Dispersive optical solitons of Biswas–Arshed equation with a couple of novel approaches

M-truncated soliton solutions of the fractional (4+1)-dimensional Fokas equation

“…Figs. (1)(2) show some of the obtained solutions. We display 3D and contour graphs of ϑ1,1 (x, y, z, s, t) in Eq.…”

“…Because of its wide application, investigation of the analytical and soliton solutions of the NLPDEs with integer or fractional order has been very popular among authors over the past few decades. Numerous techniques consisting of the analytical and numerical methods have been improved to gain the soliton and analytical solutions of the PDEs such as the combined improved Kudryashov-new extended auxiliary sub equation method [1], the enhanced modified extended tanh-expansion approach [2,3], the sine-Gordon equation approach [4], F-expansion method [5], the tanh-coth function, the modified kudryashov expansion and rational sine-cosine approaches [6], the Riccati equation method [7], the tan(Θ/2) expansion approach [8], the Jacobi elliptic functions methodology [9], the generalized Bernoulli sub-ODE scheme [10], the extended ( G ′ G 2 )-expansion scheme [11], Nucci's reduction method [12], the new Kudryashov method [13][14][15], the sub-equation method based on Riccati equation [16], and the modified Sardar subequation method [17].…”

“…Thus, various and substantial definitions of fractional derivatives types have been enhanced in the literature such as: Grunwald-Letnikov, Riemann-Liouville, Caputo [20], Caputo-Fabrizio [21], Atangana-Baleanu [22,23], the conformable fractional derivative [24], and the M-truncated derivative [11]. The (4 + 1)-dimensional Fokas equation is expressed by the following structure [25]: 4ϑ tx −ϑ xxxy +ϑ xyyy +12ϑ x ϑ y +12ϑϑ xy −6ϑ zs = 0, (1) in which ϑ(x, y, z, s, t). The model was firstly presented by A. S. Fokas [25] and the FE is the extension form of Davey-Stewartson and Kadomtsev-Petviashvili equations to some higher-dimensional nonlinear wave equations [25].…”

Solitary wave solutions of the (4+1)-dimensional Fokas equation via an efficient integration technique

“…have been developed for both numerical and symbolic programming and thus great progress has been made in solving many non-linear problems waiting to be solved before. Many researchers started to work in this field and developed new equations (for example, Kadomtsev Petviasvili (KP) and its variants, Kdv forms, Boussinessq forms, Maccari systems, Kawahara [11][12][13][14][15][16][17][18][19][20]. The equations listed above reflect only a fraction of the equations developed in this field and studied by many researchers.…”

Soliton solutions of (2+1)-dimensional non-linear reaction-diffusion model via Riccati-Bernoulli approach

“…In this way, both numerical and analytical computational techniques, methods and models have been developed and successfully applied in the solution of many non-linear equations. As an example of these equations, we can give: Schrodinger equation [13], Camassa-Holm equation [15], Kundu-Muhherjee-Naskar model [16], Biswas-Milovic equation [17], Triki-Biswas equation [18], Radhakrishnan-Kundu-Lakshmanan model [19], Kdv forms [20], Zoomeron equation [21], Fornberg-Whitham equation [22], Mikhailov-Novikov-Wang equation [23], Kawahara equation [24], Biswas-Arshed equation [25] and many more. One of these equations is nonlinear reaction-diffusion equation which has a large application area.…”