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Short-Lived Large-Amplitude Pulses in the Nonlinear Long-Wave Model Described by the Modified Korteweg–De Vries Equation


Address for correspondence: Roger Grimshaw, Loughborough University, Loughborough, LE11 3TU UK; e-mail:


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.