  Recent years have seen the development of a rich phenomenology beyond the
Luttinger Liquid model of one dimensional quantum fluids, arising from
interactions between the elementary phonon excitations. It has been known for
some time, however, that the straightforward inclusion of these interactions
presents technical difficulties that have necessitated approaches based on
refermionization or effective impurity models.
  In this work we show that the nonlinear extensions of the Luttinger model are
tractable in the phonon basis. We present a calculation of the singularities
present in the zero temperature dynamical structure factor in the semiclassical
limit where the phonon dispersion is strong.
  A unitary transformation decouples interactions between left-- and
right--moving phonons, leaving a nonlinear chiral Hamiltonian. At low momenta,
this Hamiltonian has a spectrum bounded above and below by thresholds
identified with phonon and soliton excitations in the semiclassical limit. The
chiral dynamical structure factor therefore has support only in this region,
with power law singularities at the thresholds originating in the Anderson
orthogonality catastrophe, which we calculate analytically. The dynamical
structure factor for the original nonchiral Hamiltonian is a convolution of
this chiral correlator with a power law arising from the left--right
decoupling.
