Abstract
This paper aims to present an application of the Riemann–Hilbert approach to treat higher-order nonlinear differential equation that is an eighth-order nonlinear Schrödinger equation arising in an optical fiber. Starting from the spectral analysis of the Lax pair, a matrix Riemann–Hilbert problem is formulated strictly. Then, by solving the obtained Riemann–Hilbert problem under the reflectionless case, N-soliton solution is generated for the eighth-order nonlinear Schrödinger equation. Finally, the localized structures and dynamic behaviors of one- and two-soliton solutions are illustrated by some figures.
Highlights
The infinite integrable nonlinear Schrödinger (NLS) equation hierarchy [1] reads as ipt + A2K2 p(x, t) – iA3K3 p(x, t) + A4K4 p(x, t) – iA5K5 p(x, t) + · · · = 0, (1)which is used to investigate the higher-order dispersive effects and nonlinearity
5 Conclusion In this investigation, the aim was to explore multi-soliton solutions for an eighth-order nonlinear Schrödinger equation arising in an optical fiber
The method we resort to was the Riemann–Hilbert approach which is based on a matrix Riemann–Hilbert problem
Summary
The infinite integrable nonlinear Schrödinger (NLS) equation hierarchy [1] reads as ipt + A2K2 p(x, t) – iA3K3 p(x, t) + A4K4 p(x, t) – iA5K5 p(x, t) + · · · = 0,. Which is used to investigate the higher-order dispersive effects and nonlinearity. P(x, t) denotes a normalized complex amplitude of the optical pulse envelope. The coefficients Al are arbitrary real constants, and Kl[p(x, t)] are the lth-order operators in the NLS hierarchy. K2 p(x, t) = pxx + 2p|p|2, K3 p(x, t) = pxxx + 6|p|2px, K4 p(x, t) = pxxxx + 6p∗p2x + 4p|px|2 + 8|p|2pxx + 2p2p∗xx + 6|p|4p, K5 p(x, t) = pxxxxx + 10|p|2pxxx + 30|p|4px + 10ppxp∗xx + 10pp∗xpxx.
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