Ben Doyon

The semiclassical Bethe equations, and where solitons are in a soliton gas

Integrable dispersive wave equations, such as the Korteweg-de Vries (KdV) equation, show remarkable mathematical properties. One of them is that they often support solitons: localised waves which propagate like particles. Solitons keep their shapes and velocities after interactions and are just affected by simple two-body scattering shifts, displaying elastic and factorised scattering. This particle-like behaviour suggests that, for a KdV field with a lot of solitons, there should be a “kinetic theory” describing its time evolution. This goes under the name of soliton gas, something proposed by Zakharov already in 1971, with kinetic equation first proposed by Gennady El in 2003. However, solitons really exist only asymptotically in time, when they are well separated. Solitons are not manifestly identifiable at times during which many interact. In order to have a good kinetic theory and prove the kinetic equation, one would have to have a notion of where solitons are at all times. This is the quasi-particle problem of many-body integrability: identifying asymptotic objects even at finite times and finite densities. I will explain how to solve this problem for the KdV solitons. Surprisingly, it turns out that equations that arise as a semiclassical limit from the Bethe wave functions, describe solitons’ positions. I will explain in what sense soliton’s positions are correctly described by semiclassical Bethe equations (SBE), how this arises with mathematical rigour from appropriate tau function representations, and how this connects to the kinetic equation. The SBE are believed to give a universal solution to the quasi-particle problem. The wider context is that of generalised hydrodynamics (GHD), and this offers a first step towards showing the predictions from GHD in the KdV soliton gas.