GMPDE: Difference between revisions
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It admits embedded solitons of the form <math> A\left(z\right) = \ell sech^2\left(k z\right) </math> when <math> \frac{ c^2 - b }{\delta\left(\beta c^2 - \gamma\right) } > 0</math>. The constants <math>k</math> and <math>\ell</math> depend on <math>\beta, \delta, \gamma, \mu</math> and <math>c</math> is the wave speed after the coordinate transformation <math> z = x - c t</math>. | It admits embedded solitons of the form <math> A\left(z\right) = \ell sech^2\left(k z\right) </math> when <math> \frac{ c^2 - b }{\delta\left(\beta c^2 - \gamma\right) } > 0</math>. The constants <math>k</math> and <math>\ell</math> depend on <math>\beta, \delta, \gamma, \mu</math> and <math>c</math> is the wave speed after the coordinate transformation <math> z = x - c t</math>. | ||
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. | |||
[[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6X3D-4SFG4MB-1&_user=10&_coverDate=05%2F31%2F2009&_rdoc=25&_fmt=high&_orig=browse&_srch=doc-info(%23toc%237296%232009%23999859994%23703262%23FLA%23display%23Volume)&_cdi=7296&_sort=d&_docanchor=&_ct=76&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=0785293922ab7d5140ab1c50972c026d Abtstract for Solitary wave families of a generalized microstructure PDE]] | [[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6X3D-4SFG4MB-1&_user=10&_coverDate=05%2F31%2F2009&_rdoc=25&_fmt=high&_orig=browse&_srch=doc-info(%23toc%237296%232009%23999859994%23703262%23FLA%23display%23Volume)&_cdi=7296&_sort=d&_docanchor=&_ct=76&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=0785293922ab7d5140ab1c50972c026d Abtstract for Solitary wave families of a generalized microstructure PDE]] | ||
Latest revision as of 09:14, 14 December 2008
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 when . The constants and depend on and is the wave speed after the coordinate transformation .
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]
