Self-similar solutions for the generalized fractional Korteweg-de Vries equation

We consider the Cauchy problem for the generalized fractional Korteweg-de Vries equation $$u_t+D^{\alpha }{u}_x+u^p{u}_x=0,1<\alpha \le 2,p\in \mathbb{N}\setminus \{0\},$$ with homogeneous initial data $\mathrm{\Phi }$. We show that, under smallness assumption on $\mathrm{\Phi }$, and for a wide range of $(\alpha ,p)$, including $p=3$, we can construct a self-similar solution of this problem.