A (2+1) modified KdV equation with time-dependent coefficients: exploring soliton solution via Darboux transformation and artificial neural network approach

This paper explores the modified Korteweg-de Vries equation with a time-dependent coefficient in (2+1) dimensions. We establish a robust theoretical framework by identifying the Lax pair and constructing infinitely many conservation laws. Using the strategy of covariance, we derive the Darboux transformation with quasideterminants, enabling the construction of multisoliton solutions. Additionally, we employ the Levenberg–Marquardt artificial neural network to train one- and two-soliton solutions for various time-variable coefficients. Our symbolic computations reveal that while the soliton amplitude remains unaffected by higher dimensions and time-dependent coefficients in the evolution equation, the soliton velocity changes during propagation across different planes. The accuracy of these findings is demonstrated through various graphical representations for different time-variable coefficients. This study advances the understanding of soliton dynamics in complex environments and introduces a novel intersection of neural network techniques with classical soliton theory.