Conditional asymptotic stability of solitary waves of the Euler-Poisson system on the line

We apply the idea of using a combination of virial inequalities and Kato smoothing, previously applied to NLS and generalized KdV pure power equations to Euler-Poisson: we assume that a solution remains very close for all times to a soliton in an appropriate space and then we prove an asymptotic convergence to a soliton for $t\to +\mathrm{\infty }$.