Asymptotic stability of <i>N</i>-soliton solutions for the modified nonlinear Schrödinger equation

The Cauchy problem of the modified nonlinear Schrödinger (mNLS) equation with the finite density type initial data is investigated via ∂̄-steepest descent method. In soliton region, the long-time asymptotic behavior of the mNLS equation is derived for large times. Furthermore, for general initial data in a non-vanishing background, the soliton resolution conjecture for the mNLS equation is verified, which means that the asymptotic expansion of the solution can be characterized by finite number of soliton solutions as the time goes to infinity, and error order O(t−3/4) from the ∂̄-equation. The results of this work also indicate soliton solutions of the mNLS equation are asymptotically stable.