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We consider the integrable generalization of the nonlinear Schrödinger equation that arises as a model for nonlinear pulse propagation in monomode optical fibers. The existent conditions for its modulational instability to form the rogue waves is given from its plane-wave solutions. We propose a generalized (n,N−n)$(n,N-n)$-fold Darboux transformation for this system by using the Nth-order Darboux matrix, Taylor expansion, and a limit procedure. As an application, we use the generalized perturbation (1,N−1)$(1,N-1)$-fold Darboux transformation to generate higher-order rogue wave solutions of this system. The dynamics behavior of the first-, second-, and third-order rouge wave solutions are shown graphically. These results may be useful for understanding some physical phenomena in optical fibers.

advances in difference equations

Modulational instability and higher-order rogue wave solutions for an integrable generalization of the nonlinear Schrödinger equation in monomode optical fibers