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Lump Interactions and Collapse in the Modified Zakharov–Kuznetsov Equation
Author(s) -
Sipcic R.,
Benney D. J.
Publication year - 2000
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00157
Subject(s) - gravitational singularity , singularity , generalization , physics , mathematical physics , field (mathematics) , mathematical analysis , magnetic field , kadomtsev–petviashvili equation , classical mechanics , mathematics , partial differential equation , characteristic equation , quantum mechanics , pure mathematics
Two‐dimensional solitary wave (lump) interactions and the formation of singularities in the modified Zakharov–Kuznetsov (mZK) equation are considered. The mZK equation represents an anisotropic two‐dimensional generalization of the Korteweg–deVries equation and can be derived in a magnetized plasma for small amplitude Alfvén waves at a critical angle to the undisturbed magnetic field. The radially symmetric positive solutions (lumps) for the mZK equation are computed. It is shown that when two lumps interact, the initial energy exchange between them is followed by the emergence of a single collapsing lump and a radiation field behind it. Connections to the analytic theory of collapse for the mZK equation are discussed.