Traveling wave solutions of the Schrödinger map equation

We first construct traveling wave solutions for the Schrödinger map in ℝ 2$\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{{\partial m}\over{\partial t}}= m \times (\Delta m - m_3{\vec{e}}_3) \quad {\rm in} \;\R^2 \times \R $ of the form m ( x 1 , x 2 − ϵ t ), where m has exactly two vortices at approximately $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}(\pm {{1}\over{2 \epsilon}}, 0) \in \R^2$ of degree ±1. We use a perturbative approach that gives a complete characterization of the asymptotic behavior of the solutions. With a few modifications, a similar construction yields traveling wave solutions of Schrödinger map equations in higher dimensions. © 2010 Wiley Periodicals, Inc.