  In conventional fluids, it is well known that Euler-scale equations are
plagued by ambiguities and instabilities. Smooth initial conditions may develop
shocks, and weak solutions, such as for domain wall initial conditions (the
paradigmatic Riemann problem of hydrodynamics), are not unique. The absence of
shock formation experimentally observed in quasi-one-dimensional cold-atomic
gases, which are described by the Lieb-Liniger model, provides perhaps the
strongest pointer to a modification of the hydrodynamic equation due to
integrability. Generalised hydrodynamics (GHD) is the required hydrodynamic
theory, taking into account the infinite number of conserved quantities
afforded by integrability. We provide a new quadrature for the GHD equation --
a solution in terms of a Banach fixed-point problem where time has been
explicitly integrated. In the Lieb-Liniger model, this allows us to rigorously
show for the first time that, from a wide class of initial conditions including
zero- and finite-entropy states, no shock may appear at all times, and weak
solutions are unique and entropy-preserving. This extends known results for
finite-component linearly-degenerate hydrodynamics, to the continuum of
hydrodynamic modes present in GHD. The new quadrature is also an efficient
solution tool, which, combined with recent hydrodynamic fluctuation theories,
gives rise to new expressions for correlations in non-stationary states.
