### Abstract

We consider the global Cauchy problem for a system of Schrödinger equations with quadratic interaction. Two types of analytic smoothing effect for the solutions are formulated in the small data setting under the mass resonance condition. One is the usual analytic smoothing effect in space variables in terms of the generator of Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to Galilei generators for sufficiently small data with exponential decay at infinity in space ℝ^{n} with n ≥ 3. The other is analytic smoothing effect in space-time variables in terms of generator of pseudo-conformal and Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to pseudo-conformal and Galilei generators for sufficiently small data with exponential decay in ℝ^{4}. We also discuss the associated Lagrange structure.

Original language | English |
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Article number | 091513 |

Journal | Journal of Mathematical Physics |

Volume | 56 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2015 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Analytic smoothing effect for a system of Schrödinger equations with three wave interaction.** / Hoshino, Gaku; Ozawa, Tohru.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 56, no. 9, 091513. https://doi.org/10.1063/1.4931659

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TY - JOUR

T1 - Analytic smoothing effect for a system of Schrödinger equations with three wave interaction

AU - Hoshino, Gaku

AU - Ozawa, Tohru

PY - 2015

Y1 - 2015

N2 - We consider the global Cauchy problem for a system of Schrödinger equations with quadratic interaction. Two types of analytic smoothing effect for the solutions are formulated in the small data setting under the mass resonance condition. One is the usual analytic smoothing effect in space variables in terms of the generator of Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to Galilei generators for sufficiently small data with exponential decay at infinity in space ℝn with n ≥ 3. The other is analytic smoothing effect in space-time variables in terms of generator of pseudo-conformal and Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to pseudo-conformal and Galilei generators for sufficiently small data with exponential decay in ℝ4. We also discuss the associated Lagrange structure.

AB - We consider the global Cauchy problem for a system of Schrödinger equations with quadratic interaction. Two types of analytic smoothing effect for the solutions are formulated in the small data setting under the mass resonance condition. One is the usual analytic smoothing effect in space variables in terms of the generator of Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to Galilei generators for sufficiently small data with exponential decay at infinity in space ℝn with n ≥ 3. The other is analytic smoothing effect in space-time variables in terms of generator of pseudo-conformal and Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to pseudo-conformal and Galilei generators for sufficiently small data with exponential decay in ℝ4. We also discuss the associated Lagrange structure.

UR - http://www.scopus.com/inward/record.url?scp=84943147548&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84943147548&partnerID=8YFLogxK

U2 - 10.1063/1.4931659

DO - 10.1063/1.4931659

M3 - Article

VL - 56

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

M1 - 091513

ER -