Rogue waves of the nonlinear Schrödinger equation with an external linear potential in waveguides with Helmholtz resonators: Transmission-line approach

This study explores rational soliton solutions in an acoustic waveguide system incorporating Helmholtz resonators, modeled using a transmission-line approach. A perturbation technique is used to derive a cubic-quintic nonlinear Schrödinger equation with a linear external potential. Analytical investigations consider various forms of the linear potential, including constant linear density, temporally periodic modulation, and exponentially growing or decaying profiles, to examine the characteristics of first- and second-order rogue waves. By analyzing the combined influence of wavenumber and the different external potentials, it is observed that the typical features of rogue waves such as crests, humps, and holes disappear under certain conditions. Furthermore, for sufficiently high wavenumber values, the Peregrine soliton transitions into a composite bright-dark soliton structure. Finally, a modulation instability analysis is conducted to identify both stable and unstable modes within specific angular frequency ranges.