Localized modes of the Hirota equation: Nth order rogue wave and a separation of variable technique

• Published in: Communications in nonlinear science & numerical simulation Vol. 39; pp. 118 - 133
• Main Authors: Mu, Gui, Qin, Zhenyun, Chow, Kwok Wing, Ee, Bernard K.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2016
• Subjects: Derivatives; Evolution; Generalized Darboux transformation; Mathematical analysis; Mathematical models; Nonlinearity; Rogue waves; Schroedinger equation; Separation; Stability of rogue waves; Transformations; Variable separation technique; Rogue waves; Variable separation technique; Stability of rogue waves; Generalized Darboux transformation
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2016.02.028

•Rogue waves of the Hirota equation are derived by a Darboux transformation.•Separation of variables and Taylor series are used, instead of taking derivatives.•Structural stability of rogue waves is studied by numerical simulations.•Second order rogue waves are generally less stable than first order ones. The Hirota equation is a special extension of the intensively studied nonlinear Schrödinger equation, by incorporating third order dispersion and one form of the self-steepening effect. Higher order rogue waves of the Hirota equation can be calculated theoretically through a Darboux-dressing transformation by a separation of variable approach. A Taylor expansion is used and no derivative calculation is invoked. Furthermore, stability of these rogue waves is studied computationally. By tracing the evolution of an exact solution perturbed by random noise, it is found that second order rogue waves are generally less stable than first order ones.