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Compacton equations and integrability: The Rosenau-Hyman and Cooper-Shepard-Sodano equations

  • * Corresponding author: R. Hernández Heredero

    * Corresponding author: R. Hernández Heredero 
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  • We study integrability –in the sense of admitting recursion operators– of two nonlinear equations which are known to possess compacton solutions: the $ K(m, n) $ equation introduced by Rosenau and Hyman

    $ D_t(u) + D_x(u^m) + D_x^3(u^n) = 0 \; , $

    and the CSS equation introduced by Coooper, Shepard, and Sodano,

    $ D_t(u) + u^{l-2}D_x(u) + \alpha p D_x (u^{p-1} u_x^2) + 2\alpha D_x^2(u^p u_x) = 0 \; . $

    We obtain a full classification of integrable $ K(m, n) $ and CSS equations; we present their recursion operators, and we prove that all of them are related (via nonlocal transformations) to the Korteweg-de Vries equation. As an application, we construct isochronous hierarchies of equations associated to the integrable cases of CSS.

    Mathematics Subject Classification: Primary: 37K05, 37K10; Secondary: 35B10.


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