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This paper presents numerical modeling and investigation for the Ripa system. This model is derived from a shallow water model by merging the horizontal temperature gradients. We applied the non-homogeneous Riemann solver (NHRS) method for solving the Ripa model. This scheme contains two stages named predictor and corrector. The first one is made up of a control parameter that is responsible for the numerical diffusion. The second one recuperates the balance conservation equation. One of the main features of the NHRS scheme, it can determine the numerical flux corresponding to the real state of solution in the non-attendance of Riemann solution. Various test cases of physical interest are considered. These case studies display the high resolution of the NHRS scheme and emphasize its ability to produce accurate results for the Ripa model. The presented solutions are very critical in superfluid applications of energy and many others. Finally, the NHRS technique can be used to solve a wide range of additional models in applied research.

The two-dimensional Maccari nonlinear system performs the energy and wave dynamical features in fiber communications and modern physical science as hydrodynamic and space plasma. Several new forms of solutions for the Maccari’s model are constructed by a unified solver method that mainly depends on He’s variations method. The obtained solutions identify new wave stochastic structures with important features in energy physics such as rational explosive, breather, dispersive, explosive dissipated, dark solitons and blow-up (shock structure). It was elucidated that the random effects amend the energy wave strength or the collapsing due to model medium turbulence. Finally, the produced stochastic structures may be vital in some of these relationships between dispersions, nonlinearity and dissipative effects. The predominant energy waves that are collapsing or being forced may be applied to electrostatic auroral Langmuir structures and energy-generating ocean waves.

The nonlinearity form of the Schrödinger equation (NLSE) gives a sterling account for energy and solitary transmission properties in modern communications with optical-fiber energ- reinforcement actions. The solitary representation during fiber transmissions was regulated by NLSE coefficients such as nonlinear Kerr, evolutions, and dispersions, which controlled the energy changes through the model. Sometimes, the energy values predicted from the NLSEs computations may diverge due to variations in the amplitude and width caused by scattering, dispersive, and dissipative features of fiber materials. Higher-order nonlinear Schrödinger equations (HONLSEs) should be explored to alleviate these implications in energy and wave features. The unified solver approach is employed in this work to evaluate the HONLSEs. Steepness, HO dispersions, and nonlinearity self-frequency influences have been taken into consideration. The energy and solitary features were altered by higher-order actions. The unified solver approach is employed in this work to reform the HONLSE solutions and its energy properties. The steepness, HO dispersions, and nonlinearity self-frequency influences have been taken into consideration. The energy and soliton features in the investigated model were altered by the higher-order impacts. Furthermore, the new HONLSE solutions explain a wide range of important complex phenomena in wave energy and its applications.