GMPDE

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Generalized Microstructure PDE

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

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

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

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]

GMPDE

From DispersiveWiki

Generalized Microstructure PDE

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