Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour

The first main aim of this article is to derive an explicit solution formula for the scalar 2d-Toda lattice depending on three independent operator parameters, ameliorating work in [29]. This is achieved by studying a noncommutative version of the two-dimensional Toda lattice, generalizing its soliton solution to the noncommutative setting. The purpose of the applications part is to show that the family of solutions obtained from matrix data exhibits a rich variety of asymptotic behaviour. The first indicator is that web structures, studied extensively in the literature, see [4] and references therein, are a subfamily. Then three further classes of solutions (with increasingly unusual behaviour) are constructed, and their asymptotics are derived.