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On the periodic Cauchy problem for a coupled system of third-order nonlinear Schrödinger equations

  • M. ScialomEmail author
  • L. M. Bragança
Special Section: Nonlinear Dispersive Equations

Abstract

We investigate some well-posedness issues for the initial value problem (IVP) associated to the system
$$\begin{aligned} \left\{ \begin{array}[c]{l} 2i\partial _{t}u+q\partial _{x}^{2}u+i\gamma \partial _{x}^{3}u=F_{1}(u,w)\\ 2i\partial _{t}w+q\partial _{x}^{2}w+i\gamma \partial _{x}^{3}w=F_{2}(u,w), \end{array} \right. \end{aligned}$$
where \(F_{1}\) and \(F_{2}\) are polynomials of degree 3 involving u, w and their derivatives. This system describes the dynamics of two nonlinear short-optical pulses envelopes u(xt) and w(xt) in fibers (Hasegawa and Kodama in IEEE J Quantum Electron 23(5):510–524, 1987; Porsezian et al. in Phys Rev E 50:1543–1547, 1994). We prove periodic local well-posedness for the IVP with data in Sobolev spaces \(H^{s}(\mathbb {T)\times } H^{s}(\mathbb {T)}\), \( s\ge 1/2\) and global well-posedness result in Sobolev spaces \(H^{1}(\mathbb {T)\times }H^{1}(\mathbb {T)}\).

Keywords

Coupled system of third-order nonlinear Schrödinger equations Periodic Cauchy problem Local and global well-posedness 

Mathematics Subject Classification

35Q35 35Q53 

Notes

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Copyright information

© Instituto de Matemática e Estatística da Universidade de São Paulo 2019

Authors and Affiliations

  1. 1.IMECC-UNICAMPCampinasBrazil
  2. 2.DMA-UFVViçosaBrazil

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