Abstract
Given any wave speed c ∈ R , we construct a traveling wave solution of u t = Δ u + | ∇ u | 2 u in an infinitely long cylinder, which connects two locally stable and axially symmetric steady states at x 3 = ± ∞ . Here u is a director field with values in S 2 ⊂ R 3 : | u | = 1 . The traveling wave has a singular point on the cylinder axis. In view of the bistable character of the potential, the result is surprising, and it is intimately related to the nonuniqueness of the harmonic map flow itself. We show that for only one wave speed the traveling wave behaves locally, near its singular point, as a symmetric harmonic map.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
