Modulational instability and homoclinic orbit solutions in vector nonlinear Schrödinger equation

Abstract

Modulational instability has been used to explain the formation of breather and rogue waves qualitatively. In this paper, we show modulational instability can be used to explain the structure of them in a quantitative way. We develop a method to derive general forms for Akhmediev breather and rogue wave solutions in a N -component nonlinear Schrödinger equations. The existence condition for each pattern is clarified clearly. Moreover, the general multi-high-order rogue wave solutions and multi-Akhmediev breather solutions for N -component nonlinear Schrödinger equations are constructed. The results further deepen our understanding on the quantitative relations between modulational instability and homoclinic orbits solutions.

Darboux transformation and solitonic solution to the coupled complex short pulse equation

“…The rogue waves are given in the above subsection. The higher order rogue waves can be obtained through a similar procedure as in [28]. (4): α 1 α 2 = 0 and κ = −1: Breather solutions, dark-dark solitons [26], rogue wave solutions and combinations thereof.…”

“…The coupled NLS (CNLS) equation also known as Manakov system [35], can be used as a model in nonlinear birefringent optics and for two modes of Bose-Einstein condensate [44]. The localized wave solutions in the coupled NLS equation are more complicated and richer than the scalar one [13,28,36,37]. There are some differences in compared with the scalar one.…”

Integrability and Linear Stability of Nonlinear Waves

“…(h) S x for r = 0, p = −4.7, i.e., s 1 = s 2 = −1, a 1 = a 2 = 1.533, as an example of a degenerate case of a 0G 2B 0L spectrum: this case (which also appears in [36]) when projected back onto the stereographic sphere, can be completely explained in terms of the classification scheme provided in Proposition 9 (see [17]). model, that we recall here for convenience:…”

“…As far as the standard methods are concerned, the linear stability of CW solutions has been investigated only for the focusing and defocusing regimes, but not for the mixed one (s 1 = −s 2 ), and only in the integrable cases, by means of the Fourier transform [21]. Conversely, as far as the integrability methods are concerned, it has been partially discussed in [36] to mainly show that instability may occur also in defocusing media, in contrast to scalar waves which are modulationally unstable only in the focusing case. In the following we approach the linear stability problem of the CW solutions of (1) within the integrability framework to prove that the main object to be computed is a spectrum (to be defined below) as a curve in the complex plane of the spectral variable, together with the eigenmodes wave numbers and frequencies defined on it.…”

Strong and weak interactions of rational vector rogue waves and solitons to anyn-component nonlinear Schrödinger system with higher-order effects

“…[37][38][39][40][41][42][43][44][45][46][47][48][49][50] and references therein). Since the rational vector RWs (RVRWs) of the focusing two-component NLS equations were found [6,34,51], other two-component [52][53][54][55][56][57][58][59][60][61][62][63][64] and three-component [65][66][67][68] nonlinear wave equations have been found to possess the distinct types of RVRWs or rational solitons. More recently, as the vectorization of the scalar NLS equation, any n-component NLS equations appearing in the DNA and nonlinear optics [42][43][44][45][46] were found to possess a novel multi-parameter family of higher-order rational vector RW solutions [69,70].…”