  We construct many self-similar and translating solitons for Lagrangian mean
curvature flow, including self-expanders and translating solitons with
arbitrarily small oscillation on the Lagrangian angle. Our translating solitons
play the same role as cigar solitons in Ricci flow, and are important in
studying the regularity of Lagrangian mean curvature flow.
  Given two transverse Lagrangian planes R^n in C^n with sum of characteristic
angles less than pi, we show there exists a Lagrangian self-expander asymptotic
to this pair of planes. The Maslov class of these self-expanders is zero. Thus
they can serve as local models for surgeries on Lagrangian mean curvature flow.
Families of self-shrinkers and self-expanders with different topologies are
also constructed. This paper generalizes the work of Anciaux, Joyce, Lawlor,
and Lee and Wang.
