On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves

Ademir Pastor

doi:10.3934/cpaa.2019100

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2019, Volume 18, Issue 5: 2217-2242. Doi: 10.3934/cpaa.2019100

This issue Previous Article Ground state solutions for fractional scalar field equations under a general critical nonlinearity Next Article On the existence and uniqueness of solution to a stochastic simplified liquid crystal model

On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves

Ademir Pastor

IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária, 13083-859, Campinas-SP, Brazil

Received: May 31, 2017

Revised: August 31, 2017

Published: March 2019

The author is supported by CNPq grants 402849/2016-7 and 303098/2016-3

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Abstract

Reported here are results concerning the global well-posedness in the energy space and existence and stability of standing-wave solutions for 1-dimensional three-component systems of nonlinear Schrödinger equations with quadratic nonlinearities. For two particular systems we are interested in, the global well-posedness is established in view of the a priori bounds for the local solutions. The standing waves are explicitly obtained and their spectral stability is studied in the context of Hamiltonian systems. For more general Hamiltonian systems, the existence of standing waves is accomplished with a variational approach based on the Mountain Pass Theorem. Uniqueness results are also provided in some very particular cases.

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Mathematics Subject Classification: Primary: 35Q55, 35J50, 37K45; Secondary: 35Q51.

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