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• The Schrödinger equation for polynomial law medium with new local fractional derivative. • The fractional sub-equation method is used. • The obtained results are given by 2D and 3D figures. • The bright, dark, combined optical solitons are obtained. Novel solutions for the nonlinear dynamics of Schrödinger equation for polynomial law medium with third-order dispersion (TOD), fourth-order dispersion (FOD), and self-steepening are investigated based in a novel local fractional derivative of order α and the Jacobi elliptic function method which are combined into a novel fractional sub-equation method. The Jacobi elliptic function method will provide different types of analytical solutions and not only soliton type solutions for the propagation of ultra-short optical signals through a polynomial law medium. Solutions are illustrated in 3-D graphs, contour plots and 2-D plots under the obtained constraint conditions. [ABSTRACT FROM AUTHOR]