Bright and peaklike pulse solitary waves and analogy with modulational instability in an extended nonlinear Schrödinger equation

Abstract

The modulational instability (MI) phenomenon in the nonlinear Schrödinger equation (NLSE) extended by two different nonlinear dispersion terms and the gradient term is investigated. We find that the possibility of instability of plane waves depends on the sign of the nonlinear dispersion parameters with regard to the linear dispersion coefficient. In contrast to the basic NLSE, the system may exhibit instability in the defocusing media for amplitude exceeding a critical value depending on the magnitude of the nonlinear dispersion. An additional feature, namely the higher order or the infinite gain band, absent in the NLSE case, may appear and in which MI induces the birth of the nonlinear localized wave (NLW) of different carrier wave numbers. The result of the qualitative investigations of the system's dynamics indicates the existence of the NLW, such as peak, bright, dark, and compact dark solitary waves which can be well predicted by the MI criteria. In addition the nonlinear dispersion induces the existence of a pair of bright-dark solitary waves which is usually exhibited by the coupled NLSEs only, and the pairs of peak-dark and compact dark-bright solitary waves.

Vector solitons in an extended coupled Schrödinger equations with modulated nonlinearities

“…It is interesting to note that the coefficients of cubic terms determine the similarity variables and the external potential. The expressions in (22) show that if Γ j = 0 the coefficients of quintic terms must be constant for the above conversion to work.…”

“…Theorem 1: The governing equations of dynamics of the ultra-short pulse propagation along orthogonal polarization axes in nonlinear optical fibers with or without Kerr nonlinearities given by NLSEs (1)-(2) can be transformed under the transformation (5) into an autonomous system given by (3)-(4) provided that X, T , θ j , and ρ j (j = 1, 2) are given by (21), and the external potential is given by (17) and the coefficients of quintic terms satisfy the expressions given in (22).…”

“…Naturally, such transformation is possible only for a a specific class of coefficients and external potentials satisfying derived constraints (17), (21), and (22). In fact, if (17), and (21) give the expressions of the parameters in transformation (5), and define the external potentials v j (x, t) respectively, Eq.…”

“…In fact, if (17), and (21) give the expressions of the parameters in transformation (5), and define the external potentials v j (x, t) respectively, Eq. (22) consists of the balances between the amplification/absorption…”

Travelling wave solutions and soliton solutions for the nonlinear transmission line using the generalized Riccati equation mapping method

“…Among all these fields, there are only a few systems where experiment studies are easily observed. In physics, precisely, nonlinear electrical transmission lines (NETL) [2][3][4] are good examples to provide a useful way to check how the nonlinear excitations behave inside the nonlinear medium. They are discrete systems but approximate the continuum systems quite well, as will be seen below [5].…”