Bilinear neural network computation for lump interactions and rogue waves in the (2+1)-dimensional Kairat-II-X-type model

This work presents the application of the bilinear neural network method (BNNM) to the challenging task of obtaining exact analytical solutions for the [Formula: see text]-dimensional Kairat-II-X-type (K-II-X-type) equation. As a model for nonlinear wave dynamics in complex, multidimensional settings, this equation requires sophisticated solution techniques, which the BNNM effectively provides. The approach starts with the Hirota bilinear method to transform the K-II-X-type equation into its bilinear form. Subsequently, BNNM, which integrates classical bilinear theory with neural network architectures, is employed to construct closed-form solutions. In this framework, trial functions are represented using layered neural networks, where activation functions and weight matrices dictate the solution characteristics. The “4-3-1”, “4-3-2”, “4-3-3”, and “4-3-4” network architectures are employed to generate a variety of waveforms, including M-lump solutions, lump–kink interactions, lump solitons, and rogue wave structures. The resulting solutions are illustrated through 2D and 3D surface plots, as well as contour and density plots, to highlight their geometric and dynamical properties. This work presents the power of BNNM in not only reproducing the known nonlinear pattern of waves but also in generating novel structures of their solution, offering a flexible framework for the computational exploration of multidimensional nonlinear wave phenomena, being at the same time computationally efficient. The versatility of the method makes it pertinent to applications in the fields of nonlinear optics, plasma physics, fluid mechanics, geophysical flows and acoustics, where an understanding of the formation, interaction and stability of localized and periodic waves is of particular importance.

Hali tarjima qilinmagan