Modified scattering for the higher-order KdV–BBM equations

We study the Cauchy problem for the higher-order KdV–BBM type equation {[ ∂ _tu+iΛu=Θ∂ _xu^3, t>0, x∈ℝ,; u( 0,x) =u_0( x) , x∈ℝ, ]. where Λ =ℱ^-1Λℱ and Θ =ℱ^-1Θℱ are the pseudodifferential operators, defined by their symbols Λ( ξ) and Θ( ξ) , respectively. The aim of the present paper is to develop a general approach through the Factorization Techniques of evolution operators which can be applied for finding the large time asymptotics of small solutions to a wide class of nonlinear dispersive KdV- type equations including the KdV or the improved version of the KdV with higher order dispersion terms.