We study the sonic horizon formation problem for quantum system incorporating septic nonlinearity, which is modeled by the nonlinear Schrödinger equation (NLSE) with nonlinearity up to septic order. Based on the F-expansion method combined with modulus-phase transformation, we derived the soliton solutions of such NLSE for the one-dimensional and three-dimensional scenarios, from which the sonic horizon formation dynamical variables are derived. We identify that the distribution of system flow velocity and sound velocity, which determine the occurrence of the sonic horizon, agree well with the corresponding quantities obtained from pure numerical evaluation, demonstrating the applicability of the theoretical approach adopted in this study.

For power-law nonlinear system in higher-dimensional setting like the three-dimensional case, we derived its bright soliton solution with specific power-law feature based on the self-similar method. The dynamical evolution pattern of the derived solution is pictorially demonstrated which illustrates the typical bright soliton feature. The theoretical results presented in this work can be used to guide experimental observation of soliton behavior in higher-dimensional system with power-law nonlinearity.

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In this study, we investigate the typical systems modeled by the (3 + 1)-dimensional as well as (1 + 1)-dimensional Schrödinger equations incorporating third-order dispersion effects, higher-order scattering effects, and cubic–fifth–seventh degree nonlinear interactions. We use the F-expansion method and the self-similar method to solve the higher-order Schrödinger equation for one-dimensional and three-dimensional settings, respectively, identifying typical bright soliton solutions under appropriate system settings. The bright soliton features are demonstrated analytically in regions around the soliton peak region. Pictorial bright soliton features are demonstrated for the three-dimensional setting as well as one-dimensional setting. Our work shows the applicability of the theoretical treatment utilized in studying bright soliton dynamics for systems with third-order dispersion and seventh degree nonlinearity.