Appendix B:    Some Seventh-Order Equations

We consider when the equation

$${\varphi_t\over\varphi_x}+{\partial^4\over\partial x^4}\{\var... ...l x}\{\varphi;x\} \right)^2+\lambda\{\varphi;x\}^3=0\eqno (B1)$$

has a transformation

$$\varphi_x=\psi_x^m\eqno (B2)$$

preserving the form of (B1).

Directly,

$$\{\varphi;x\}=m{\psi_{xxx}\over\psi_x}-\left({m^2\over 2}+m\right) {\psi_{xx}^2\over\psi_x^2}\eqno (B3)$$

and

$$\varphi_{xt}=m\psi_x^{m-1}\psi_{xt}\ .\eqno (B4)$$

We note that

$$m\psi_x^{m-1}\psi_{xt}={\partial\over\partial x}(\psi_x^m F)=... ...{\partial\over\partial x}F+m\psi_x^{m-1} \psi_{xx} F\eqno (B5)$$

or Therefore, for Eq. (B6) to be of the form (B1)

$$\psi_x{\partial\over\partial x}F+m\psi_{xx} F={\partial\over\partial x}G\ ,\eqno (B7)$$

where $G$ is a functional of $\psi_x$. Expressions on the lhs of (B7) that are not ``gradients'' must vanish. In this case, we find:

(i) Term $\psi_{xx}\psi_{xxxx}^2/\psi_x^2$ obtains the condition

$$2m+7+2m(\alpha-\beta)=0\ .\eqno (B8)$$

(ii) Term $\psi_{xx}\psi_{xxx}^2/\psi_x^3$ obtains the condition

$$17m+42+\textstyle{1\over 2}\alpha m(9m+28)-6\beta m(m+3)-3\lambda m^2=0\ .\eqno (B9)$$

(iii) Term $\psi_{xxx}^3\psi_{xxx}^2/\psi_x^4$ obtains the condition

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...beta m(m^2-5m-16)+3\lambda m^2(m+2)=0\ .\cr\crcr}}\,\eqno (B10)$$

(iv) Term $\psi_{xx}^7/\psi_x^6$ obtains the condition

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...3\over 4}\lambda m^2(m^3+m^2-8m-12)=0\ .\cr\crcr}}\,\eqno (B11)$$

Equation (B8)-(B11) have the following solutions:

(i) $m=-1\ ,\qquad\alpha=\beta+ {5\over 2}\ ,\qquad 6\lambda=5\beta+{5\over 2}\ ,$(B12)

(ii) $m=-2\ ,\qquad\alpha=\beta+ {3\over 4}\ ,\qquad 6\lambda=\beta+{1\over 4}\ ,$(B13)

(iii) $m=-{1\over 2}\ ,\qquad\alpha =12\ ,\qquad\beta=6\ ,\qquad\lambda={32\over 3}\ ,$(B14)

(iv) $m=-{1\over 3}\ ,\qquad\alpha =26\ ,\qquad\beta={33\over 2}\ ,\qquad\lambda=48\ ,$(B15)

(v) $m=-{2\over 3}\ ,\qquad\alpha =5\ ,\qquad\beta={3\over 4}\ ,\qquad\lambda={3\over 4}\ ,$(B16)

Further calculation obtains that Eq. (B6) will be of the form (B1) when

(i) $m=-1\ ,\qquad\alpha=5\ ,\qquad \beta={5\over 2}\ ,\qquad\lambda={5\over 2}\ ,$

(ii) $m=-2\ ,\qquad\alpha= {3\over 2}\ ,\qquad\beta={3\over 4}\ ,\qquad\lambda={1\over 6}\ ,$(B17)

(iii) $m=-{1\over 2}\ ,\qquad\alpha =12\ ,\qquad\beta=6\ ,\qquad\lambda={32\over 3}\ ,$

The transformations defined by (B15) and (B16) do not preserve the form of Eq. (B1).