New Complex and Hyperbolic Forms for Ablowitz–Kaup–Newell–Segur Wave Equation with Fourth Order

Abstract

Researching different solutions of nonlinear models has been interesting in different fields of science and application. In this study, we investigated different solutions of fourth-order nonlinear Ablowitz– Kaup–Newell–Segur wave equation. We have used the sine-Gordon expansion method (SGEM) during this research. We have given the 2D, 3D, and contour graphs acquired from the values of the solutions obtained using strong SGEM.

Complex and Real Optical Soliton Properties of the Paraxial Non-linear Schrödinger Equation in Kerr Media With M-Fractional

“…In this subsection, we apply the extended sinh-Gordon method to the M-fractional paraxial wave equation that labeled Equation (12). Consider the Equation 17and applying the principal homogeneous balance between the between U ′′ and U 3 , we find n = 1.…”

“…the dispersion, diffraction, and Kerr non-linearity, respectively. In Equation (12) if ab > 0 we get elliptic non-linear Schrödinger equation and if ab < 0, Equation (12) becomes hyperbolic nonlinear Schrödinger equation. Now assume the following wave transformations:…”

“…Inserting Equation (13) into Equation (12), and separate the result into the real and imaginary part, we get…”

“…In the research papers, researchers have been noted several computational methods for solving NPDEs, building separate solitons, and other alternatives for distinct types of NPDEs such as, the Haar wavelet method [1], the homotopy perturbation method [2], the Adomian decomposition method [3,4], the shooting method [5][6][7][8], the sine-Gordon expansion method [9][10][11][12], the inverse scattering method [13], the sinh-Gordon expansion method [14][15][16], the tan(φ (ξ ) /2)-expansion method [17,18], the inverse mapping method [19], modified exp (−ϕ (ξ ))-expansion function method [20][21][22][23], the decomposition-Sumudu-like-integral-transform method [24], a functional variable method [25], the Bernoulli sub-equation function method [26][27][28], modified exponential function method [29], the modified auxiliary expansion method [30], the Riccati-Bernoulli sub-ODE method [31], the extended trial equation method [32,33], and tanh function method [34,35]. Also, different methods have been used to solve fractional differential equation such as, the finite difference method [36], the improved Adams-Bashforth algorithm [37,38], Adams-Bashforth-Moulton method [39], the extended fractional sinh-Gordon expansion method …”

Analytical Insights into a Generalized Semidiscrete System with Time-Varying Coefficients: Derivation, Exact Solutions, and Nonlinear Soliton Dynamics

“…Recently, some new analytical approaches [16][17][18][19][20][21][22][23][24][25], novel nonlinear results [26][27][28][29][30][31][32][33][34], and soliton dynamics in inhomogeneous media [35][36][37] for differential equations have been reported. Generally speaking, extending the existing methods for differential equations encounters the difficulty in searching for the iterative relations of discrete variables.…”