The Boussinesq Equation

The Boussinesq equation

$$U_{tt}=-{\partial^2\over\partial x^2}\left({U_{xx}\over 3}+U^2\right)\eqno (2.1)$$

is known to possess the Painlevé property [1], [2]. That is, about a ``manifold" of ``movable" singularities determined by the expression

$$\varphi(x,t)=0\ ,\eqno (2.2)$$

the Boussinesq equation has the expansion

$$U=\varphi^{-2}\sum_{j=0}^{\infty} U_j\varphi^j\ ,\eqno (2.3)$$

where $(\varphi,U_4,U_5,U_6)$ are ``arbitrary", locally analytic functions of $(x,t)$. In general, for the expansion (2.3) to be well defined about the manifold (2.2), it is required that (2.2) be ``noncharacteristic" for the equation (2.1) (i.e., the Cauchy-Kovalevskaya theorem). In the present case, this requires that $\varphi_x\ne 0$ when $\varphi=0$. With this provision, (2.3) defines the general (meromorphic) expansion of the solution about (2.2).

From the recursion relations for $U_j$ [substituting (2.3) into (2.1)] it is found that

$$U_0=-2\varphi_x^2\ ,\eqno (2.4)$$

$$U_1=2\varphi_{xx}\ ,\eqno (2.5)$$

$$\varphi_t^2-\varphi_{xx}^2+\textstyle{4\over 3}\varphi_x\varphi_{xxx}+ 2\varphi_x^2 U_2=0\ ,\eqno (2.6)$$

$$\varphi_{tt}+\textstyle{1\over 3}\varphi_{xxxx}+2\varphi_{xx}U_2-2\varphi_x^2 U_3=0\ ,\eqno (2.7)$$

and $(U_4,U_5,U_6)$ are ``arbitrary" [1]. We note that the noncharacteristic condition is (essentially) $U_0\ne 0$ when $\varphi-0$.

We now attempt to define a Bäcklund transformation for Eq. (2.1) by truncating the expansion (2.3) at the ``constant" level. That is, let

$$U=U_0\varphi^{-2}+U_+\varphi^{-1}+U_2\ ,\eqno (2.8)$$

and require, in the expressions defined by the recursion relations, that

$$U_j\equiv 0\ ,\eqno (2.9)$$

for $j\ge 3$. In general, we would expect to obtain an overdetermined system of equations for $(\varphi,U_0,U_1,U_2)$. In this case, the system is not overdetermined. The $(U_0,U_1)$ are determined by (2.4) and (2.5), and the $(\varphi,U_2)$ are defined by (2.6) and (2.7), with $U_3=0$. Since $(U_4,U_5,U_6)$ are arbitrary they may be set to zero without restriction. The system terminates at the condition $U_6=0$, obtaining that $U_2$ satisfies Eq. (2.1) as a (trivial) consequence of Eqs. (2.4)-(2.7) (with $U_3=0$). Solving for $(U_2, \varphi)$, the Bäcklund transformation reads

$$U=2{\partial^2\over\partial x^2}\ln\varphi+U_2\ ,\eqno (2.10)$$

where $U=(U,U_2)$ satisfy Eq. (2.1),

$$2U_2+{\varphi_t^2\over\varphi_x^2}-{\varphi_{xx}^2\over\varphi_x^2}+ {4\over 3}{\varphi_{xxx}\over\varphi_x}=0\ ,\eqno (2.11)$$

and

$${\partial\over\partial t}\left({\varphi_t\over\varphi_x}\righ... ...eft({\varphi_t\over\varphi_x}\right)^2\right)=0\ .\eqno (2.12)$$

The expression

$$\{\varphi;x\}={\partial\over\partial x}\left({\varphi_{xx}\ov... ...ver 2} \left({\varphi_{xx}\over\varphi_x}\right)^2\eqno (2.13)$$

is the Schwarzian derivative [2], which is invariant under the Moebius group

$$\varphi=(a\psi+b)/(c\psi+d)\ .\eqno (2.14)$$

By this Eq. (2.12) is also invariant under (2.14). Note that (2.11) is a Miura-type transformation from Eq. (2.12) to Eq. (2.1). In effect, Eq. (2.12) is a form of ``modified" Boussinesq equation. If we let

$$\nu=\varphi_{xx}/\varphi_x\ ,\eqno (2.15)$$

$$\omega=\varphi_t/\varphi_x\eqno (2.16)$$

and use the identity

$$\nu_t={\partial\over\partial x}\left({\partial\over\partial x}+\nu\right)\omega\ ,\eqno (2.17)$$

then $(\nu,\omega)$ satisfy the system of modified Boussinesq equations [7]:

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ... 2}\nu^2+{3\over 2}\omega^2 \right)\ .\cr\crcr}}\,\eqno (2.18)$$

The Miura transformation (2.11) is

$$2U_2+\omega^2+\textstyle{4\over 3}(\nu_x+\textstyle{1\over 4}\nu^2)=0\ .\eqno (2.19)$$

Since (2.19) maps the system (2.18) into the scalar equation (2.1), it is convenient to reformulate (2.1) as the system of equations

$$U_t=H_x\ ,\qquad H_t={\partial\over\partial x}\left(-{U_{xx}\over 3}-U_2\right)\ ,\eqno (2.20)$$

with the Miura transformation

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...x\omega+3\nu\omega_x+\nu^2\omega=0\ .\cr\crcr}}\, \eqno (2.21)$$

Now, the modified Boussinesq equations (2.18) are invariant under the transformation

$$\left(\matrix{\nu\cr \omega\cr}\right)=A_{\pm}\left(\matrix{\theta\cr z\cr}\right)\ ,\eqno (2.22)$$

where

$$A_{\pm}=\left(\matrix{-\textstyle{1\over 2}&\mp\textstyle{3\o... ...style{1\over 2}&-\textstyle{1\over 2}\cr}\right)\ ,\eqno (2.23)$$

and

(i) $\vert A_{\pm}\vert=1$ ,

(ii) $A_{\pm}^{-1}=A_{\mp}$ ,(2.24)

(iii)$A_{\pm}^3=I$ .

