This study reaches the dark, bright, mixed dark-bright, and singular optical solitons to the fractional Schrödinger-Hirota equation with a truncated M-fractional derivative via the extended sinh-Gordon equation expansion method. Dark soliton describes the solitary waves with lower intensity than the background, bright soliton describes the solitary waves whose peak intensity is larger than the background, and the singular soliton solutions is a solitary wave with discontinuous derivatives; examples of such solitary waves include compactions, which have finite (compact) support, and peakons, whose peaks have a discontinuous first derivative. The constraint conditions for the existence of valid solutions are given. We use some suitable values of the parameters in plotting 3-dimensional surfaces to some of the reported solutions.

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In this paper, the (2+1)-dimensional resonant Davey–Stewartson equations are solved by using two methods; namely, [Formula: see text]-expansion and [Formula: see text]-expansion methods. A wave transform is used to convert the (2+1)-dimensional resonant Davey–Stewartson (RDS) equations with M-derivative into a system of nonlinear ordinary differential equations. Different forms of solutions, such as dark, bright, singular and periodic singular solutions are successfully constructed. The obtained solutions are plotted in 3D for both M- derivative and classical derivative to more understand the effect of M-derivative on the studied equation.

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