Existence of a family of soliton-like solutions for the Kawahara equation

Existence is proved for a family of soliton-like solutions for the nonlinear evolution equation $\mathbf{u}_t+\mathbf{uu}_x+\mathbf{u}_{xxx}-\mathbf{u}_{xxxxx}=0$. The problem is reduced to investigating the fixed points of the operator
$$ (Au)(x)=\int_{-\infty}^{\infty}k(x-y)u^2(y)\,dy, \quad \int_{-\infty}^{\infty}k(x)=1, $$
whose action is considered in a cone of Frechet functions that are continuous on the real axis.