2021/06/05 by Justin Holmer, Holmer, Justin, Katherine Zhiyuan Zhang +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Nonlinear Photonic Systems
paper · pdf · doi:10.48550/arxiv.2106.02971
The Benjamin Ono equation with a slowly varying potential is (pBO) ut + (Hux-Vu + \tfrac12 u2)x=0 with V(x)=W(hx), 0< h ≪ 1, and W∈ Cc^∞(ℝ), and H denotes the Hilbert transform. The soliton profile is Qa,c(x) = cQ(c(x-a)) , where Q(x) = (4)/(1+x2) and a∈ ℝ, c>0 are parameters. For initial condition u0(x) to (pBO) close to Q0,1(x), it was shown in a previous work by Z. Zhang that the solution u(x,t) to (pBO) remains close to Qa(t),c(t)(x) and approximate parameter dynamics for (a,c) were provided, on a dynamically relevant time scale. In this paper, we prove exact (a,c) parameter dynamics. This is achieved using the basic framework of the previous work by Z. Zhang but adding a local virial estimate for the linearization of (pBO) around the soliton. This is a local-in-space estimate averaged in time, often called a local smoothing estimate, showing that effectively the remainder function in the perturbation analysis is smaller near the soliton than globally in space. A weaker version of this estimate is proved in a paper by Kenig & Martel as part of a "linear Liouville" result, and we have adapted and extended their proof for our application.