THE BOUSSINESQ EQUATION

The analysis for the Boussinesq equation [10]

$$u_{tt} + \frac{\partial^2}{\partial x^2}(\frac13 u_{xx} + u^2)=0$$ (44)

finds the Bäcklund - Darboux transformation

$$u = 2\frac{\partial^2}{\partial x^2}\ln\phi + u_2$$ (45)

where $\phi = v_1/v_2$ and $v_1,\;v_2$ satisfy

$$4v_{xxx} + 6u v_x + 3(u_x + h)v = 0$$

$$v_t=v_{xx} + (u + \lambda)v$$

$$h_x=u_t.$$

The Schwarzian modified equation is

$$\frac{\partial}{\partial t}(\phi_t/\phi_x) + \frac13\frac{\partial}{\partial x}(\{\phi;x\} + \frac32(\phi_t/\phi_x)^2 ) = 0 .$$ (46)

The discrete symmetries of the modified equations induce the invariant transformation of the Schwarzian equations

$$\frac{\phi_{xx}}{\phi_x} = -\frac12\frac{\psi_{xx}}{\psi_x} \mp \frac32\frac{\psi_t}{\psi_x}$$

$$\frac{\phi_t}{\phi_x} = \pm\frac12\frac{\psi_{xx}}{\psi_x} - \frac12\frac{\psi_t}{\psi_x}.$$ (47)