Rogue wave solutions up to the fifth-order are obtained by solving the modified nonlinear Schrödinger equation, also known as the Dysthe equation which provides a better representation of the wave evolution in space. Using the fourth-order split-step pseudo-spectral method during the integral process, more accurate results with a smaller conservation error were obtained. One of the key features is the adoption of the wavelet analysis method to analyse the time-frequency energy distribution during the generation and evolution of the hierarchy of so-called super rogue waves. It is revealed that a strong energy density surges instantaneously and is seemingly carried over to the high frequency components at the instant when a large, rogue wave occurs. Super rogue waves tend to be the consequence of the nonlinear superposition and focusing of waves in the time and space domain, where transient energy moves between the fundamental frequency and sideband frequencies.
- nonlinear Schrödinger equation
- rational solution
- super rogue wave