Quintic time-dependent-coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics: bilinear forms and dark/anti-dark/gray solitons

Studies on the water waves contribute to the design of the related industries, such as the marine and offshore engineering, while the media with the negative refractive index can be applied as the carrier media in fiber optics. In consideration of the inhomogeneities of the media and nonuniformities of the boundaries in the real physical backgrounds, a quintic time-dependent-coefficient derivative nonlinear Schrodinger equation for certain hydrodynamic wave packets or medium with the negative refractive index is investigated in this paper. Bilinear forms and the N-soliton solutions with respect to the nonzero background, which are different from those in the existing studies, are derived under the certain constraints. Conditions for the dark/anti-dark/gray solitons are deduced due to the properties of the solitons derived via the asymptotic analysis. Effects of the dispersion coefficient \(\lambda (t)\), self-steepening coefficient \(\alpha (t)\), cubic nonlinearity \(\mu (t)\) and quintic nonlinearity \(\nu (t)\) on the interactions between the anti-dark and gray solitons under the certain condition are investigated. Interactions among the dark, anti-dark and gray solitons are discussed under two cases: when \({\alpha (t)}/{\lambda (t)}\) and \({\mu (t)}/{\lambda (t)}\) are the constants, whether the interaction is elastic or not depends on whether \(\lambda (t)\), \(\alpha (t)\) and \(\mu (t)\) are the constants or the functions of t; when \({\alpha (t)}/{\lambda (t)}\) and \({\mu (t)}/{\lambda (t)}\) are related to t, if the velocity of the soliton is a periodic function of t, the propagation of the corresponding soliton is periodic and the corresponding interaction is inelastic. Interactions among the three/four solitons are described to be elastic or inelastic based on the changes in the velocities and waveforms of the three/four solitons after the interactions.