Hasibun Naher

The fractional nonlinear Schrödinger equation (FNLSE) describes the wave propagation in nonlinear optical fibers, ion-acoustic waves in plasmas, propagation of ultrashort laser pulses, and signal processing. In this study, we investigate the (3 + 1)-dimensional FNLSE with beta fractional derivative to explore soliton dynamics in diverse physical and engineering domains. A couple of analytical approaches, namely the extended sinh-Gordon expansion method and the two-variable (G′/G, 1/G)-expansion method, are used to determine the assorted soliton solutions expressed in trigonometric, hyperbolic, and rational forms. The obtained solutions include bright, bell-shaped, anti-peakon-shaped, anti-bell-shaped, periodic, singular, and singular periodic solitons. We examine the impact of the beta fractional parameter on soliton dynamics through graphical simulations and test the stability analysis using linear stability theory. We also conduct bifurcation analysis to investigate the qualitative behavior of the derived solutions, highlighting the emergence of static solitons via saddle–center bifurcation. The results indicate that the introduced methods generate a wide range of soliton structures and provide insights into their stability and dynamic properties. The results contribute to the theoretical understanding of soliton propagation in nonlinear optical fibers and other dispersive media.

Bangladesh is one of the most vulnerable countries in the world with around 718,000 deaths in the past fifty years. This country is especially in danger for cyclones because of its location in the triangular-shaped Bay of Bengal. The scientific scenario suggests that enlarged sea surface temperature will intensify cyclone movement. Tropical cyclones generate storm surges. Storm surges severely change the coastal environment, damage coastal structures, destroy forests and crops, inundate the coastline with saltwater and cause loss of lives. Due to overcrowding in the mainland in Bangladesh, poor and landless people live on the coast and they face frequent cyclones and associated surges. They affect food and drinking water and there is danger of transmission risks of infectious diseases, such as diarrhea, malaria, eye infections, skin diseases, etc. Some problems following a cyclone are usually created for their low literacy rate and poor knowledge of the environment. The tangible monitoring and warning of cyclones and associated surges should be given more priority for the region. The main objectives of this paper are to highlight the existing activities of the model in storm surges and related areas in the Bay of Bengal. We will explain the progress of a location-specific real-time standpoint prediction system for providing effective and timely surge forecasts. We will also introduce a model through numerical experiments using a severe cyclone in April 1991 to predict the storm surges that will be used to reduce economic losses and the number of deaths during a strong storm surge in the coastal area of Bangladesh.

In this article, the nonlinear partial fractional differential equation, namely the KdV equation is renewed with the help of modified Riemann-Liouville fractional derivative. The equation is transformed into the nonlinear ordinary differential equation by using the fractional complex transformation. The goal of this paper is to construct new analytical solutions of the space and time fractional nonlinear KdV equation through the extended () '/ G G-expansion method. The work produces abundant exact solutions in terms of hyperbolic, trigonometric, rational, exponential, and complex forms, which are new and more general than existing results in literature. The newly generated solutions show that the executed method is a well-organized and competent mathematical tool to investigate a class of nonlinear evolution fractional order equations.

We have generated many new non-travelling wave solutions by executing the new extended generalized and improved (G'/G)-Expansion Method. Here the nonlinear ordinary differential equation with many new and real parameters has been used as an auxiliary equation. We have investigated the Fisher equation to show the advantages and effectiveness of this method. The obtained non-travelling solutions are expressed through the hyperbolic functions, trigonometric functions and rational functional forms. Results showing that the method is concise, direct and highly effective to study nonlinear evolution equations those are in mathematical physics and engineering.

This paper presents the implementation of the new extended-expansion method to the Zhiber-Shabat equation and the Liouville equation. Using this method, a variety of explicit travelling wave solutions has been generated. Results also show that the obtained solutions are exclusively new and, the implemented scheme has significant improvement against other methods. Finally, the work presents here the profound implications for future studies of nonlinear evolution equations in visualizing the dynamics of the equations and may help to describe physical phenomena in mathematical physics.

In this article, the generalized and improved (G′/G)-expansion method has been proposed for further extension to generate many new travelling wave solutions. In addition, nonlinear ordinary differential equation is implemented as auxiliary equation including many parameters instead of linear ordinary differential equation. Moreover, the presentation of the travelling wave solutions is quiet new. The effectiveness and reliability of the method are shown by its application to the Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation. Some of our generated solutions turned into some known solutions, when parameters consider specific values and others are new

In this study, new extension of the (G'/G)-expansion method with nonlinear ODE has been introduced to investigate the generalized regularized long wave equation. Therefore, many new travelling wave solutions with many parameters are generated. The obtained solutions are more general and solutions include the solutions, the hyperbolic function, the trigonometric function and the rational forms. In addition, some of our solutions are identical with already published results, which validate newly generated solutions and others solutions have not been reported in the previous literature.

In this article, an improved (G′/G)-expansion method is implemented for the simplified Modified Camassa-Holm (MCH) equation involving parameters, with an aim to construct many new traveling wave solutions. In this method, second order linear ordinary differential equation with constant coefficients has been implemented as an auxiliary equation. The generated solutions including solitons and periodic solutions are demonstrated by the hyperbolic function, the trigonometric function and the rational forms. If the parameters take particular values, the solutions become in special functional form. Moreover, it is worth mentioning that, some of our solutions are in good agreement with already published results in the open literature by setting appropriate values of constants, which proves our other solutions. In addition, some of obtained solutions are described in the figures with the aid of commercial software Maple 13.

The generalized and improved -expansion method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. In this article, we investigate the higher dimensional nonlinear evolution equation, namely, the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation via this powerful method. The solutions are found in hyperbolic, trigonometric and rational function form involving more parameters and some of our constructed solutions are identical with results obtained by other authors if certain parameters take special values and some are new. The numerical results described in the figures were obtained with the aid of commercial software Maple.

In this article, the generalized Riccati equation mapping together with the basic (G′/G)-expansion method is implemented which is advance mathematical tool to investigate nonlinear partial differential equations. Moreover, the auxiliary equation G′(ϕ) = h + f G(ϕ) + g G2(ϕ) is used with arbitrary constant coefficients and called the generalized Riccati equation. By applying this method, we have constructed abundant traveling wave solutions in a uniform way for the Sawada-Kotera equation. The obtained solutions of this equation have vital and noteworthy explanations for some practical physical phenomena.