  We study properties of solutions of the initial value problem for the
nonlinear and nonlocal equation u_t+(-\partial^2_x)^{\alpha/2} u+uu_x=0 with
alpha in (0,1], supplemented with an initial datum approaching the constant
states u+/u- (u_-smaller than u_+) as x goes to +/-infty, respectively. It was
shown by Karch, Miao & Xu (SIAM J. Math. Anal. 39 (2008), 1536--1549) that, for
alpha in (1,2), the large time asymptotics of solutions is described by
rarefaction waves. The goal of this paper is to show that the asymptotic
profile of solutions changes for alpha \leq 1. If alpha=1, there exists a
self-similar solution to the equation which describes the large time
asymptotics of other solutions. In the case alpha \in (0,1), we show that the
nonlinearity of the equation is negligible in the large time asymptotic
expansion of solutions.
