The fourth-order dispersion effect on the soliton waves and soliton stabilities for the cubic-quintic Gross–Pitaevskii equation

A generalized nonlinear Gross–Pitaevskii equation with the third-, fourth-order dispersions, and the cubic, quintic nonlinear terms is considered, which shows that the wave velocity, amplitude, inverse temporal width and wave number of propagating waves are uniquely dependent on the parameters of second-, third- and fourth-order dispersions. We discuss the anomalous dispersion relations of the cubic-quintic Gross–Pitaevskii(CQGP) equation with second-, third- and fourth-order dispersions, and find that the fourth-order dispersion has a significant influence on the characteristics of the obtained soliton wave solutions. In particular, we analyze the effect of the fourth-order dispersion function r(t) on periodic, bright and dark wave solutions of the CQGP model. We report several families of non-autonomous soliton solutions with different amplitude surfaces, and see that the fourth-order dispersion is essential in a broad range of real physical media as it influences the wave dynamics. Some numerical simulations of bright and dark solitons for the CQGP equation with the different external potentials are considered, which shows the external potential plays a certain stabilizing role for the coexistence state of bright and dark solitons. These dynamic analyses play a significant role in improving the dispersive transport of the nonlinear wave in parabolic-law media.