Abstract
The nonlinear Schrödinger hierarchy has a wide range of applications in modeling the propagation of light pulses in optical fibers. In this paper, we focus on the integrable nonlinear Schrödinger (NLS) equation with quintic terms, which play a prominent role when the pulse duration is very short. First, we investigate the spectral signatures of the spatial Lax pair with distinct analytical solutions and their periodized wavetrains by Fourier oscillatory method. Then, we numerically simulate the wave evolution of the quintic NLS equation from different initial conditions through the symmetrical split-step Fourier method. We find many localized high-peak structures whose profiles are very similar to the analytical solutions, and we analyze the formation of rouge waves (RWs) in different cases. These results can be helpful to understand the excitation of nonlinear waves in some nonlinear fields, such as the Heisenberg ferromagnetic spin system in condensed matter physics, ultrashort pulses in nonlinear optical fibers, and so on.
Highlights
The analytical solutions of the nonlinear evolution equations are widely applied in many fields, such as nonlinear optical fiber, plasma physics and so on [1, 2]
We focus on the quintic nonlinear Schrodinger (NLS) equation
We numerically study the spectral signatures of the spatial Lax pair with distinct analytical solutions [i.e., solitons, Akhmediev (AB), Kuznetsov-Ma (KM) breathers, and rogue waves (RWs)] of the quintic NLS equation through the Fourier oscillatory method
Summary
The analytical solutions of the nonlinear evolution equations are widely applied in many fields, such as nonlinear optical fiber, plasma physics and so on [1, 2]. (1) can be used to model the propagation of light pulses in optical fibers and the quintic terms play a prominent role when the pulse duration is very short. It is a compatibility condition, Φxt = Φtx, of the system of two linear partial differential equations with variable coefficients ( known as the Lax pair) [25], Φx = XΦ, X(x, t, λ) = iλσ + Q,.
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