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arXiv · 1006.4938

Dynamical systems theory for nonlinear evolution equations

Amitava Choudhuri·B Talukdar·Umapada Das
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Abstract

We observe that the fully nonlinear evolution equations of Rosenau and Hymann, often abbreviated as $K(n,\,m)$ equations, can be reduced to Hamiltonian form only on a zero-energy hypersurface belonging to some potential function associated with the equations. We treat the resulting Hamiltonian equations by the dynamical systems theory and present a phase-space analysis of their stable points. The results of our study demonstrate that the equations can, in general, support both compacton and soliton solutions. For the $K(2,\,2)$ and $K(3,\,3)$ cases one type of solutions can be obtained from the other by continuously varying a parameter of the equations. This is not true for the $K(3,\,2)$ equation for which the parameter can take only negative values. The $K(2,\,3)$ equation does not have any stable point and, in the language of mechanics, represents a particle moving with constant acceleration.

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Amitava Choudhuri, B Talukdar, Umapada Das. 2010-06-25. Dynamical systems theory for nonlinear evolution equations. https://doi.org/10.1103/physreve.82.036609

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