Inverse scattering transform and high-order pole solutions for the NLS equation with quartic terms under vanishing/non-vanishing boundary conditions

The aim of this article is to investigate the inverse scattering transform and high-order pole solutions for the focusing nonlinear Schrodinger equation with quartic terms. Under the vanishing and non-vanishing boundary conditions, the scattering data are obtained respectively for one high-order pole and multi high-order poles. The formulas of soliton solutions are established by Laurent expansion of the Riemann-Hilbert problem. The method used here is different from the usual method of computing residues since the coefficients of higher negative power of $\lambda-\lambda_{j}$ and $\lambda-\overline{\lambda}_{j}$ are difficult to be obtained. The determinant formula of high-order pole solution for non-vanishing boundary condition is given. The dynamical properties and characteristics for high-order pole solutions are further analyzed.