On the Global Well-Posedness and Orbital Stability of Standing Waves for the Schrodinger Equation with Fractional Dissipation

In this paper, we are concerned with the nonlinear fractional Schr & ouml;dinger equation. We extend the result of Guo and Huo and prove that the Cauchy problem of the nonlinear fractional Schr & ouml;dinger equation is global well-posed in H3/2-?(R) with 1/2 = ? < 1. In view of the complexity of the nonlinear fractional Schr & ouml;dinger equation itself, the local smoothing effect and maximal function estimates are not enough for presenting the global well-posedness for the nonlinear fractional Schr & ouml;dinger equation. In this paper, we use a suitably iterative scheme and complete the global well-posed result for Equation (R). Moreover, we obtain the orbital stability of standing waves for the above equations via establishing the profile decomposition of bounded sequences in H-s (R-N) (0 < s < 1) with N = 2.

Automated summary

This paper focuses on the nonlinear fractional Schr & ouml;dinger equation. The result of Guo and Huo is extended and it is proved that the Cauchy problem of the nonlinear fractional Schr & ouml;dinger equation is globally well-posed in H3/2-?(R) with 1/2 = ? < 1. Due to the complexity of the nonlinear fractional Schr & ouml;dinger equation itself, the local smoothing effect and maximal function estimates are insufficient for presenting the global well-posedness. In this paper, a suitably iterative scheme is used to complete the global well-posed result for Equation (R). Furthermore, the orbital stability of standing waves for the above equations is obtained by establishing the profile decomposition of bounded sequences in H-s (R-N) (0 < s < 1) with N = 2.