Zeitschrift für angewandte Mathematik und Physik

, Volume 66, Issue 5, pp 2343–2377 | Cite as

Factorization technique for the fourth-order nonlinear Schrödinger equation

  • Nakao HayashiEmail author
  • Pavel I. Naumkin


We consider the Cauchy problem for the cubic fourth-order nonlinear Schrödinger equation
$$\left\{\begin{array}{ll}i\partial_{t}u+\frac{1}{4}\partial_{x}^{4}u = i \lambda \partial _{x}(\left| u \right| ^{2}u),&\quad t > 0,\, x \in \mathbf{R},\\ u \left( 0,x\right) = u_{0}\left( x\right) ,&\quad x \in \mathbf{R},\end{array}\right.$$
where \({\lambda \in \mathbf{R}.}\) We introduce the factorization formula for the free evolution group to prove the global existence of solutions. Also we show that the large time asymptotics of solutions has a logarithmic correction in the phase comparing with the corresponding linear case.


Fourth-order nonlinear Schrödinger equation Factorization formula Large time asymptotics of solutions Logarithmic phase correction Critical order of nonlinearity 

Mathematics Subject Classification

35Q55 35Q35 35Q51 


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© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityOsakaJapan
  2. 2.Centro de Ciencias MatemáticasUNAM Campus MoreliaMoreliaMexico

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