Orbital Stability of Domain Walls in Coupled Gross--Pitaevskii Systems

Domain walls are minimizers of energy for coupled one-dimensional Gross--Pitaevskii systems with nontrivial boundary conditions at infinity. It has been shown in [S. Alama, L. Bronsard, A. Contreras, and D. E. Pelinovsky, Arch. Ration. Mech. Anal., 215 (2015), pp. 579--610] that these solutions are orbitally stable in the space of complex $\dot{H}^1$ functions with the same limits at infinity. In the present work we adopt a new weighted $H^1$ space to control perturbations of the domain walls and thus to obtain an improved orbital stability result. A major difficulty arises from the degeneracy of linearized operators at the domain walls and the lack of coercivity.