Modified scattering for the nonlinear Klein–Gordon equation

We consider the modified scattering problem for the nonlinear Klein–Gordon equation with cubic nonlinear interactions in one spatial dimension 1 $$\begin{aligned} {\left\{ \begin{array}{ll} w_{tt}-\Delta w+w=\mu |w|^{2}w,\\ w\left( 0,x\right) =w_{0}\left( x\right) , w_{t}\left( 0,x\right) =w_{1}(x), \end{array}\right. } \end{aligned}$$ w tt - Δ w + w = μ | w | 2 w , w 0 , x = w 0 x , w t 0 , x = w 1 ( x ) , for $$\mu \in \mathbb {R}$$ μ ∈ R . We show that for any small real valued initial data $$w_{0},$$ w 0 , $$w_{1}$$ w 1 low-regularity weighted Sobolev spaces there exists a unique modified final state $$W_{+}\in \textbf{L}^{\infty }$$ W + ∈ L ∞ such that the corresponding solution exhibits a logarithmic phase correction as $$t\rightarrow \infty .$$ t → ∞ . Our proof is based on the Factorization Technique. We present more simple and robust method for the proof of this result based on the Factorization Technique and on the $$\textbf{L}^{2}$$ L 2 - estimates for the transformed evolution operators, which considerably simplifies the proof of the asymptotics of solutions.