# GMPDE

### Generalized Microstructure PDE

One dimensional wave propagation in microstructured solids has recently been modeled by an equation

$\displaystyle v_{tt} - b v_{xx} − \frac{\mu }{2}\left( v \right)_{xx} − \delta \left(\beta v_{tt} - \gamma v_{xx} \right)_{xx} = 0$

It admits embedded solitons of the form ${\displaystyle A\left(z\right)=\ell sech^{2}\left(kz\right)}$ when ${\displaystyle {\frac {c^{2}-b}{\delta \left(\beta c^{2}-\gamma \right)}}>0}$. The constants ${\displaystyle k}$ and ${\displaystyle \ell }$ depend on ${\displaystyle \beta ,\delta ,\gamma ,\mu }$ and ${\displaystyle c}$ is the wave speed after the coordinate transformation ${\displaystyle z=x-ct}$.

This equation has the property of being a reversible system, and therefore one may find the exact solution without resorting to the Inverse Scattering Transform by using properties of reversible systems to recast the equation in terms of a bilinear operator.