On the periodic Cauchy problem for a coupled system of third-order nonlinear Schrödinger 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 \(H^{1}(\mathbb {T)\times }H^{1}(\mathbb {T)}\).

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Correspondence to M. Scialom.

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Scialom, M., Bragança, L.M. On the periodic Cauchy problem for a coupled system of third-order nonlinear Schrödinger equations. São Paulo J. Math. Sci. 13, 475–498 (2019). https://doi.org/10.1007/s40863-019-00143-6

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Keywords

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

Mathematics Subject Classification

  • 35Q35
  • 35Q53