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• W3172133145 abstract "The Benjamin--Ono equation with a slowly varying potential is $text{(pBO)} u_t + (Hu_x-Vu + tfrac12 u^2)_x=0$ with $V(x)=W(hx)$, $0< h ll 1$, and $Win C_c^infty(mathbb{R})$, and $H$ denotes the Hilbert transform. The soliton profile is $Q_{a,c}(x) = cQ(c(x-a)) ,, text{ where } Q(x) = frac{4}{1+x^2}$ and $ain mathbb{R}$, $c>0$ are parameters. For initial condition $u_0(x)$ to (pBO) close to $Q_{0,1}(x)$, it was shown in [K. Z. Zhang, Nonlinearity, 33 (2020), pp. 1064--1093] that the solution $u(x,t)$ to (pBO) remains close to $Q_{a(t),c(t)}(x)$ 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 paper [K. Z. Zhang, Nonlinearity, 33 (2020), pp. 1064--1093] 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 [C. E. Kenig and Y. Martel, Rev. Mat. Iberoam., 25 (2009), pp. 909--970] as part of a “linear Liouville” result, and we have adapted and extended their proof for our application." @default.
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• W3172133145 date "2022-04-01" @default.
• W3172133145 modified "2023-09-23" @default.
• W3172133145 title "Benjamin--Ono Soliton Dynamics in a Slowly Varying Potential Revisited" @default.
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• W3172133145 doi "https://doi.org/10.1137/21m1425177" @default.
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