This paper retrieves some new optical solutions to the Kundu–Mukherjee–Naskar (KMN) equation in the context of nonlinear optical fiber communication systems. In this regard, the generalized Kudryashov and new auxiliary equation methods are applied to the KMN equation and consequently, dark, bright, periodic U-shaped and singular soliton solutions are explored. The discrepancies between the present obtained solutions and the previously obtained solutions by using different methods are discussed. The time fractional derivative and an oblique wave transformation in coordination with the methods of interest are considered for acquiring new optical wave solutions of the KMN equation in the sense of conformable derivative and wave obliqueness, respectively. The effects of obliqueness and fractionality on the attained solutions are demonstrated graphically along with its physical descriptions. It is found that the optical wave phenomena are changed with the increase of obliqueness as well as fractionality. All the obtained optical solutions are found to be new in the sense of conformable derivative, wave obliqueness, and the applied methods. Finally, it is found that the utilized methods and the relevant transformation are powerful over the other methods and it can be applicable for further studies to explain the pragmatic phenomena in optical fiber communication systems.
The paper investigates a class of exactly solvable third order nonlinear evolution equation . A list of unknown function F(u) is reported for which considered equation contains the nontrivial Lie point symmetries. Moreover, nonlinear self-adjointness is discussed and it is examined that it is not strictly self-adjoint equation for physical parameter A 6= 0 but quasi self-adjoint or more generally nonlinear self-adjoint. Additionally, it is observed that Calogero-Degasperis-Fokas (CDF) equation admits a minimal set of Lie algebra under invariance criteria of Lie groups. These classes are utilized one by one to construct the similarity variables to reduce the dimension of the discussed equation. Additionally, Lie symmetries are used to exhibit the associated conservation laws. Henceforth, Lie symmetry reductions of CDF equation are reported with the help of an optimal system. Meantime, this Lie symmetry method reduces the considered equation into ordinary differential equations. Moreover, well known (G’/G)-expansion method is used to get the exact solutions. The obtained new periodic and solitary wave solutions can be widely used to provide many attractive complex physical phenomena in the different fields of sciences.
In this work, we implement some analytical techniques such as Exp--Function method, Tan and Tanh methods, Extended Tan and Tanh methods, and Sech method for solving the fractional nonlinear partial differential equation with a truncated M-fractional derivative, which contain exponential terms its name, General Modified fractional Degasperis-Procesi-Camassa-Holm equation with a truncated M-fractional derivative. These methods can be used as an alternative to obtain exact solutions of different types of differential equations applied in engineering mathematics. Finally, we obtain the analytical solution of the M-fractional heat equation and present a graphical analysis.