Diverse geometric shape solutions of the time-fractional nonlinear model used in communication engineering

Various nonlinear characteristics of breather/rogue waves and controllable interaction phenomena for a new KdV equation with variable coeffcients

“…We will investigate the breather/rogue wave solutions for equation (8) in this section. In order to obtain the bilinear form of equation (8), we suppose that…”

“…where Φ = Φ(T, X, Y). Substituting equation (12) into equation (8), we obtain the following bilinear form…”

“…. In relation to equation (13), we adopt a '3-3-1' model [28] for exploring the exact solutions of equation (8). This model consists of 3 neurons in the input layer l 0 , 3 neurons in the hidden layer l 1 , and 1 neuron in the output layer.…”

“…Nonlinear phenomena exist widely in nature, and nonlinear partial differential equations (NLPDEs) are indispensable tools to describe the nonlinear phenomena. Nowadays, NLPDEs are widely used in water waves [1], elastic mechanics [2], quantum mechanics [3], plasma physics [4,5], nonlinear optics [6,7], communication engineering [8,9], biomedicine [10] and other fields [11][12][13]. It is well known that in some real-world problems, NLPDEs with variable coefficients provide a more realistic perspective on the inhomogeneities of media and non-uniformities of boundaries in comparison than NLPDEs with constant coefficients, and have applications in various fields such as optical fibers [14], propagations of wave and rogue waves [15], and various other physical systems [16][17][18][19].…”

The Extended Direct Algebraic Method for Extracting Analytical Solitons Solutions to the Cubic Nonlinear Schrödinger Equation Involving Beta Derivatives in Space and Time

“…As of now, the beta derivative has not been found to have any limitations and it fulfills all the properties associated with integerorder derivatives. Furthermore, it exhibits the property of yielding a derivative of zero for constant functions [40][41][42]. The beta derivative is a non-local derivative that exhibits its distinctiveness when applied to functions that embody the entire characteristic of the function itself.…”