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Short‐Lived Large‐Amplitude Pulses in the Nonlinear Long‐Wave Model Described by the Modified Korteweg–De Vries Equation
Author(s) -
Grimshaw Roger,
Pelinovsky Efim,
Talipova Tatiana,
Ruderman Michael,
Erdélyi Róbert
Publication year - 2005
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/j.0022-2526.2005.01544.x
Subject(s) - modulational instability , breather , korteweg–de vries equation , inverse scattering transform , wave packet , nonlinear system , physics , dispersionless equation , rogue wave , instability , amplitude , inverse scattering problem , nonlinear schrödinger equation , classical mechanics , mathematical physics , quantum electrodynamics , kadomtsev–petviashvili equation , scattering , quantum mechanics , burgers' equation
The appearance and disappearance of short‐lived large‐amplitude pulses in a nonlinear long wave model is studied in the framework of the modified Korteweg–de Vries equation. The major mechanism of such wave generation is modulational instability leading to the generation and interaction of the breathers. The properties of breathers are studied both within the modified Korteweg–de Vries equation, and also within the nonlinear Schrödinger equations derived by an asymptotic reduction from the modified Korteweg–de Vries for weakly nonlinear wave packets. The associated spectral problems (AKNS or Zakharov‐Shabat) of the inverse‐scattering transform technique also are utilized. Wave formation due to this modulational instability is investigated for localized and for periodic disturbances. Nonlinear‐dispersive focusing is identified as a possible mechanism for the formation of anomalously large pulses.