Near Soliton Evolution for 2-Equivariant Schrödinger Maps in Two Space Dimensions

We consider equivariant solutions for the Schrödinger Map equation in 2 + 1 2+1 dimensions, with values into S 2 \mathbb {S}^2 . Within each equivariance class m ∈ Z m \in \mathbb {Z} this admits a lowest energy nontrivial steady state Q m Q^m , which extends to a two dimensional family of steady states by scaling and rotation. If | m | ≥ 3 |m| \geq 3 then these ground states are known to be stable in the energy space H ˙ 1 \dot H^1 , whereas instability and even finite time blow-up along the ground state family may occur if | m | = 1 |m| = 1 . In this article we consider the most delicate case | m | = 2 |m| = 2 . Our main result asserts that small H ˙ 1 \dot H^1 perturbations of the ground state Q 2 Q^2 yield global in time solutions, which satisfy global dispersive bounds. Unlike the higher equivariance classes, here we expect solutions to move arbitrarily far along the soliton family; however, we are able to provide a time dependent bound on the growth of the scale modulation parameter. We also show that within the equivariant class the ground state is stable in a slightly stronger topology X