Bright, Dark, and Rogue Wave Soliton Solutions of the Quadratic Nonlinear Klein–Gordon Equation

Abstract

This article reflects on the Klein–Gordon model, which frequently arises in the fields of solid-state physics and quantum field theories. We analytically delve into solitons and composite rogue-type wave propagation solutions of the model via the generalized Kudryashov and the extended Sinh Gordon expansion approaches. We obtain a class of analytically exact solutions in the forms of exponential and hyperbolic functions involving some arbitrary parameters with the help of Maple, which included comparing symmetric and non-symmetric solutions with other methods. After analyzing the dynamical behaviors, we caught distinct conditions on the accessible parameters of the solutions for the model. By applying conditions to the existing parameters, we obtained various types of rogue waves, bright and dark bells, combing bright–dark, combined dark–bright bells, kink and anti-kink solitons, and multi-soliton solutions. The nature of the solitons is geometrically explained for particular choices of the arbitrary parameters. It is indicated that the nonlinear rogue-type wave packets are restricted in two dimensions that characterized the rogue-type wave envelopes.

Dynamical interaction of solitary, periodic, rogue type wave solutions and multi-soliton solutions of the nonlinear models

“…Analytical solutions were found for Klein–Gordon equation with a combined potential in [35] . Bright, Dark and rogue wave soliton solution existed from quadratic nonlinear Klein–Gordon equation [36] .…”

Training Neural Networks by Time-Fractional Gradient Descent

“…On the other hand, for the complex time-fractional Schrödinger equation and the space-time fractional differential equation, novel precise solitary wave solutions are obtained by the modified simple equation scheme [21]. A number of solitary envelope solutions of the quadratic nonlinear Klein-Gordon equation also are constructed by the generalized Kudryashov and extended Sinh-Gordon expansion schemes [22].…”

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]…”