Analytical methods and Riccati neural networks method of Kudryashov’s quintuple self-phase modulation with dual-form of generalized nonlocal nonlinearity

In this paper, the exact soliton solutions of the nonlinear Schrödinger’s equation having Kudryashov’s quintuple power law of refractive index together with dual form of generalized nonlocal nonlinearity are studied. The proposed system is a non-integrable equation. Riccati subequation technique based on the neural network model is considered. The Riccati subequation neural networks model as activation functions in the first hidden layer of the neural network architecture to reach to output as an exact solution is taken. The diverse forms of solitary wave solutions by utilizing the improved $$\tan (\phi /2)$$-expansion technique and the $$\exp (-\phi (\eta ))$$-Expansion scheme are delivered. In particular, the key types found (e.g., dark solitons, singular solitons, combined hyperbolic function solutions). To achieve this, an illustrative example of the Schrödinger’s equation is provided to demonstrate the feasibility and reliability of the procedure which used in this study. The effect of the free parameters on the behavior of acquired figures to a few obtained solutions for two cases of rational exact solutions are also analyzed due to the nature of nonlinearities. The dynamic properties of the obtained results are shown and analyzed by some density, two and three-dimensional images. The results may provide a further detailed physical understanding of nonlinear plasma wave interactions, contributing to the development of advanced models for high-energy plasma wave modulation, energy transport, and optical solitons.