Bright–Dark Soliton Waves’ Dynamics in Pseudo Spherical Surfaces through the Nonlinear Kaup–Kupershmidt Equation

Abstract

The soliton waves’ physical behavior on the pseudo spherical surfaces is studied through the analytical solutions of the nonlinear (1+1)–dimensional Kaup–Kupershmidt (KK) equation. This model is named after Boris Abram Kupershmidt and David J. Kaup. This model has been used in various branches such as fluid dynamics, nonlinear optics, and plasma physics. The model’s computational solutions are obtained by employing two recent analytical methods. Additionally, the solutions’ accuracy is checked by comparing the analytical and approximate solutions. The soliton waves’ characterizations are illustrated by some sketches such as polar, spherical, contour, two, and three-dimensional plots. The paper’s novelty is shown by comparing our obtained solutions with those previously published of the considered model.

Soliton: A dispersion-less solution with existence and its types

“…The dark solitons are more stable and less affected by background noise and interference compared to the light solitons. Apart from NLS equation the Chaffee–Infante equation [ 22 ] and Kaup–Kupershmidt equation [ 23 ] also plays an important role in bright and dark soliton. The envelope soliton resulting from the NLS equation has been presented in Section 4.…”

“…In addition to the above equations, there are many equations which have important applications in field of science and technology and yields soliton type solutions i.e. Kolmogorov–Petrovskii–Piskunov equation [ 73 ], generalized (2 + 1)-dimensional shallow water waves equation [ 74 ], human immunodeficiency virus (HIV)-1 infection of CD4+ T-cells fractional biomathematical model for constructing novel solitary wave solutions [ 75 ], Fokas–Lenells equation [ 49 ], Klein–Fock–Gordon equation [ 76 ] relates to Schrodinger equation, phi-four equation [ 77 ] which is a particular case of the Klein–Fock–Gordon equation, telegraph equation [ 52 ], Chaffee–Infante equation [ 22 ], Benjamin–Bona–Mahony (BBM) [ 38 ], Cahn–Allen equation [ 78 ], Klein–Gordon–Zakharov equation [ 36 , 79 ], Kaup–Kupershmidt equation [ 23 ], Fisher-Kolmogorov-Petrovskii-Piskunov [ 80 ], Kadomtsev−Petviashvili equation [ 81 ], Ginzburg–Landau equation [ 82 ], Hirota–Satsuma–Shallow Water Wave Equation [ 83 ] (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation [ 84 ], cubic–quintic nonlinear Helmholtz model [ 85 ], Ostrovsky equation [ 13 ], Vakhnenko–Parkes equation which is reduced from the Ostrovsky equation [ 86 ], complex perturbed Gerdjikov–Ivanov (CPGI) equation [ 51 ], etc.…”

Novel and accurate solitary wave solutions of the conformable fractional nonlinear Schrödinger equation

“…18 As a result, numerous computational, semi-analytical, and numerical techniques have been developed, including the extended simplest equation method, the extended tanh expansion method, the generalized Kudryashov method, the modified Khater method, the Sin-Cos expansion method, the Sech-Tanh expansion method, the Sine-Gordon expansion method, and the new auxiliary equation method. [19][20][21][22][23] However, there is no one-sizefits-all analytical technique for all nonlinear evolution problems. [24][25][26] In this context, we used the Khater II scheme to the 3-FNLS [27][28][29][30][31] to evaluate an unlimited number of solutions and then used the TQBS scheme 32,33 to determine the absolute value of error between the precise and numerical solutions.…”

“…18 As a result, numerous computational, semi-analytical, and numerical techniques have been developed, including the extended simplest equation method, the extended tanh expansion method, the generalized Kudryashov method, the modified Khater method, the Sin–Cos expansion method, the Sech–Tanh expansion method, the Sine-Gordon expansion method, and the new auxiliary equation method. 19-23 However, there is no one-size-fits-all analytical technique for all nonlinear evolution problems. 24-26…”

Diverse Variety of Exact Solutions for Nonlinear Gilson–Pickering Equation

“…Therefore, we need different and new techniques to solve such kinds of NLEEs. For this aspect, researchers have developed different, unique, and powerful techniques to solve NLEEs, which include the modified simple equation technique [13][14][15], the variational iteration method [16,17], the variational method [18], the first integral method [19], the perturbation method [20], method of integrability [21], the nonperturbative technique [22], the modified F-expansion method [23][24][25], the exp-function method [26,27], the sine-cosine method [28][29][30], the Riccatti-Bernoulli sub-ODE method [31,32], the Jacobi elliptic function method [33,34], the generalized Kudryashov method [35,36], the functional variable method [37,38], the modified Khater method [39,40], the new extended direct algebraic method [41,42], the Lie symmetry technique [43,44], the (G /G)-expansion method [45], the tanh-coth method [46,47], the new auxiliary equation method [48,49], the (G /G, 1/G)expansion method [50], the technique of (m + 1, G ) [51], the addendum to Kudryashov's method [52], and many others [53]…”