Using the unified solver technique, the rigorous and effective new novel optical progressive and stationary structures are established in the aspects of hyperbolic, trigonometric, rational, periodical and explosive types. These types are concrete in the stochastic nonlinear Schrödinger equations (NLSEs) with operative physical parameters. The obtained stochastic solutions with random parameters that are founded in the form of rational, dissipative, explosive, envelope, periodic, and localized soliton can be utilized in fiber applications. The stochastic modulations of structures' amplitude and frequency caused by dramatic instantaneous influences of both fibers nonlinear, dispersive, losing and noise term effects maybe very important in new fiber communications.

This article investigates the nonlinear Maccari model with multiplicative noise using the unified technique. Numerous new important solitary wave solutions are presented with free physical parameters. These solutions play a vital role in various domains, including nonlinear optics, plasma physics, and hydrodynamics. The investigation shows that the solution process is quick and clear, where a comparatively higher number of novel solutions are obtained. The performance of the used approach is compared with that of other methods. We create 2D and 3D graphs for certain solutions of the study, utilizing suitably selected values for the physical parameters. We also address the impact of model parameters on the solution characteristics. We observe that our results may help to resolve some physical problems in the actual world by determining the motion of a single wave in a tiny region. Finally, the outcomes show how simple and effective this method is at producing rich, accurate solutions to nonlinear models in mathematical physics as well as complex nonlinear wave structures.

In this work, we consider the model of shallow water equation with horizontal density gradients. We develop the modified Rusanov (mR) scheme to solve this model in one and two dimensions. Predictor and corrector are the two stages of the suggested scheme. The predictor stage is dependent on a local parameter $({\alpha }_{i+\frac{1}{2}}^n)$ that allows for diffusion control. The balance conservation equation is recovered in the corrector stage. The proposed approach is well-balanced, conservative, and straightforward. Several 1D and 2D test cases are produced after presenting the shallow water model and the numerical technique. In the 1D case, we compared the proposed scheme with the Rusanov scheme, mR with constant $\alpha$ and analytical solutions. The numerical simulation demonstrates the mR's great resolution and attests to its capacity to produce accurate simulations of the shallow water equation with horizontal density gradients. Our results demonstrate that the mR technique is a highly effective instrument for solving a variety of equations in applied science and developed physics.

The nonlinear Maccari's systems depict the dynamics of isolated waves, detained in a small part of space, in optical communications, hydrodynamics and plasma physics. In this paper, we construct some new solutions for the Maccari's systems, using the unified solver technique based on He's variations technique. These solutions prescribe some vital complex phenomena in plasma physics. The proposed solver will be used as a box solver for considering various models in applied science and new physics. Some graphs are presented in order to display the dynamical behaviour of the gained solutions.

The main motive of this work is to introduce a numerical investigation for the one and two-dimensional (1D/2D) Chaplygin gas model. Namely, we developed the non homogeneous Riemann solver (NHRS) method to solve these models. After discussing the Chaplygin gas models and the numerical scheme, various 1D and 2D test problems are introduced. In order to complete the numerical investigation in a completely unified way, Rusanov scheme, modified Lax-Friedrichs and analytical solution are compared with NHRS scheme in 1D case. The acquired results clarify the high resolution of the NHRS technique. The NHRS technique is efficacious and robust. Finally, our study displays that the NHRS scheme is a very powerful tool to solve many other models arising in applied science.

Nonlinear Schrödinger equations are a key paradigm in nonlinear research, attracting both mathematical and physical attention. This work was primarily concerned with the usage of a reliable analytic technique in order to solve two models of (2+1)-dimensional nonlinear Schrödinger equations. By applying a comprehensible wave transformation, every nonlinear model was simplified to an ordinary differential equation. A number of critical solutions were observed that correlated to various parameters. The provided approach has various advantages, including reducing difficult computations and succinctly presenting key results. Some 2D and 3D graphical representations regarding presented solitons were considered for the appropriate values of the parameters. We also showed the effect of the physical parameters on the dynamical behavior of the presented solutions. Finally, the proposed approach may be expanded to tackle increasingly complicated problems in applied science.

The nonlinear long-short wave interaction system serves as a fundamental nonlinear model that characterizes the resonant interactions occurring between high-frequency (short) waves and low-frequency (long) waves. This system is especially useful for comprehending energy transfer, wave modulation, and the development of localized structures such as solitons or breathers. This paper proposes innovative stochastic solutions for the nonlinear long-short wave interaction model within the context of the Brownian motion process. This stochastic process takes into consideration the intrinsic randomness and variations found in real-world systems, including nonlinear optical fibers, Bose-Einstein condensates, fluid dynamics, and plasma environments. We derive stochastic traveling wave solutions using an extended tanh function approach and examine the resulting stochastic dynamics concerning amplitude variation, phase shifts, and noise-induced modulation. We consider the impact of noise intensity on the stability and coherence of wave structures. Our findings suggest that Brownian forcing can fulfill two roles; either aiding in the preservation of localized structures or leading to their collapse and dispersion, depending on the system parameters and initial conditions. To depict the behavior of the designated stochastic solutions, various wave profiles were generated utilizing the MATLAB software. Finally, the proposed method holds the promise of being adapted for various other practical models.