  We develop a formalism for computing the non-linear response of interacting
integrable systems. Our results are asymptotically exact in the hydrodynamic
limit where perturbing fields vary sufficiently slowly in space and time. We
show that spatially resolved nonlinear response distinguishes interacting
integrable systems from noninteracting ones, exemplifying this for the
Lieb-Liniger gas. We give a prescription for computing finite-temperature Drude
weights of arbitrary order, which is in excellent agreement with numerical
evaluation of the third-order response of the XXZ spin chain. We identify
intrinsically nonperturbative regimes of the nonlinear response of integrable
systems.
