We consider the following nonlinear Schr\"{o}dinger equation of derivative type: \begin{equation}i \partial_t u + \partial_x^2 +i |u|^{2} \partial_x +b|u|^4u=0 , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in\mathbb{R}. \end{equation} If $b=0$, this is known as a gauge equivalent form well-known (DNLS), which mass critical and completely integrable. The can be considered generalized DNLS w...

In this work, we have applied Elzaki transform and He's homotopy perturbation method to solvepartial dierential equation (PDEs) with time-fractional derivative. With help He's homotopy per-turbation, we can handle the nonlinear terms. Further, we have applied this suggested He's homotopyperturbation method in order to reformulate initial value problem. Some illustrative examples aregiven in ord...

In this study, a numerical solution of singular nonlinear differential equations, stemming from biology and physiology problems, is proposed. The methodology is based on the shifted Chebyshev polynomials operational matrix of derivative and collocation. To assess the accuracy of the method, five numerical problems, such as the human head, Oxygen diffusion and Bessel differential equation, were ...

A polynomial-in-time growth bound is established for global Sobolev $$H^s({\mathbb {T}})$$ solutions to the derivative nonlinear Schrödinger equation on circle with $$s>1$$ . These bounds are derived as a consequence of smoothing effect an appropriate gauge-transformed version periodic Cauchy problem, according which solution its linear part removed possesses higher spatial regularity than init...

We implement the dressing method for a novel integrable generalization of the nonlinear Schrödinger equation. As an application, explicit formulas for the N-soliton solutions are derived. Moreover, as a by-product of the analysis, we find a simplification of the formulas for the N-solitons of the derivative nonlinear Schrödinger equation given by Huang and Chen. AMS Subject Classification (2000...

We consider Darboux transformations for the derivative nonlinear Schr\"odinger equation. A new theorem of operators with no term are presented and proved. The solution is expressed in quasideterminant forms. Additionally, parabolic soliton solutions equation given as explicit examples.

We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for th...

In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (Gʹ/G)-expansion method has been implemented, to celeb...

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space ...

The Richards equation is widely used as a model for the flow of water in unsaturated soils. For modelling one-dimensional flow in a homogeneous soil, this equation can be cast in the form of a specific nonlinear partial differential equation with a time derivative and one spatial derivative. This paper is a survey of recent progress in the pure mathematical analysis of this last equation. The e...