Nonlinear Schrödingers equations with cubic nonlinearity: M-derivative soliton solutions by $\exp(-\Phi(\xi))$-Expansion method

Abstract

This paper uses the $\exp(-\Phi(\xi))$-Expansion method to investigate solitons to the M-fractional nonlinear Schrödingers equation with cubic nonlinearity.  The results obtained are dark solitons, trigonometric function solutions, hyperbolic solutions and rational solutions. Thus, the constraint relations between the model coefficients and the traveling wave frequency coefficient for the existence of solitons solutions are also derived.

Optical Soliton Solutions of the Cubic-Quartic Nonlinear Schrödinger and Resonant Nonlinear Schrödinger Equation with the Parabolic Law

“…Therefore, one can observe that the wave transformation considered in this paper in Equation (17) satisfies these conditions. We substituted all solutions to the main equations Equations (3) and (4), and they verified it; the constraint conditions Equations (20) and (29) were also used to verify this existence. The optical soliton solutions obtained in this research paper may be of concern and useful in many fields of science, such as mathematical physics, applied physics, nonlinear science, and engineering.…”

“…Various numeric and analytic techniques have been used to seek solutions for nonlinear differential equations such as the homotopy perturbation scheme [5], the Adams-Bashforth-Moulton method [6], the shooting technique with fourth-order Runge-Kutta scheme [7][8][9][10], the group preserving method [11], the finite forward difference method [12,13], the Adomian decomposition method [14,15], the sine-Gordon expansion method [16][17][18], the modified auxiliary expansion method [19], the modified exp(−ϕ (ξ)) expansion function method [20,21], the improved Bernoulli sub-equation method [22,23], the Riccati-Bernoulli sub-ODE method [24], the modified exponential function method [25], the improved tan(φ (ξ) /2) [26,27], the Darboux transformation method [28,29], the double G G , 1…”

Exact Soliton Solutions to the Cubic-Quartic Non-linear Schrödinger Equation With Conformable Derivative

“…The non-linear Schrödinger equation is a generalized (1+1)-dimensional version of the Ginzburg-Landau equation presented in 1950 in their study on supraconductivity and has been specifically reported by Chiao et al [1] in their research of optical beams. In the past several years, various methods have been proposed to obtain the exact optical soliton solutions of the non-linear Schrödinger equation [2][3][4][5][6][7][8][9][10][11][12]. Dispersion and non-linearity are two of the essential components for the distribution of solitons across inter-continental regions.…”

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

“…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 …”