• DOI:10.1007/s11005-018-1061-4
• Corpus ID: 119615952

Asymptotic behavior of the Schrödinger–Debye system with refractive index of quadratic wave amplitude

@article{Corcho2017AsymptoticBO,
  title={Asymptotic behavior of the Schr{\"o}dinger–Debye system with refractive index of quadratic wave amplitude},
  author={Ad{\'a}n J. Corcho and Juan C. Cordero},
  journal={Letters in Mathematical Physics},
  year={2017},
  volume={108},
  pages={2031-2054}
}

• A. Corcho, J. C. Cordero
• Published 2 May 2017
• Mathematics
• Letters in Mathematical Physics

We obtain local well-posedness for the one-dimensional Schrödinger–Debye interactions in nonlinear optics in the spaces $$L^2\times L^p,\; 1\le p < \infty $$L2×Lp,1≤p<∞. When $$p=1$$p=1 we show that the local solutions extend globally. In the focusing regime, we consider a family of solutions $$\{(u_{\tau }, v_{\tau })\}_{\tau >0}$${(uτ,vτ)}τ>0 in $$ H^1\times H^1$$H1×H1 associated to an initial data family $$\{(u_{\tau _0},v_{\tau _0})\}_{\tau >0}$${(uτ0,vτ0)}τ>0 uniformly bounded in $$H^1\times L^2$$H1×L2, where $$\tau $$τ is a small response time parameter. We prove that $$\left( u_{\tau }, v_{\tau }\right) $$uτ,vτ converges to $$\left( u, -|u|^2\right) $$u,-|u|2 in the space $$L^{\infty }_{[0, T]}L^2_x\times L^1_{[0, T]}L^2_x$$L[0,T]∞Lx2×L[0,T]1Lx2 whenever $$u_{\tau _0}$$uτ0 converges to $$u_0$$u0 in $$H^1$$H1 as long as $$\tau $$τ tends to 0, where u is the solution of the one-dimensional cubic nonlinear Schrödinger equation with the initial data $$u_0$$u0. The convergence of $$v_{\tau }$$vτ for $$-|u|^2$$-|u|2 in the space $$L^{\infty }_{[0, T]}L^2_x$$L[0,T]∞Lx2 is shown under compatibility conditions of the initial data. For non-compatible data, we prove convergence except for a corrector term which looks like an initial layer phenomenon.