Rogue waves and rational solutions of the nonlinear Schrödinger equation

Abstract

We present a method for finding the hierarchy of rational solutions of the self-focusing nonlinear Schrödinger equation and present explicit forms for these solutions from first to fourth order. We also explain their relation to the highest amplitude part of a field that starts with a plane wave perturbed by random small amplitude radiation waves. Our work can elucidate the appearance of rogue waves in the deep ocean and can be applied to the observation of rogue light pulse waves in optical fibers.

Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation

“…More sophisticated, higher order SFBs, are possible (see e.g. [2]), which represent degenerate fNLS solutions with genus g > 2.…”

Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents

“…Consequently, the amplitude of the full grown RW described by the lump soliton can be changed continuously by changing parameter c (with A rogue (c) increasing with decreasing c) and could therefore be adjusted to fit the heights of any observed RW. Consequently, the maximum RW amplitude in our 2) and its generalizations [25], as mentioned above, we conclude that, in the well-known class of PB solutions, the maximum amplitudes reachable by the one-dimensional RW are given by the fixed discrete odd numbers 2j + 1, with j = 1, 2, 3, . .…”

“…The modular inclination of this wave as well as the fastness of its appearance is also fixed, as solution (1.2) admits no free parameters. This situation can be improved to obtain higher amplitude and modular inclination of the PB model by using higher order rational solutions [25]. For example, the next higher order PB known also as Akhmediev-PB can enhance the maximum wave elevation by a factor of five, while the next one by a factor of seven and so on, with an intriguing enhancement of factors by increasing odd numbers.…”

“…For example, the next higher order PB known also as Akhmediev-PB can enhance the maximum wave elevation by a factor of five, while the next one by a factor of seven and so on, with an intriguing enhancement of factors by increasing odd numbers. Such increments in amplitude, however, are discrete and could be achieved at the cost of going to new solutions with increasingly complicated structures involving higher and higher order polynomials [25]. The maximum amplitude and modular inclination reachable by this class of solutions are fixed owing to the absence of relevant free tunable parameters, making it difficult to adjust to the continuously varied range of shape and sizes of the observed oceanic RWs.…”

“…However such modifications of the NLS equation (1.1) make the system non-integrable, allowing only numerical solutions. The most popular one-dimensional RW model is a unique analytic rational solution of the original NLS equation (1.1) [24], given by the Peregrine breather (PB) [12,14,17] or its higher order versions [25][26][27] and the trigonometric variants [28,29]. However, as the RW is an aperiodic event with a single appearance, the trigonometric breather solutions, owing to their periodic nature, are not very suitable for a direct description of the RW.…”