This work utilizes Lax integrability and the Hirota bilinear approach to examine cold bosons in a zig-zag optical lattice. Such a configuration is frequently employed to elucidate various intricate phenomena in fluid dynamics, quantum optics, and plasma physics. We validated the integrability of the underlying nonlinear evaluation equation by formulating a Lax pair. Subsequently, the Hirota bilinear approach is employed to simplify the derivation of solitons. solutions by converting nonlinear partial differential equations into bilinear form. This method further facilitates the construction of multi-soliton solutions through disturbance expansion. Various soliton solutions are determined and visually demonstrated through two- and threedimensional representations. The derived soliton solutions provide essential insights that enable applied mathematicians, theoretical physicists, and engineering researchers to gain a deeper understanding of the physical significance of the model. To the best of our knowledge, the Lax pair integrability and the obtained solutions for the studied equation are new and have not been reported before.

The generalized Schrödinger equation with the power-law of nonlinearity, which describes propagation pulses in optical fibers, is analyzed. The well-known unified auxiliary equation approach is used to study the solitary wave solutions of the proposed equation. The solutions of solitary waves and periodic waves at zero or unity modulus of ellipticity are obtained by employing Jacobi elliptic function solutions. The obtained novel solutions which come in a variety of forms, including kink, periodic, dark, bright, W-shaped, singular, and singular periodic are extracted by the hyperbolic, trigonometric, and exponential functions. In addition, the impact of the degree of nonlinearity on the periodic and solitary wave structures is examined. Furthermore, a comprehensive bifurcation analysis is performed to explore the structural transitions and qualitative behaviors of the solutions. Phase portraits are constructed to visualize the dynamical nature of the equilibrium points and to elucidate the global behavior of the system in the phase space

Schrödinger's nonlinear equation is a fundamental model in fiber optics and many other areas of science. Using the Jacobi elliptic expansion function method, the time-fractional cubic-quartic nonlinear Schrödinger equation and cubic-quartic resonant nonlinear Schrödinger equation are investigated. By applying the effective Jacobi elliptic expansion function method, optical soliton solutions such as bright, dark, singular, periodic singular, exponential, and Jacobi elliptic function solutions have been obtained. The effect of the time-fractional derivative on the solutions is also revealed. Graphical representations are illustrated to showcase the physical properties of raised solutions, providing a comprehensive understanding of the solutions' functionality.

In this paper, we study the stochastic Fokas–Lenells equation with multiplicative white noise in the Itô sense. For this purpose, we will use the Jacobi elliptic expansion function method. The nonlinear partial diferential equation is transformed into ordinary diferential equations through the symmetry reduction technique. New optical solutions are constructed as dark, combined hyperbolic, periodic, bright, and rational solutions. Depending on the parameters, these optical solutions exhibit varying periods of qualitative characteristics. Furthermore, the efects of time on the solutions of the nonlinear stochastic Fokas– Lenells equation are examined. In addition, graphical representations of a few solutions are included to supplement our analysis.

In this paper, a mathematical model is constructed to describe a two dimensional incompressible flow in a symmetric horizontal thin liquid film for unsteadies flow. We apply the Navier-Stokes equations with specified boundary conditions and we obtain the equation of the film thickness by using the similarity method in which we can isolate the explicit time dependence and then the shape of the film will depend on one variable only.

In this study, The steady flow of and within horizontal thin liquid symmetric double-sided film is considered. The nonlinear differential equations that govern such flow are derived from the Navier-Stokes equation. The solution curves are obtained numerically by using MATLAB software for two nondimensional cases. Numerical method and MATLAB are used to obtain the solutions and plot the curves for the result nonlinear differential equation and for number of fluids