Applications of nonlinear integrable equations in artificial intelligence

This study begins with the nonlinear coupled Korteweg-de Vries equation and successfully solves its Lax pair using the prolongation structure method. Based on the obtained Lax pair, the Bäcklund transformation and superposition formula are further derived. These results provide a theoretical foundation for understanding the dynamic behavior of nonlinear systems and offer new approaches for constructing exact solutions. The study highlights the potential of the Bäcklund transformation and superposition formula in improving computational efficiency, enhancing model generalization capabilities, and increasing prediction accuracy. Furthermore, this paper explores the application prospects of these mathematical tools in the field of AI demonstrating significant advantages in simulating complex physical phenomena, such as waves and turbulence. Experimental results show that the AI model incorporating Bäcklund transformation improves computational efficiency by 40 percent and reduces prediction error by 25 percent in turbulence forecasting. Integrating traditional mathematical methods with modern AI technologies not only significantly enhances the performance of existing algorithms but also opens up new possibilities for addressing complex real-world problems. This research lays the groundwork for future interdisciplinary studies, encouraging more scholars to engage in this cutting-edge field. Through cross-disciplinary integration, it is expected to promote the joint advancement of nonlinear science and AI technology.