Standing waves of the complex Ginzburg–Landau equation

We prove the existence of nontrivial standing wave solutions of the complex Ginzburg–Landau equation ϕt=eiθΔϕ+eiγ|ϕ|αϕ+kϕϕt=eiθΔϕ+eiγ|ϕ|αϕ+kϕ on a bounded domain with Dirichlet boundary conditions. Our result requires that α>0α>0 be sufficiently small.