We present a gauged Lifshitz Lagrangian including second and forth order
spatial derivatives of the scalar field and a Chern-Simons term, and study
non-trivial solutions of the classical equations of motion. While the
coefficient beta of the forth order term should be positive in order to
guarantee positivity of the energy, the coefficient alpha of the quadratic one
need not be. We investigate the parameter domains finding significant
differences in the field behaviors. Apart from the usual vortex field behavior
of the ordinary relativistic Chern-Simons-Higgs model, we find in certain
parameter domains oscillatory solutions reminiscent of the modulated phases of
Lifshitz systems.Comment: 13 pages, 6 figure