The integrable nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential [M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 110, 064105 (2013)] is investigated, which is an integrable extension of the standard nonlinear Schrödinger equation. Its novel higher-order rational solitons are found using the nonlocal version of the generalized perturbation (1,N-1)-fold Darboux transformation. These rational solitons illustrate abundant wave structures for the distinct choices of parameters (e.g., the strong and weak interactions of bright and dark rational solitons). Moreover, we also explore the dynamical behaviors of these higher-order rational solitons with some small noises on the basis of numerical simulations.
In this paper, we investigate higher-order rogue wave solutions of the Kundu-Eckhaus equation, which contains quintic nonlinearity and the Raman effect in nonlinear optics. By means of a gauge transformation, the Kundu-Eckhaus equation is converted to an extended nonlinear Schrödinger equation. We derive the Lax pair, the generalized Darboux transformation, and the Nth-order rogue wave solution for the extended nonlinear Schrödinger equation. Then, by using the gauge transformation between the two equations, a concise unified formula of the Nth-order rogue wave solution with several free parameters for the Kundu-Eckhaus equation is obtained. In particular, based on symbolic computation, explicit rogue wave solutions to the Kundu-Eckhaus equation from the first to the third order are presented. Some figures illustrate dynamic structures of the rogue waves from the first to the fourth order. Moreover, through numerical calculations and plots, we show that the quintic and Raman-effect nonlinear terms affect the spatial distributions of the humps in higher-order rogue waves, although the amplitudes and the time of appearance of the humps are unchanged.
We analytically study rogue-wave (RW) solutions and rational solitons of an integrable fifth-order nonlinear Schrödinger (FONLS) equation with three free parameters. It includes, as particular cases, the usual NLS, Hirota, and Lakshmanan-Porsezian-Daniel equations. We present continuous-wave (CW) solutions and conditions for their modulation instability in the framework of this model. Applying the Darboux transformation to the CW input, novel first- and second-order RW solutions of the FONLS equation are analytically found. In particular, trajectories of motion of peaks and depressions of profiles of the first- and second-order RWs are produced by means of analytical and numerical methods. The solutions also include newly found rational and W-shaped one- and two-soliton modes. The results predict the corresponding dynamical phenomena in extended models of nonlinear fiber optics and other physically relevant integrable systems.
In this paper, a novel, simple, and constructive method is presented to find the generalized perturbation (n, M )-fold Darboux transformations (DTs) of the modified nonlinear Schrödinger (MNLS) equation in terms of fractional forms of determinants. In particular, we apply the generalized perturbation (1, N − 1)-fold DTs to find its explicit multi-rogue wave solutions. The wave structures of these rogue wave solutions of the MNLS equation are discussed in detail for different parameters, which display abundant interesting wave structures including the triangle and pentagon, etc. and may be useful to study the physical mechanism of multi-rogue waves in optics. The dynamical behaviors of these multi-rogue wave solutions are illustrated using numerical simulations. The same Darboux matrix can also be used to investigate the Gerjikov-Ivanov equation such that its multirogue wave solutions and their wave structures are also found. The method can also be extended to find multi-rogue wave solutions of other nonlinear integrable equations.
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