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Completely resonant collision of lumps and line solitons in the Kadomtsev–Petviashvili I equation
Author(s) -
Rao Jiguang,
Chow Kwok Wing,
Mihalache Dumitru,
He Jingsong
Publication year - 2021
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/sapm.12417
Subject(s) - rogue wave , soliton , integrable system , physics , kadomtsev–petviashvili equation , line (geometry) , bilinear interpolation , transverse plane , nonlinear system , elastic collision , phase space , dispersion (optics) , classical mechanics , mathematical physics , quantum mechanics , mathematics , geometry , statistics , structural engineering , burgers' equation , engineering , electron
Resonant collisions among localized lumps and line solitons of the Kadomtsev–Petviashvili I (KP‐I) equation are studied. The KP‐I equation describes the evolution of weakly nonlinear, weakly dispersive waves with slow transverse variations. Lumps can only exist for the KP equation when the signs of the transverse derivative and the weak dispersion in the propagation direction are different, that is, in the KP‐I regime. Collisions among lumps and solitons for “integrable” equations are normally elastic, that is, wave shapes are preserved except possibly for phase shifts. For resonant collisions, mathematically the phase shift will become indefinitely large. Physically a lump may be detached (or emitted) from a line soliton, survives for a brief transient period in time, and then merges with the next adjacent soliton. This special lump is thus localized in time as well as the two spatial dimensions, and can be termed a “rogue lump.” By employing a reduction method for the KP hierarchy in conjunction with the Hirota bilinear technique, a general L ‐lump/ ( L + 1 ) ‐line‐soliton solution is obtained for the KP‐I equation for an arbitrary positive integer L . For L ≥ 2 , collisions among modes become increasingly complicated, with multiple lumps detaching from one single or different line solitons and then disappearing into the remaining solitons. In terms of applications, such “rogue lumps” represent a mode truly localized both in space and time, and will be valuable in modeling physical problems.