Symmetry reduction and soliton-like solutions for the generalized Korteweg-de Vries equation

We analyze the gKdV equation, a generalized version of Korteweg-de Vries with an arbitrary function $f(u)$. In general, for a function $f(u)$ the Lie algebra of symmetries of gKdV is the $2$-dimensional Lie algebra of translations of the plane $xt$. This implies the existence of plane wave solutions. Indeed, for some specific values of $f(u)$ the equation gKdV admits a Lie algebra of symmetries of dimension grater than $2$. We compute the similarity reductions corresponding to these exceptional symmetries. We prove that the gKdV equation has soliton-like solutions under some general assumptions, and we find a closed formula for the plane wave solutions, that are of hyperbolic secant type.