The $$\left( \frac{\boldsymbol{G}^{\prime }}{\boldsymbol{G}},\frac{\boldsymbol{1}}{\boldsymbol{G}}\right)$$ G ′ G , 1 G -expansion method and its applications for constructing many new exact solutions of the higher-order nonlinear Schrödinger equation and the quantum Zakharov–Kuznetsov equation

Solitons and Other Exact Solutions for Two Nonlinear PDEs in Mathematical Physics Using the Generalized Projective Riccati Equations Method

“…and, differentiating (18) and substituting the resulting equation in (17), we have the following nonlinear ODE:…”

“…Nonlinear waves appear in various scientific fields, especially in physics such as fluid mechanics, plasma physics, optical fibers, and solid state physics. In recent years, many powerful tools have been established to determine soliton and periodic wave solutions of nonlinear PDEs, such as the ( / )-expansion method [1][2][3][4][5][6], the extended auxiliary equation method [7,8], the new mapping method [9][10][11], the generalized projective Riccati equations method [12][13][14][15][16][17], and the ( / , 1/ )-expansion method [18]. Conte and Musette [12] presented an indirect method to find solitary wave solutions of some nonlinear PDEs that can be expressed as polynomials in two elementary functions which satisfy a projective Riccati equation [19].…”

Wave Structures for Nonlinear Schrodinger Types Fractional Partial Differential Equations Arise in Physical Sciences

“…Recently, the search for a different type of solution of the fractional nonlinear Schrodinger's model has represented numerous scientists and researchers (Saxena & Kalla, 2010;Abdel-Salam et al, 2016;Younis et al, 2017;Rizvi et al, 2017). The established methods available in the literature are 𝐺 ′ /𝐺expansion method for nonlinear fractional differential equations by Bekir and Guner (2013), the meshless method of lines (Mohyud-Din, 2012), Adomian decomposition scheme (Guo, 2019), the new generalized 𝐺 ′ /𝐺-expansion method (Alam et al, 2014;Alam, 2015;Alam & Li, 2019), the Darcy law method (Sheikholeslami, 2017), the reproducing kernel algorithm (Omar, 2019a), topological solitons for certain differential equation by Biswas et al (2013a), Biswas et al (2013b), Laplace-Adomian decomposition method (Shah et al, 2019), reproducing kernel Hilbert space method (Omar, 2019b), homotopy perturbation method (Golmankhaneh & Baleanu, 2011), improved sub-equation method (Karaagac, 2019), Schrodinger's equation (Rizvi et al, 2017;Li et al, 2019), generalized exponential rational function method (Ghanbari et al, 2019), the Kudryashov methods (Saha, 2016;Kudryashov, 2012), integral transform based decomposition methods are used for Schrodinger and other differential equations by Nuruddeen (2017); Nuruddeen & Nass (2017), Various phenomena such as shallow water waves and multicellular biology dynamics arising in the nonlinear physical science (Lu et al, 2017;Bazyar & Song, 2017), the (𝐺 ′ /𝐺, 1/𝐺)-expansion method (Zayed & Abdelaziz, 2012;Zayed et al, 2018;Zayed & Alurrfi, 2016), the Jacobi collocation method (Doha et al, 2014;Bhrawy e...…”

Exact Traveling Wave Solutions of Certain Nonlinear Partial Differential Equations Using the G′/G2Sirisubtawee1, Koonprasert2 2018Advances in Mathematical Physics460190View full textAdd to dashboardCite

“…Over the last few decades, exact solutions, analytical approximate solutions, and numerical solutions of many NPDEs have been successfully obtained. The methods for obtaining exact explicit solutions of NPDEs are, for example, the ( / )-expansion method [6][7][8], the ( / , 1/ )-expansion method [9][10][11], the novel ( / )-expansion method [12], the tanh-function method [13], the exp-function method [14,15], the F-expansion method [16], Hirota's direct method [17,18], Kudryashov method [19,20], and the extended auxiliary equation method [21]. Examples of the methods for obtaining analytical approximate solutions to NPDEs are the variational iteration method [22,23] (VIM), the Adomian decomposition method [24,25] (ADM), the homotopy perturbation method [26,27] (HPM), and the reduced differential transform method [28].…”