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

Abstract

In this paper, the cubic-quartic nonlinear Schrödinger and resonant nonlinear Schrödinger equation in parabolic law media are investigated to obtain the dark, singular, bright-singular combo and periodic soliton solutions. Two powerful methods, the m + G ′ G improved expansion method and the exp − φ ξ expansion method are utilized to construct some novel solutions of the governing equations. The obtained optical soliton solutions are presented graphically to clarify their physical parameters. Moreover, to verify the existence solutions, the constraint conditions are utilized.

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

“…The exact solutions of Equation (12) may be given as sinh ( θ ) = ± csch (ζ ) or sinh ( θ ) = ± isech (ζ ) , (13) and cosh (θ ) = ± coth (ζ ) or cosh (θ ) = ± tanh (ζ ) . (14) Letting solutions of Equation (10) along with Equations (13) and (14) as the form…”

“…In fact it might happen that the GVD is tiny and thus totally ignored, in this case the dispersion effect is determined by third and fourth order dispersion effects. Subsequently, this equation has been studied in a variety of ways, such as the Lie symmetry [13], both the m + G ′ G -improved expansion, and the exp (−ϕ (ξ )) −expansion methods [14], and the semiinverse variation principle method [4]. In this study, the extended sinh-Gordon expansion method (ShGEM) is applied to the non-linear cubic-quartic Schrödinger equations with the Parabolic law of fractional order, which is given by…”

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

“…In 2015, Caputo and Fabrizio discovered a new operator of arbitrary order, namely, Caputo‐Fabrizio (CF) operator with arbitrary order and enforced to the several linear and nonlinear physical problems . In 2016, Atangana and Baleanu introduced another nonsingular derivative based on Miitag‐Leffler kernel and applied to the many problems . The parabolic heat equation was first developed and introduced Joseph Fourier in 1822.…”

On the New Wave Behaviors of the Gilson-Pickering Equation

“…Among the most profitable strategies for examining such nonlinear physical phenomena is to seek for the exact solutions of NLPDEs [1][2][3][4][5]. In recent years, a variety of effective methods have been implemented to investigate the exact solutions of nonlinear partial differential equations, such as Hirota's bilinear method [6], the Adomian decomposition method [7], the exp(− (ξ ))-expansion method [8], the sine-Gordon expansion method [9], the Bernoulli sub-equation method [10,11], the shooting method with the fourth-order Runge-Kutta scheme [12,13], the generalized exponential rational function method [14][15][16][17][18], the modified exponential function method [19], the modified auxiliary expansion method [20], the homotopy perturbation Sumudu transform method [21], the homotopy perturbation transform method [22,23], and the fractional homotopy analysis transform method [24].…”