Well-posedness and inviscid limit behavior of solution for the generalized 1D Ginzburg–Landau equation

AbstractThe Cauchy problem of the one-dimensional generalized Ginzburg–Landau (GGL) equation is considered. The local well-posedness is obtained for initial data in Hs(R) with s>0, and global result in Hs(R) with s>0 is also obtained under some conditions. Moreover, the relation between the solution for GGL equation and the solution for the derivative nonlinear Schrödinger (DNLS) equation is studied. It is proved that for some T>0, the solution of Cauchy problem for the GGL equation converge to the solution of Cauchy problem for the DNLS in the natural space C([0,T];Hs) with s>12 if some coefficients tend to zero. Moreover, if initial data belong to H2, the convergence holds in C([0,T];H1) for any T>0