Citation: Ullah, N.; Asjad, M.I.; Almusawa, M.Y.; Eldin, S.M. Dynamics of Nonlinear Optics with Different Analytical Approaches. Fractal Fract. 2023, 7, 138. https:// doi.org/10.3390/fractalfract7020138 Academic Editor: Carlo Cattani Received: 10 January 2023 Revised: 20 January 2023 Accepted: 24 January 2023 Published: 2 February 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). fractal and fractional Article Dynamics of Nonlinear Optics with Different Analytical Approaches Naeem Ullah 1 , Muhammad Imran Asjad 1, * , Musawa Yahya Almusawa 2 and Sayed M. Eldin 3 1 Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan 2 Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia 3 Center of Research, Faculty of Engineering and Technology, Future University in Egypt, New Cairo 11835, Egypt * Correspondence: imran.asjad@umt.edu.pk Abstract: In this article, we investigate novel optical solitons solutions for the Lakshmanan–Porsezian– Daniel (LPD) equation, along with group velocity dispersion and spatio-temporal dispersion, via three altered analytical techniques. A variety of bright, singular, dark, periodic singular, and kink solitons solutions are constructed via the Kudryashov method, the generalized tanh method and the Sardar-subequation method. The dynamical behavior of the extracted solutions is demonstrated in graphical form such as 3D plots, 2D plots, and contour plots. The originality of the obtained solutions is recognized by comparison with each other and solutions previously stated in the literature for the LPD model, which displays the efficiency of the methods under consideration. Keywords: Lakshmanan–Porsezian–Daniel equation; Sardar-subeqaution method; Kudryashov method; tanh method; optical solitons 1. Introduction Nonlinear evolution equations (NLEEs) have become more significant because many physical phenomena are demonstrated in nonlinear nature. The study of NLEEs is a main field of research in physics, mathematics, optics, biology, engineering, and other areas of natural sciences. A differential model is considered one of the best sources to examine the characteristics of physical phenomena thoroughly. The LPD equation is one of the well-known nonlinear partial differential equations to study physical phenomena. In many research problems, the application of nonlinear models has great interest for mathematical modeling. The LPD equation has the capability to perform the dynamics of optical solitons similarly to the Schrödinger equation [1,2]. The LPD equation de- fines the communication of solitons using different waveguides. Many mathematicians and researchers have worked on the LPD equation using different mathematical tech- niques, for instance, the undetermined coefficients method [3], extended trial equation method [4], semi-inverse variational principle [5], etc. In the current study, we examined the dynamics of solitons along with group velocity dispersion (GVD) and spatio-temporal dispersion (STD). Recently, many analytical schemes have been established to find the solutions of NLEEs, such as the modified exp-function scheme [6], extended tanh func- tion scheme [7], variational iteration method [8], first integral method [9], Hirota’s direct method [10], new extended generalized Kudryashov method [11], improved subequation scheme [12], modified (G ′ /G)-expansion approach [13], fractional reduced differential transform scheme [14], Sine–Gordon expansion approach [15], extended modified mapping scheme [16], iterative approach [17], extended trial equation method [18], simplest equa- tion method [19], F-expansion approach [20], ansatz scheme [21], modified Kudryashov scheme [22], homoseparation analysis scheme [23], extended mapping method [24], mod- ified simple equation method [25], modified extended mapping method [26], reduced differential transform approach [27], extended direct algebraic approach [28], functional Fractal Fract. 2023, 7, 138. https://doi.org/10.3390/fractalfract7020138 https://www.mdpi.com/journal/fractalfract