Modified scattering for the higher-order KdV–BBM equations

We study the Cauchy problem for the higher-order KdV–BBM type equation \(\begin{aligned} \left\{ \begin{array}{c} \partial _{t}u+i\varvec{\Lambda }u=\varvec{\Theta }\partial _{x}u^{3}, \ t>0, \ x\in \mathbb {R}, \\ u\left( 0,x\right) =u_{0}\left( x\right) , \ x\in \mathbb {R}, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}u+i\mathbf{\Lambda }u=\mathbf{\Theta }{\partial }_x{u}^3,t>0,x\in \mathbb{R},\\ u\left(0,,,x\right)=u_0\left(x\right),x\in \mathbb{R},\end{array}\right)\end{array}$$ where \(\varvec{\Lambda }\) $\mathbf{\Lambda }$ \(=\mathcal {F}^{-1}\Lambda \mathcal {F}\) $={\mathcal{F}}^{-1}\mathrm{\Lambda }\mathcal{F}$ and \(\Theta \) $\mathrm{\Theta }$ \(=\mathcal {F}^{-1}\Theta \mathcal {F}\) $={\mathcal{F}}^{-1}\mathrm{\Theta }\mathcal{F}$ are the pseudodifferential operators, defined by their symbols \(\Lambda \left( \xi \right) \) $\mathrm{\Lambda }\left(\xi \right)$ and \( \Theta \left( \xi \right) \) $\mathrm{\Theta }\left(\xi \right)$ , 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.