The modified extended tanh technique ruled to exploration of soliton solutions and fractional effects to the time fractional couple Drinfel'd–Sokolov–Wilson equation

Exploring Novel Soliton Solutions to the Time-Fractional Coupled Drinfel’d–Sokolov–Wilson Equation in Industrial Engineering Using Two Efficient Techniques

“…In this section, we examine the novelty and scientific contributions of our paper by comparing our findings with those in [6]. This comparison is organized into two parts, focusing on the similarities and differences between our studies.…”

“…Reference [6] employed the modified extended tanh method (one variable) to derive exact solutions, whereas this study utilizes two efficient methods: the Sardar subequation method (one variable) and the Ω ′ Ω , 1 Ω -expansion technique (two variables); ii. While they only obtained multiple singular and bell-shaped solitons, this study includes a wider variety, such as kink, dark, multi-dark, bell-shaped, bright, multiple singular, and singular solitons; iii.…”

“…Their applications span various fields including fluid mechanics, optical physics, finance, plasma physics, biology, and many others. Unlike classical NLEs, FNLDEs pose challenges in solving due to their non-local nature [5,6]. Nevertheless, researchers employ fractional models to find exact solutions because they more precisely represent complex systems with memory and hereditary properties.…”

“…By comprehensively studying traveling wave propagation within the DSW equation, we gain valuable insights into their behavior across these systems, thereby deepening our understanding of their dynamics. The DSW equations, with fractional derivative concerning time, are given as follows [6,11]:…”

Dynamical exploration of optical soliton solutions for M-fractional Paraxial wave equation

“…Most of such phenomena in real life can be represented as nonlinear PDEs. To investigate various exact and explicit solutions there are many schemes established, such as bilinear Bäcklund transformation [ 8 ], transformed rational function method [ 9 ], tanh method [ 10 ], Bifurcation analysis [ 11 ], tanh-coth method [ 12 ], extended tanh–coth expansion method [ 13 ], multiple exp-function algorithm [ 14 ], the Kudryashov-expansion method [ 15 ], the extended Kudryashov method [ 16 ], Advanced exp (- ϕ ( ξ ))-Expansion Scheme [ 17 ], modified extended tanh scheme [ 18 , 19 ], MSE scheme [ 20 ], constraint and complexification method [ 21 ], Hirota bilinear [ 22 – 25 ], Darboux transformation [ 26 ], MEAEM [ 27 ], Generalized Darboux transformation [ 28 ], the extended direct algebraic method [ 29 , 30 ], Square operator method [ 31 , 32 ], the extended unified method [ 33 ], new auxiliary equation method [ 34 , 35 ], Sardar sub-equation method [ 36 , 37 ], Evans function method [ 38 ], a new adaptive numerical method [ 39 ], sine-Gordon expansion method [ 40 ], advanced generalized ( G ′/ G )–expansion scheme [ 41 ], the double variable expansion method [ 42 ], the double expansion method [ 43 ], exp (− ϕ ( η ))-expansion method [ 44 ], etc.…”