The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.
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| http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5582171 | PMC |
| http://dx.doi.org/10.1098/rspa.2016.0695 | DOI Listing |
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Whitham modulation theory for the Kadomtsev- Petviashvili equation.
Proc Math Phys Eng Sci
August 2017
Department of Mathematics, State University of New York, Buffalo, NY 14260, USA.
The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.
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