GMPDE

Generalized Microstructure PDE

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

Failed to parse (syntax error): {\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 $A\left(z\right)=\ell sech^{2}\left(kz\right)$ when ${\frac {c^{2}-b}{\delta \left(\beta c^{2}-\gamma \right)}}>0$. The constants $k$ and $\ell$ depend on $\beta ,\delta ,\gamma ,\mu$ and $c$ is the wave speed after the coordinate transformation $z=x-ct$.

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.

[Abtstract for Solitary wave families of a generalized microstructure PDE]