W-shaped, dark and grey solitary waves in the nonlinear Schrodinger equation competing dual power-law nonlinear terms and potentials modulated in time and space

This paper explores the higher-order nonlinear Schrodinger equation with dual power-law nonlinearities and external potential that can depend on space and time coordinates. We constructed exact soliton-like solutions with arbitrary power-law nonlinearities using the similarity transformation approach. We obtained a family of soliton-like solutions: the so-called W-shaped, dark and the grey solitons. These solutions are at first, found by considering just the time modulated nonlinearities, where the frame variable coincides with the characteristic coordinate xi. Then the nonlinearities coefficients and potential are space-time modulated with a bounded function . In both cases, a numerical study based on appropriate choices of parameters is achieved. For a fixed absolute value of nonlinearities, we examine how different soliton profiles are influenced when increasing the power-law index n. These variations of parameters display some substantial changes in solitary wave profiles; paving the way for many applications.