Jacobian-elliptic-function and rogue-periodic-wave solutions of a high-order nonlinear Schrödinger equation in an inhomogeneous optical fiber

Abstract

Optical fiber communication plays an important role in the modern communication. Under investigation is a high-order nonlinear Schrödinger equation with the additional high-order dispersion and nonlinear terms in an inhomogeneous optical fiber. We give the Jacobian-elliptic-function solutions of that equation. Based on the nonlinearization of the existing Lax pair, we obtain the values of the spectral parameters corresponding to that Lax pair. We construct the linearly-independent and non-periodic solutions of that Lax pair. We take the Jacobian-elliptic-function solutions as the seed solutions, and substitute the linearly-independent and non-periodic solutions into the Darboux transformation to obtain the rogue-periodic-wave solutions. For the one-fold rogue-dn-periodic wave, when the parameter γ of the equation increases, we find that the period and maximum amplitude of the one-fold rogue-dn-periodic wave remain unchanged, while the minimum amplitude decreases. When the parameter η of the equation increases, we find that the maximum amplitude of the one-fold rogue-dn-periodic wave remains unchanged, while the period becomes the smaller. For the one-fold rogue-cn-periodic wave, the same conclusions can be observed. For the two-fold rogue-cn-periodic wave, when the parameter γ of the equation increases, we find that the period and maximum amplitude of the two-fold rogue-cn-periodic wave remain unchanged, while the minimum amplitude rises. When the parameter η of the equation increases, we find that the maximum amplitude of the two-fold rogue-cn-periodic wave remains unchanged, while the period becomes the smaller.