Nonlinear asymptotic stability of smooth 1-solitons of the Degasperis–Procesi equation

Establish nonlinear asymptotic stability, in exponentially weighted spaces L^2_α with 0<α<sqrt((c−4k)/(c−k)), of the smooth 1-soliton traveling waves u_0(·;k,c) of the Degasperis–Procesi equation u_t − u_{txx} = 3 u_x u_{xx} − 4 u u_x + u u_{xxx}. Precisely, determine that for initial data sufficiently close to u_0(·;k,c) in L^2_α, the corresponding solution exists globally and converges as t→∞ to a modulated solitary wave u_0(x−(c+β)t+γ;k,c+β) with suitable phase γ and speed shift β, with exponential decay in L^2_α orthogonal to the generalized kernel.

Background

The paper proves strong spectral stability and linear asymptotic stability (with a spectral gap) for the linearized operator about smooth 1-solitons of the Degasperis–Procesi (DP) equation in exponentially weighted spaces, and characterizes the generalized kernel generated by translation and speed modulation.

However, the quasilinear structure of the DP equation and the presence of the nonlinear dispersive term uu_{xxx} cause a loss of derivatives in a nonlinear iteration scheme. Unlike the generalized KdV setting, no linear smoothing estimate is available here because the weighted essential spectrum is asymptotically vertical, preventing recovery of the lost derivatives. As a result, the authors cannot yet promote the linear decay to a full nonlinear asymptotic stability theorem.