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Solutions to nonlinear higher order Schrödinger equations with small initial data on modulation spaces. (English) Zbl 1375.35493

Summary: In this paper, we consider the Cauchy problem for the nonlinear higher order Schrödinger equations on modulation spaces \(M_{p,q}^s\) and show the existence of a unique global solution by using integrability of time decay factors of time decay estimates. As a result, we are able to deal with wider classes of a nonlinearity and a solution space. Moreover, we study time decay estimates of a semi-group \(e^{it\phi(\sqrt{-\Delta})}\) with a polynomial symbol \(\phi\). Considering multiplicities of critical points and inflection points of \(\phi\) carefully, we have time decay estimates with better time decay rate.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35G25 Initial value problems for nonlinear higher-order PDEs
35G05 Linear higher-order PDEs
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Full Text: Euclid