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Fractional nonlinear models involving the underlying mechanisms of numerous complicated physical phenomena arising in nature of real world have been taken major place of research arena during the couple of years for their significant roles. The study about the nonlinear optical and quantum context connecting to mostly Kerr law media as well as power law, dual-power law, triple-power law, saturable law, logarithm law and polynomial low is increasing at an inconceivable rate. In this exploration, the integrable generalized (2 + 1)-dimensional nonlinear Schrödinger system of equations in the sense of conformable fractional derivative is considered to unravel by means of two innovative schemes namely improved tanh method and rational G ′ / G -expansion method. The advised techniques are employed to seek for appropriate analytic wave solutions after converting the mentioned equation to an ordinary differential equation by introducing a wave variable alteration. The hyperbolic, trigonometric and rational function solutions are successfully gained and put forwarded for graphical representations. The assembled solutions are figured out in 3-D, 2-D and contour formats to illustrate their different views which appeared as kink type, anti-kink type, singular kink type, bell shape, anti-bell shape, singular bell shape, cuspon, peakon, periodic and singular periodic etc. The entire study bears the diversity and novelty of found solutions and applied techniques after making a comparable study with recent work recorded in the literature. [ABSTRACT FROM AUTHOR]

Numerous tangible incidents in physics, chemistry, applied mathematics and engineering are described successfully by means of models making use of the theory of derivatives of fractional order and research in this area has grown significantly. In this article, we establish exact solutions to some nonlinear fractional differential equations. The recently established rational ( G ' / G )-expansion method with the help of fractional complex transform is used to examine abundant further general and new closed form wave solutions to the nonlinear space-time fractional mBBM equation, the space-time fractional Burger's equation and the space-time fractional ZKBBM in the sense of the Jumarie modified Riemann-Liouville derivative. The fractional complex transform reduces the nonlinear fractional differential equations into nonlinear ordinary differential equations and then the theories of ordinary differential equations are implemented effectively. It is observed that the performance of this method is reliable, useful and gives new and broad-ranging closed form solutions than the existing methods. [ABSTRACT FROM AUTHOR]

Nonlinear evolution equations of arbitrary order bearing a significantly broad range of capability to illustrate the underlying behavior of naturalistic structures relating to the real world, have become a major source of attraction of scientists and scholars. In quantum mechanics, the nonlinear dynamical system is most reasonably modeled through the Schrödinger-type partial differential equations. In this paper, we discuss the (2+1)-dimensional time-fractional nonlinear Schrödinger equation and the (1+1)-dimensional space–time fractional nonlinear Schrödinger equation for appropriate solutions by means of the recommended enhanced rational (G ′ / G) -expansion technique adopting Cole–Hopf transformation and Riccati equation. The considered equations are turned into ordinary differential equations by implementing a composite wave variable replacement alongside the conformable fractional derivative. Then a successful execution of the proposed method has been made, which brought out supplementary innovative outcomes of the considered equations compared with the existing results found so far. The well-generated solutions are presented graphically in 3D views for numerous wave structures. The high performance of the employed technique shows the acceptability which might provide a new guideline for research hereafter. [ABSTRACT FROM AUTHOR]

Arbitrary order partial differential equations involving nonlinearity have mostly been utilized to portray interior behavior of numerous real-world phenomena during the couple of years. The research about the nonlinear optical context relating to saturable law, power law, triple-power law, dual-power law, logarithm law, polynomial low and mostly visible Kerr law media is increasing at a remarkable rate. In this exploration, the space and time fractional nonlinear Schrodinger equation with the quadratic-cubic nonlinearity is taken into account for optical solitons and other solutions by means of the improved tanh method and the rational G ′ / G -expansion method. An alteration of wave variable with the assistance of conformable fractional derivative reduces the suggested equation into an ordinary differential equation. A successful adaptation of the mentioned techniques makes available plentiful solitons and other types solutions of the above equation. The originated solutions might be accommodating to analyze the underlying structures of nonlinear optics. We bring out the diverse 3-D and 2-D shapes for solitons to depict the physical appearances of the achieved solutions. The performance of the adopted methods is mentionable which claimed to be eligible for using to unravel any other nonlinear partial differential equations emerging in nature sciences. [ABSTRACT FROM AUTHOR]