Benjamin-Ono Soliton Dynamics in a Slowly Varying Potential

2019/05/07 by Zhang, Katherine Zhiyuan
#35Q53 35Q51 37K40 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1905.02348

Abstract

We consider the Benjamin-Ono equation with a slowly varying potential u_t + (Hu_x-Vu + \tfrac12 u²)_x=0 with V(x)=W(hx), 0< h ≪ 1, and W∈ C_c^∞(ℝ), and H denotes the Hilbert transform. The soliton profile is Q_a,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 in H_x^1/2 to Q_0,1(x), we show that the solution u(x,t) to (pBO) remains close in H_x^1/2 to Q_a(t),c(t)(x) and specify the (a,c) parameter dynamics on an O(h^-1) time scale.

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