdots$			$\vdots$		$\vdots$		$\vdots$
$m+1$		$m$		$\pm m$			$-m-1$		$-m-2$		$\mp m$
$m+4$		$m+3$		$\pm 1$
$\vdots$		$\vdots$		$\vdots$
$m+4$		$m+3$		$\pm(m+1)$
$m+4$		$m+3$		$\pm(m+3)$

The preceding remarks show that Tables III and IV contain allowed forms of singularities [values of $(k\beta)$] for the $(j+2)$-th equation. We show now that, according to the degrees of various polynomials in $\beta$ defined by conditions (3.102), (3.103), and (3.105), the tables contain every solution $(k,\beta)$ of these conditions.

For singularities of type 1, it is found from (3.93)-(3.95) and (3.103) that $\det\vert C_j\vert$ vanishes when

$$k+1=\pm(3\beta+2m+5)\ ,\eqno (3.110)$$

Table IV.    $(j+1)$-th equation, $m=3j+1$ odd.

$\alpha_+$		$k$		$\beta$			$\alpha_-$		$k$		$\beta$
$-2m-5$		$-2m-6$		0			$2m+5$		$2m+4$		0
$-2m-2$		$-2m-3$		$\pm 1$			$2m+2$		$2m+1$		$\pm 1$
$-2m+1$		$-2m$		$\pm 2$			$2m-1$		$2m-2$		$\mp 2$
$\vdots$		$\vdots$		$\vdots$			$\vdots$		$\vdots$		$\vdots$
$m+1$		$m$		$\pm(m+2)$			$-m-1$		$-m-2$		$\pm(m+2)$
$m+1$		$m$		$\pm 1$			$-m-1$		$-m-2$		$\mp 1$
$m+1$		$m$		$\pm 3$			$-m-1$		$-m-2$		$\mp 3$
$\vdots$		$\vdots$		$\vdots$			$\vdots$		$\vdots$		$\vdots$
$m+1$		$m$		$\pm m$			$-m-1$		$-m-2$		$\mp m$
$m+4$		$m+3$		0
$\vdots$		$\vdots$		$\vdots$
$m+4$		$m+3$		$\pm(m+1)$
$m+4$		$m+3$		$\pm(m+3)$

and then $\det\vert C_j^{\ast}\vert$ vanishes when

$$k+1=\pm(3\beta+2m-5)\ .\eqno (3.111)$$

For singularities of type 2, $\det\vert\Omega_j\vert$ vanishes when

$$k+1=\pm(m+1)\ ,\eqno (3.112)$$

and for singularities of type 3, by (3.106),

$$k+1=m+4\ .\eqno (3.113)$$

Therefore $k$ is either a linear or constant function of $\beta$ and substitution for $k$ determines (3.102), (3.103), and (3.105) as polynomial conditions for $\beta$, which depend on the index $m$.

In all cases, by (3.90),

$$\widehat{V}_{0j}\simeq\left(\matrix{\beta\cr \noalign{\vskip 5pt} \beta^2\cr}\right)\ ,\eqno (3.114)$$

where the equivalence indicates the highest power of $\beta$ in an expression. For type 1, by (3.93) to (3.97); and (3.110), (3.111),

$$A_i\simeq\left(\matrix{\beta^3&\beta^2\cr \noalign{\vskip 5pt} \beta^4&\beta^3\cr}\right)\ ,\eqno (3.115)$$

for $i=0,1,\ldots,j-1$. Now, for (3.110) with $\det\vert C_j\vert=0$,

$$C_j\Omega_j\simeq\left(\matrix{\beta^2&\beta\cr \noalign{\vskip 5pt} \beta^3&\beta^2\cr}\right)\ ,\eqno (3.116)$$

and by the above,

$$A_j\widehat{P}_j\simeq \left(\matrix{\beta^{3j+3}\cr \noalign{\vskip 5pt} \beta^{3j+3}\cr}\right)\ ,\eqno (3.117)$$

When $\det\vert C_j^{\ast}\vert=0$, by (3.93) and (3.111),

$$C_j^{\ast}\Omega_{ij}^{-1}\simeq \left(\matrix{\beta^3&\beta^... ...\noalign{\vskip 5pt} \beta^4&\beta^3\cr}\right)\ ,\eqno (3.118)$$

and

$$A_j\widehat{P}_j\simeq \left(\matrix{\beta^{3j+4}\cr \noalign{\vskip 5pt} \beta^{3j+5}\cr}\right)\ ,\eqno (3.119)$$

Now using the definition of $m$,

$$m=3j+1\ ,\eqno (3.120)$$

condition (3.117) determines $m+2$, and condition (3.119) $m+3$ solutions for $\beta$ which equals the number $(2m+5)$ of (allowed) solutions of type 1 in a column of Table III or IV. The separate determinations of $k+1$ in (3.110) or (3.111) complete the left or right columns.

For singularities of type 2, by (3.112),

$$A_i\simeq\left(\matrix{\beta&\beta^2\cr \noalign{\vskip 5pt} \beta^2&\beta^3\cr}\right)\ ,\eqno (3.121)$$

for $i=0,1,\ldots,j-1$, and

$$\Omega_j\simeq\left(\matrix{0&0\cr 0&1\cr}\right)\ .\eqno (3.122)$$

By the above,

$$A_j\widehat{P}_j\simeq\left(\matrix{0\cr \noalign{\vskip 5pt} \beta^{3j+2}\cr}\right)\ ,\eqno (3.123)$$

which determines $m+1=3j+2$ solutions. This is equal to the number of type 2 solutions in Tables III or IV, where the separate determinations of $k+1$ in (3.112) complete the left or right columns.

For singularities of type 3, (3.121) is valid for $i=0,1,2,\ldots,j,j+1$ and

$$\widehat{P}_{j+1}\simeq\left(\matrix{C\cr 0\cr}\right)\simeq\... ...\noalign{\vskip 5pt} \beta^{3j+5}\cr}\right)\ . \eqno (3.124)$$

This determines $3j+5=m+4$ solutions which equals the number of type 3 singularities in Table III or IV.

Therefore all singularities have been accounted for and the Boussinesq sequence has the Painlevé property.