On the factorization technique for the dispersive nonlinear equations

Abstract

We review some recent results on the large time asymptotic behavior of solutions to the Cauchy problem for the fourth-order nonlinear Schrödinger equation \[\Bigg\{ \begin{gather} i\partial_{t}u + \mu\partial_{x}^{2}u + \alpha \partial_{x}^{4}u = N (u), \ (t,x) \in \mathbb{R}^{+} \times \mathbb{R}, \\ u ( 0, x) = u_{0} ( x), \ x \in \mathbb{R},\end{gather}\] with different types of the nonlinearity $N(u)$. We use the factorization technique originated from paper [30] for the free Schrödinger evolution group. This factorization formula appears very useful for studying the global existence and the large time asymptotic behavior of solutions to the nonlinear Schrödinger equations. Also it helps us to consider the problem in lower order Sobolev spaces. Later it was found that similar factorization technique can be developed for other dispersive equations, such as the nonlinear Klein-Gordon equation, fourth-order nonlinear Schrödinger equation, fractional order cubic nonlinear Schrödinger equation, reduced Ostrovsky equation, etc. By using the factorization formula, we divide the nonlinear term into the main and remainder terms, then we find the appropriate phase corrections in the large time asymptotic formula.