The Miura transformation (2.21) is

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...theta_x\pm {\theta^2\over 2}\right)\ ,\cr\crcr}}\,\eqno (2.25)$$

where

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...er 2}\theta^2+{3\over 2}z^2 \right)\ .\cr\crcr}}\,\eqno (2.26)$$

Equations (2.25) are linearized by the substitution

$$z=\mp\beta_x/\beta\eqno (2.27)$$

to

$$4\beta_{xxx}+6U\beta_x+3(U_x\pm H)\beta=0\ ,\eqno (2.28)$$

From Eqs. (2.26) there is found

$$\pm\beta_t=\beta_{xx}+(U+\lambda)\beta\ ,\eqno (2.29)$$

where $\lambda$ is a constant of integration. Equations (2.28) and (2.29) are the Lax pair for Eq. (2.20).

We recall that, for Eq. (2.12),

$$\nu=\varphi_{xx}/\varphi_x\ ,\qquad\omega=\varphi_t/\varphi_x\ .\eqno (2.30)$$

From the symmetry (2.22) of (2.18) we identify

$$\theta=\psi_{xx}/\psi_x\ ,\qquad z=\psi_t/\psi_x\ .\eqno (2.31)$$

Thus,

$${\varphi_{xx}\over\varphi_x}=-{1\over 2}{\psi_{xx}\over\psi_x... ..._{xx}\over\psi_x}-{1\over 2}{\psi_t\over\psi_x}\ .\eqno (2.32)$$

The compatibility condition

$$\varphi_{xxt}=\varphi_{txx}\eqno (2.33)$$

is satisfied by Eqs. (2.32) if and only if $\psi$ satisfies Eq. (2.12). Thus, Eqs. (2.32) constitute a Bäcklund transformation for (2.12). As previously noted, Eq. (2.12) is also invariant under the Moebius group. This dual invariance allows certain rational solutions to be constructed iteratively for Eq. (2.12) (see Appendix B).

Equation (2.12) allows two types of singularities. For one,

(i) $${\varphi=\epsilon^{-1}\sum_{j=0}^{\infty} \varphi_j\epsilon^j}$$ ,(2.34)

and for the other

(ii) $\varphi\simeq\varphi_0(t)+\varphi_2 \epsilon^2+\cdots~,$(2.35)

where

$$\varphi_{0_t}=\pm 2\epsilon_x\varphi_2\ .$$

Singularities of the form (2.35) occur at point where $\varphi_x=0$. By direct calculation both forms of singularity are single valued. As explained in [4], the form of Eq. (2.12) is sufficient to guarantee the meromorphic behavior of the singularity, (2.34). For instance, the invariance of Eq. (2.12) [under (2.14)]

$$\psi=1/\varphi\eqno (2.36)$$

throws the simple pole of $\varphi$ into a simple zero of $\psi$:

$$\psi=\epsilon\sum_{j=0}^{\infty}\psi_j\epsilon^j\ ,\eqno (2.37)$$

where $\psi$ is locally analytic near $\epsilon=0$. We note that, by the Cauchy-Kovalevsky theorem [9], (2.37) converges in an open neighborhood at the manifold ($\epsilon=0$).

For simplicity, let

$$\epsilon\rightarrow x+\epsilon(t)\ ,\eqno (2.38)$$

and find to leading order

(i) $\nu=\varphi_{xx}/\varphi_x\simeq-2/\epsilon\ ,\qquad\omega=\varphi_t/\varphi_x\simeq O(1)\ ,$

$$\left(\matrix{\nu\cr \omega\cr}\right)=\left(\matrix{-2\cr 0\cr} \right)\epsilon^{-1}\ ,\eqno (2.39)$$

for (2.34); and

(ii) $${\left(\matrix{\nu\cr \omega\cr}\right) \simeq\left(\matrix{1\cr \pm 1\cr}\right)\epsilon^{-1}\ ,}$$(2.40)

for (2.35). From (2.22),

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...,\ \left(\matrix{-2\cr -0\cr}\right)\ .\cr\crcr}}\,\eqno (2.41)$$

Thus, the singularities of Eqs. (2.12) and (2.18) are permuted by the symmetry (2.22) and (2.32). A singularity of the form (2.35) can be transformed into the form (2.34). Therefore, by reconstruction from (2.30) and (2.22), (2.35) is single valued.

In the next section it is found that all singularities of the Boussinesq sequence can be transformed into form (2.34) by a combination of the invariances (2.32) and (2.36).

At this point it is worth remarking that Eq. (2.12) is unique among equations of the form

$${\partial\over\partial t}\left({\varphi_t\over\varphi_x}\righ... ...left({\varphi_t\over\varphi_x}\right)^2\right)=0\ ,\eqno (2.42)$$

since only equations equivalent to (2.12) under scalings of $(x,t)$ have a set of nontrivial discrete symmetries [when expressed in the form (2.18)]. This will be relevant to the analysis of the nonlinear Schrödinger equation in Appendix A.