Existence of the stable traveling wave for half-wave equation with L2-critical combined nonlinearities. Issue 7 (3rd May 2022) - Li, Yuan; Zhao, Dun; Wang, Qingxuan
Abstract : For the Cauchy problem of the half-wave equation with L 2 -critical combined nonlinearities 1 i ψ t = − Δ ψ − | ψ | 2 / N ψ − | ψ | p ψ, 0 < p < 2 N ( N ≥ 2 ), ψ ( 0, x ) = ψ 0 ( x ), we obtain the existence of traveling wave solution of the form ψ ( t, x ) = e i t μ u v ( x − v t ) with 0 < | v | < 1 . If ∥ ψ 0 ∥ 2 < ∥ Q v ∥ 2, where Q v is a solution of the equation 2 − Δ u v + i ( v ⋅ ∇ ) u v − | u v | 2 / N u v = − u v, we prove that the traveling wave is stable.