Optical Solitons of a Fractional Schrödinger Equation with Kudryashov's Law and Nonlocal Nonlinearity

Abstract

We investigate solitary-wave solutions of a time-fractional nonlinear Schrödinger equation in which the on-site nonlinearity is given by Kudryashov's generalized polynomial law and the nonlocal contribution combines second derivatives of two distinct intensity powers. Temporal evolution is described through the conformable fractional derivative of order μ∈(0,1]. After a clean traveling-wave reduction q(x,t)=U(ζ)e^(iθ(x,t)), the imaginary part fixes the wave velocity as v=-2ak, while the real part yields a fourth-order nonlinear ordinary differential equation in the real amplitude U(ζ). The substitution U=V^(1/n) converts that equation into a polynomial ODE of degree six in V to which the generalised tanh-coth expansion method applies; balancing produces M=1. Solving the resulting algebraic system in closed form, we obtain dark, singular, periodic, and rational-type solutions, together with the precise parameter constraints under which they exist. Two algebraic typos present in earlier treatments of the same model are identified, corrected, and the corrected coefficients are listed. Representative profiles are illustrated for two values of the fractional order μ, exhibiting the slowing of pulse evolution familiar from the conformable setting.