General Rogue Wave Solutions Research Articles

Rogue waves in the massive Thirring model

AbstractIn this paper, general rogue wave solutions in the massive Thirring (MT) model are derived by using the Kadomtsev–Petviashvili (KP) hierarchy reduction method and these rational solutions are presented explicitly in terms of determinants whose matrix elements are elementary Schur polynomials. In the reduction process, three reduction conditions including one index‐ and two dimension‐ones are proved to be consistent by only one constraint relation on parameters of tau‐functions of the KP‐Toda hierarchy. It is found that the rogue wave solutions in the MT model depend on two background parameters, which influence their orientation and duration. Differing from many other coupled integrable systems, the MT model only admits the rogue waves of bright‐type, and the higher order rogue waves represent the superposition of fundamental ones in which the nonreducible parameters determine the arrangement patterns of fundamental rogue waves. Particularly, the super rogue wave at each order can be achieved simply by setting all internal parameters to be zero, resulting in the amplitude of the sole huge peak of order N being times the background. Finally, rogue wave patterns are discussed when one of the internal parameters is large. Similar to other integrable equations, the patterns are shown to be associated with the root structures of the Yablonskii–Vorob'ev polynomial hierarchy through a linear transformation.

General rogue waves in the AB system

In this work, general rogue wave solutions in the AB system are constructed by means of the Kadomtsev–Petviashvili hierarchy reduction method and explicit representations of these solutions are explored in terms of determinants whose matrix elements are fundamental Schur polynomials. Moreover, the dynamics of these rogue waves are investigated graphically by different choices of the free parameters. In particular, we observe that the rogue wave in the second component B is actually a four peaky-shaped rogue wave, different form the eye-shaped rogue wave in NLSE.

General rogue wave solutions under SU(2) transformation in the vector Chen–Lee–Liu nonlinear Schrödinger equation

We obtain analytically the exact explicit rogue wave solutions up to the second order of the vector Chen–Lee–Liu nonlinear Schrödinger equation, using the generalized non-recursive Darboux transformation method. In terms of these solutions, we demonstrate the fundamental Peregrine solitons as well as their doublet, triplet, quartet, and sextet counterparts on the general periodic backgrounds caused by SU(2) transformation. We numerically confirm that, although the Peregrine solitons sitting on SU(2) periodic backgrounds may suffer from larger disturbances than what they experience on SO(2) periodic backgrounds, such rational solitons developed on either kind of backgrounds can manifest clearly in spite of strong non-integrable perturbations. Other rogue wave topics such as rogue wave coexistence and the related parametric conditions are also discussed.