Multiple soliton, M-lump and interaction solutions to the (3+1)-dimensional soliton equation

Study on the (2+1)-dimensional extension of Hietarinta equation: soliton solutions and Bäcklund transformation

“…Through the Hirota bilinear method, a large number of local wave solutions have been studied for many NLEEs, such as soliton solution [4,[18][19][20], lump solution [21,22], interaction solution [23,24], and rouge wave solution [25,26]. Moreover, it is particularly important to study the dynamic evolution of nonlinear waves and other factors that affect them [27][28][29].…”

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.…”

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

“…The NLEEs equation has not yet been solved using such a technique. Due to this, several researchers have created a variety of trustworthy, effective, and simple methods for solving NLEEs equations like as the procedure of enhanced (G ′/G)-expansion [ 13 ], the manner of modified Kudryashov [ 14 ], the procedure of modified simple equation [ 15 ], the procedure of sine-Gordon equation expansion [ 16 ], the procedure of extended sinh-Gordon equation expansion [ 17 ], the manner of exp -expansion [ 18 ], the new auxiliary equation and modified Kudryashov scheme [ 19 ], MSE scheme [ 20 ], the trial solution formula [ 21 ], the procedure of Frobenius integrable decomposition [ 22 ], – expansion and Sine-Gordon-expansion methods [ 23 ], the manner of multiple simplest equation [ 24 ], the manner of solitary wave ansatzes [ 25 ], the manner of simple equation [ 26 ], the manner of extended simplest equation [ 27 ], the scheme of modified extended tanh-function [ 28 ], the manner of Hirota [ [29] , [30] , [31] , [32] ], new extended direct algebraic method [ 33 ], MSE method [ 34 ], the tanh-coth method [ 35 ], Riemann-Hilbert problems [ [36] , [37] , [38] , [39] ] and so on. As a result, nonlinear science is emerging as one of the fundamental areas of research in the growing wave system.…”