Appendix A:    Lax Pair and Bäcklund Transformations for the Caudrey-Dodd-Gibbon Equation

In Sec. 3 the Caudrey-Dodd-Gibbon equation

$$u_t+{\partial\over\partial x}(u_{xxxx}+30uu_{xx}+60u^3)=0\eqno (A1)$$

was found to have the Bäcklund transformation

$$u={\partial^2\over\partial x^2}\ln\varphi+u_2\ ,\eqno (A2)$$

where $u_2$ satisfies (A1) and

(i) $u_2=-{1\over 6}{\varphi_{xxx} \over\varphi_x}\ ,$(A3)

(ii) ${\varphi_t\over\varphi_x} +{\partial^2\over\partial x^2}\{\varphi;x\}+4\{\varphi;x\}^2=0\ .$(A4)

Equations (A3) and (A4) may be rewritten as the following ``Lax pair'':

$$\varphi_{xxx}+6u_2\varphi_x=0\ ,\eqno (A5)$$

$$\varphi_t=-18u_{2x}\varphi_{xx}+6(u_{2xx}-6u_2^2)\varphi_x\ .\eqno (A6)$$

With the exception that the spectral parameter vanishes, this is the Lax pair found in [4].

To obtain a Lax pair with the spectral parameter, it is necessary to generalize the procedures introduced in [2]. That is, we define a Bäcklund transformation (A2), where $(u,u_2)$ satisfy (A1). In Sec. 3 the resulting expressions were ordered according to the inverse powers of $\varphi$, i.e., (3.6iii, iv, and v). Herein, other than requiring that $u_2$ satisfy (A1) the various terms are collected into a single equation, obtaining

$${\partial^2\over\partial x^2}\left({\varphi_t\over\varphi}\ri... ...t({H_5\over\varphi} +{H_4\over\varphi^2}\right)=0\ ,\eqno (A7)$$

where

$$H_4=-\varphi_x^2\left\{{\partial^2\over\partial x^2}\{\varphi... ...vartheta^2+2\{\varphi;x\}\vartheta\bigr)\right\}\ , \eqno (A8)$$

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...eta^2+10\{\varphi;x\}\vartheta\right\}\ ,\cr\crcr}}\,\eqno (A9)$$

$$\vartheta=\{\varphi;x\}+6W\ ,\eqno (A10)$$

and

$$W=u_2+{1\over 4}{\varphi_{xx}^2\over\varphi_x^2}\ .\eqno (A11)$$

Now, letting

$$\vartheta=6\lambda\varphi/\varphi_x\ ,\eqno (A12)$$

it is found from (A10) and (A11) that

$$\varphi_{xxx}+6u_2\varphi_x=6\lambda\varphi\ .\eqno (A13)$$

From (A7)-(A9) and (Al2) there results

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...2\over \varphi_x^2}\right)\right\}=0\ .\cr\crcr}}\,\eqno (A14)$$

Setting the term inside the bracket equal to 0,

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...mbda^2{\varphi^2\over\varphi_x^2}=0 \ .\cr\crcr}}\,\eqno (A15)$$

Using (A13),

$$\varphi_t=(54\lambda-18u_{2x})\varphi_{xx}+6(u_{2xx}-6u_2^2)\varphi_x +216\lambda u_2\varphi\ .\eqno (Al6)$$

Equations (A13) and (A16) constitute the Lax pair for the Caudrey-Dodd-Gibbon equation [4], where $\lambda$ is the spectral parameter. We note that Eq. (A15) is not invariant under the Moebius group.