Abstract:
We study the existence and nonlinear stability of standing waves for the periodic cubic nonlinear Schrodinger equation with a point defect determined by the periodic Dirac distribution at the origin, namely,
$$
iu_t + u_{xx} + Z\delta(x)u + |u|^3u = 0,
$$
$u = u(x,t)\in C$, $x,t\in R$, $Z\in R and <\delta,v>= v(0)$. This equation admits a smooth curve of positive periodic traveling waves with a profile determined by the Jacobi elliptic function of dnoidal type. Via a perturbation method and continuation argument, we obtain that in the case of a attractive defect (Z > 0) the standing wave solutions are stable in the energy space $H^1_{per}$. In the case of a repulsive defect (Z < 0), the standing wave solutions are stable in the subspace of even functions of $H^1_{per}$ and unstable in $H^1_{per}$.
This is a joint work with Gustavo Ponce
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