Orbital stability of domain walls in coupled Gross-Pitaevskii systems. (English) Zbl 1387.35430
Summary: 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.
MSC:
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37K45 | Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems |
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