Rapidly oscillating spatially inhomogeneous structures in coherent nonlinear optical systems (Q542311)

This paper concerns the solution behavior of the parabolic equation \[ \partial_t u+ u=\varepsilon\partial^2_x u+ K\sin u(t,x-h) \] with periodic boundary conditions \(u(t,x+ 2\pi)= u(t,x)\) close to a homogeneous steady state \(u_0= K\sin u_0\). It is supposed that \(\varepsilon\) is close to zero, \(h\) is close to a rational multiple of \(2\pi\), and \(K\cos u_0\) is close to \(-1\). For several scalings of the three small control parameters a quasinormal form is derived and analyzed.