Global existence for a coupled system of Schrödinger equations with power-type nonlinearities

Nghiem V. Nguyen; Rushun Tian; Bernard Deconinck; Natalie Sheils

doi:10.1063/1.4774149

Global existence for a coupled system of Schrödinger equations with power-type nonlinearities

Nghiem V. Nguyen, Rushun Tian, Bernard Deconinck, Natalie Sheils

School of Mathematics

Research output: Contribution to journal › Article › peer-review

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Abstract

In this manuscript, we consider the Cauchy problem for a Schrödinger system with power-type nonlinearities where for u_j u_j: R^N × R → C, Ψj0: R^N → C for j = 1, 2, ... m and a_jk = a_kjare positive real numbers. Global existence for the Cauchy problem is established for a certain range of p. A sharp form of a vector-valued Gagliardo-Nirenberg inequality is deduced, which yields the minimal embedding constant for the inequality. Using this minimal embedding constant, global existence for small initial data is shown for the critical case p = 1 + 2/N. Finite-time blow-up, as well as stability of solutions in the critical case, is discussed.

Original language	English (US)
Article number	011503
Journal	Journal of Mathematical Physics
Volume	54
Issue number	1
DOIs	https://doi.org/10.1063/1.4774149
State	Published - Jan 22 2013

Bibliographical note

Funding Information:
Bernard Deconinck and Natalie Sheils acknowledge support from the National Science Foundation (Grant No. NSF-DMS-1008001). Natalie Sheils also acknowledges this material is based upon work supported by the National Science Foundation (Grant No. NSF-DGE-0718124). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources.

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title = "Global existence for a coupled system of Schr{\"o}dinger equations with power-type nonlinearities",

abstract = "In this manuscript, we consider the Cauchy problem for a Schr{\"o}dinger system with power-type nonlinearities where for uj uj: RN × R → C, Ψj0: RN → C for j = 1, 2, ... m and ajk = akjare positive real numbers. Global existence for the Cauchy problem is established for a certain range of p. A sharp form of a vector-valued Gagliardo-Nirenberg inequality is deduced, which yields the minimal embedding constant for the inequality. Using this minimal embedding constant, global existence for small initial data is shown for the critical case p = 1 + 2/N. Finite-time blow-up, as well as stability of solutions in the critical case, is discussed.",

author = "Nguyen, {Nghiem V.} and Rushun Tian and Bernard Deconinck and Natalie Sheils",

note = "Funding Information: Bernard Deconinck and Natalie Sheils acknowledge support from the National Science Foundation (Grant No. NSF-DMS-1008001). Natalie Sheils also acknowledges this material is based upon work supported by the National Science Foundation (Grant No. NSF-DGE-0718124). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources.",

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AU - Nguyen, Nghiem V.

AU - Tian, Rushun

AU - Deconinck, Bernard

AU - Sheils, Natalie

N1 - Funding Information: Bernard Deconinck and Natalie Sheils acknowledge support from the National Science Foundation (Grant No. NSF-DMS-1008001). Natalie Sheils also acknowledges this material is based upon work supported by the National Science Foundation (Grant No. NSF-DGE-0718124). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources.

PY - 2013/1/22

Y1 - 2013/1/22

N2 - In this manuscript, we consider the Cauchy problem for a Schrödinger system with power-type nonlinearities where for uj uj: RN × R → C, Ψj0: RN → C for j = 1, 2, ... m and ajk = akjare positive real numbers. Global existence for the Cauchy problem is established for a certain range of p. A sharp form of a vector-valued Gagliardo-Nirenberg inequality is deduced, which yields the minimal embedding constant for the inequality. Using this minimal embedding constant, global existence for small initial data is shown for the critical case p = 1 + 2/N. Finite-time blow-up, as well as stability of solutions in the critical case, is discussed.

AB - In this manuscript, we consider the Cauchy problem for a Schrödinger system with power-type nonlinearities where for uj uj: RN × R → C, Ψj0: RN → C for j = 1, 2, ... m and ajk = akjare positive real numbers. Global existence for the Cauchy problem is established for a certain range of p. A sharp form of a vector-valued Gagliardo-Nirenberg inequality is deduced, which yields the minimal embedding constant for the inequality. Using this minimal embedding constant, global existence for small initial data is shown for the critical case p = 1 + 2/N. Finite-time blow-up, as well as stability of solutions in the critical case, is discussed.

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Global existence for a coupled system of Schrödinger equations with power-type nonlinearities

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