Exact Charge, Current, and Velocity Fields of Interacting Korteweg-de Vries Solitons

Abstract

Solitons in integrable systems exhibit a dual wave-particle character, yet their identification as individual objects becomes ambiguous during interactions, where they deform and delocalize. We develop a microscopic, field-based description that resolves this issue by introducing exact space-time fields of charge, current, and velocity derived from the inverse scattering transform (IST) of the Korteweg-de Vries (KdV) equation. This framework enables individual KdV solitons to be tracked throughout interactions, providing a quantitative description of their trajectories and deformation beyond the asymptotic regime. We show that the dynamics of $N$ interacting solitons can be formulated in terms of $N$ independent continuity equations, in which interactions are encoded in initial correlations that are subsequently propagated. From this microscopic viewpoint, effective velocities and hydrodynamic behavior emerge upon coarse graining, recovering kinetic theory of soliton gases and Generalized Hydrodynamics as scaling limits. Our results establish a direct connection between the wave-based IST formalism and particle-like emergent descriptions, offering a unified framework for soliton dynamics across scales.