Tropical curves and solitons in nonlinear integrable systems

Abstract

We construct families of rationally degenerating Riemann surfaces corresponding to tropical curves with trivial weights, and give explicit formulas of the associated quasi-periodic solutions and soliton solutions as their regularized limits to the KP hierarchy and the 2D Toda lattice hierarchy. This result especially gives rise to soliton solutions in extensive classes to these nonlinear integrable systems which are expected to cover all solutions expressed by compositions of elementary functions. For these quasi-periodic solutions, we also discuss their regularity and limits to the mixtures with solitons obtained from tropical curves with nontrivial weights.