Numerical analysis of the Hirota equation: Modulational instability, breathers, rogue waves, and interactions.

In this paper, we focus on the integrable Hirota equation, which describes the propagation of ultrashort light pulses in optical fibers. First, we numerically study spectral signatures of the spatial Lax pair with distinct potentials [e.g., solitons, Akhmediev-Kuznetsov-Ma (AKM) and Kuznetsov-Ma (KM) breathers, and rogue waves (RWs)] of the Hirota equation. Second, we discuss the RW generation by using the dam-break problem with a decaying initial condition and further analyze spectral signatures of periodized wavetrains. Third, we explore two kinds of noise-derived modulational instabilities: (i) the one case is based on the initial condition (one plus a random noise) such that the KM and AKM breathers, and RWs can be generated, and they agree well with analytical solutions; (ii) another case is to consider another initial condition (one plus a Gaussian wave with a random noise phase) such that some RWs with higher amplitudes can be found. Moreover, we also investigate the spectral signatures of corresponding periodic wavetrains. Finally, we find that the interactions of two waves can also generate the RW phenomena with higher amplitudes. These obtained results will be useful to understand the RW generation in the third-order nonlinear Schrödinger equation and other related models.

Generation mechanisms of fundamental rogue wave spatial-temporal structure.

We discuss the generation mechanism of fundamental rogue wave structures in N-component coupled systems, based on analytical solutions of the nonlinear Schrödinger equation and modulational instability analysis. Our analysis discloses that the pattern of a fundamental rogue wave is determined by the evolution energy and growth rate of the resonant perturbation that is responsible for forming the rogue wave. This finding allows one to predict the rogue wave pattern without the need to solve the N-component coupled nonlinear Schrödinger equation. Furthermore, our results show that N-component coupled nonlinear Schrödinger systems may possess N different fundamental rogue wave patterns at most. These results can be extended to evaluate the type and number of fundamental rogue wave structure in other coupled nonlinear systems.

High-order rogue waves in vector nonlinear Schrödinger equations.

We study the dynamics of high-order rogue waves (RWs) in two-component coupled nonlinear Schrödinger equations. We find that four fundamental rogue waves can emerge from second-order vector RWs in the coupled system, in contrast to the high-order ones in single-component systems. The distribution shape can be quadrilateral, triangle, and line structures by varying the proper initial excitations given by the exact analytical solutions. The distribution pattern for vector RWs is more abundant than that for scalar rogue waves. Possibilities to observe these new patterns for rogue waves are discussed for a nonlinear fiber.

Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions.

In this paper, we construct a generalized Darboux transformation for the nonlinear Schrödinger equation. The associated N-fold Darboux transformation is given in terms of both a summation formula and determinants. As applications, we obtain compact representations for the Nth-order rogue wave solutions of the focusing nonlinear Schrödinger equation and Hirota equation. In particular, the dynamics of the general third-order rogue wave is discussed and shown to exhibit interesting structures.