Outset of multiple soliton solutions to the nonlinear Schrödinger equation and the coupled Burgers equation

Abstract

The nonlinear Schrödinger equation and the coupled Burgers equation illustrate the status of quantum particles, shock waves, acoustic transmission and traffic flow. Therefore these equations are physically significant in their own right. In this article, the new auxiliary equation method has been contrivanced in order to rummage exact wave solutions to previously stated nonlinear evolution equations (NLEEs). We have developed ample soliton solutions and have to do with the physical importance of the acquired solutions by setting the specific values of the embodied parameters through portraying figures and deciphered the physical phenomena. It has been established that the executed method is powerful, skilled to examine NLEEs, compatible to computer algebra and provides further general wave solutions. Thus, the investigation of exact solutions to other NLEES through the new auxiliary method is prospective and deserves further research.

Diverse Soliton Structures for Fractional Nonlinear Schrodinger Equation, KdV Equation and WBBM Equation Adopting a New Technique

“…When 2 − 4 < 0 and ≠ 0, Making a comparable study of the found solutions to those existing ever it is claimed that this study ensures the diversity and novelty of our results [32][33][34][35][36][37][38].…”

“…The solutions of Eq. (2.5) are available in [38]. Now, we formulate the closed form wave solutions of the following space and time fractional nonlinear partial differential equations:…”

“…The literature carries out a lot of work with the equations we aim to explore. Instantly, the fractional nonlinear Schrodinger equation has been demonstrated by Hemida et al, (2012) via homotopy analysis method and found few series solutions [32]; the same equation for = 1 has been considered by and found three analytic solutions only by using the ( ′ / , 1/ )-expansion method and one solution by the (1/ ′)-expansion method [33]; arbitrary order nonlinear Schrodinger equation has been studied through fractional Riccati expansion method and achieved four exact wave solutions by Salam et al [34]; the new type F-expansion method has been applied to construct exact solutions of the space-time fractional cubic Schrodinger equation by Pandir and Duzgun [35]; Wazwaz and Kaur have used variational iteration method and improved ( ′ / )expansion method which left several solutions to the integer order cubic Schrodinger equation [36]; Cheema and Younis have considered the same equation for = 1 and obtained sixteen solutions by the extended Fan sub-equation method [37]; have investigated the equation of integer order by new auxiliary equation method and found soliton types solutions [38]. The conformable space-time fractional KdV equation has been studied by Yaslan and Girgin (2019) via the ( ′ / 2 )-expansion method for analytic wave solutions [39]; Odibat (2017) has employed Riccati equation approach to the KdV equation and constructed wave solutions [40]; the local variational iteration method and the local fractional series expansion method has been adopted by Jasim and Baleanu (2019) to study the fractional KdV equation [41]; Liu and Zhang (2018) have investigated the same equation by improved ( ′ / )-expansion method and found exact analytic solutions [42]; Jacobi elliptic function expansion method has been assumed to examine the KdV equation by Dascioglu et al (2017) [43].…”

Study of the parametric effects on soliton propagation in optical fibers through two analytical methods

“…In that context, scientists have been solving the problems by explaining those mathematical models through various methods have been exposed to solve the mathematical problems, the solutions from each schemes of which have given various solutions and research on these is ongoing. Among the above methods, very detailed results are found in these two methods namely, improved Bernoulli sub-equation function method (IBSEFM) [22,23] and the new auxiliary equation method (NAEM) [24] which can be further enriched by explaining our research.…”

Adequate soliton solutions to the time fractional Zakharov-Kuznetsov equation and the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation

“…Therefore a variety of powerful and modern approaches using computer algebra such as Mathematica or Maple, have recently been developed by several researchers. The available methods are: the auxiliary equation method (Akbar, Ali, & Tanjim, 2019;Akbulut, Kaplan, & Bekir, 2016;Ismail, 2016), the G 0 =G ð Þ-expansion method (Alam, Akbar, & Mohyud-Din, 2014;Bekir & Guner, 2013;Islam & Akbar, 2018a, 2018b, 2018c , the tanh-function method (Ali, 2007), the first integral method (Younis, 2013), the fractional sub-equation method (Alzaidy, 2013), the exp-function method (Zheng, 2013;Yokus, Durur, Ahmad, & Yao, 2020), the modified Kudryashov method (Ege & Misirli, 2014), the variational iteration method (He & Latifizadeh, 2020;Neamaty, Agheli, & Darzi, 2015), the homotopy perturbation method (Ain et al, 2020;Fereidoon, Yaghoobi, & Davoudabadi, 2011;He, 2019aHe, , 2019bYu, He, & Garcia, 2019), the modified simple equation method (Islam & Akbar, 2018c), the Jacobi elliptic method (Zheng & Feng, 2014), the differential transformation method (Rabtah, Erturk, & Momani, 2010), the modified trial equation method (Bulut, Baskonus, & Pandir, 2013), the finite element method (Deng, 2009), the Adomin decomposition method (Dahmani, Mesmoudi, & Bebbouchi, 2008), the Taylor series method , the variational iteration algorithm-I (Ahmad & Khan, 2019;Ahmed, Khan, & Cesarano, 2019) the variational iteration algorithm-II Ahmad, Seadawy, Khan, & Thounthong, 2020), the Riccati transformation method …”