Abstract
In this paper, we investigate the exact solutions and conservation laws of a general Hirota equation. Firstly, the N-fold Darboux transformation of this equation is proposed. Then by choosing three kinds of seed solutions, the multisoliton solutions, breather solutions, and rogue wave solutions of the general Hirota equation are obtained based on the Darboux transformation. Finally, the conservation laws of this equation are derived by using its linear spectral problem. The results in this paper may be useful in the study of ultrashort optical solitons in optical fibers.
Highlights
During the past decades, many researches have been focused on optical solitons due to their potential applications in optical fiber long-distance transmission systems
According to the theoretical report and experimental results in [, ], optical solitons are based on the balance between the group velocity dispersion and self-phase modulation in the picosecond regime, and the propagation of such a soliton is governed by the standard nonlinear Schrödinger equation (NLS) [, ], iut where u = u(x, t) denotes the slowly varying complex envelope of the wave, and subscripts x and t are the longitudinal distance and retarded time, respectively
The NLS equation ( ) cannot describe the corresponding physical characteristics, and it accounts for the following three points: first, the fourth-order dispersion should be considered when the pulse width is below femtoseconds [, ]; second, the higher-order nonlinearities should not be neglected when the optical field frequency approaches a resonant frequency of the optical fibers material [, ]; third, the self-steepening and self-frequency shift should be included when extremely narrow pulse has very high optical intensity as the fourth-order dispersion
Summary
Many researches have been focused on optical solitons due to their potential applications in optical fiber long-distance transmission systems. We remark that we can get the high-order soliton solutions of the general Hirota equation ( ) by continuing iteration of the DT in Theorem. Figures (a), (b) and (a), (b) display the structures of the one-breather solution u(b s) for the general Hirota equation ( ) with = , and Figures (c), (d) and (c), (d) display the same case except the parameter =.
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