Long-time asymptotics and conservation laws in integrable systems

One dimensional systems sometimes show pathologically slow decay of currents. This robustness can be traced to the fact that an integrable model is nearby in parameter space. In integrable models some part of the current can be conserved, explaining this slow decay. Unfortunately, although this conservation law is formally anticipated, in practice it has been difficult to find in concrete cases, such as the Heisenberg model. We investigate this issue both analytically and numerically and find that the appropriate conservation law can be a non-analytic combination of the known local conservation laws and hence is invisible to elementary assumptions.