Dynamics of the (2+1)-dimensional Yu-Toda-Sasa-Fukuyama equation

In this study, we employ Hirota’s bilinear technique to derive soliton, breather, and lump solutions for the (2+1)-dimensional Yu-Toda-Sasa-Fukuyama equation. These solutions are explicitly constructed as N×N Gram-type determinant expressions, providing a unified framework for their mathematical characterization. A systematic analysis is conducted on the dynamical behaviors of one-, two-, and three-soliton configurations, breather structures, and lump solutions, including their propagation patterns and interaction mechanisms. Furthermore, we investigate the hybrid interactions between breathers and solitons, revealing distinct collision dynamics governed by the underlying nonlinear dynamics. By strategically selecting parametric configurations, three distinct classes of breather solutions are identified: Akhmediev breathers, Kuznetsov-Ma breathers, and generalized breathers propagating along arbitrary oblique trajectories. These solutions are distinguished by their spatiotemporal periodicity and spectral properties. Through the introduction of two appropriately designed differential operators, rational solutions are formulated in terms of Schur polynomials, establishing a connection to symmetric function theory. A novel link is established between the theory of integer partitions and the construction of multi-lump solutions, providing combinatorial insights into their algebraic structure. The geometric patterns emerging from these solutions are systematically characterized via the complex root configurations of special polynomials, such as the Yablonskii-Vorob’ev polynomials. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2026.