Global Solutions and Stability Properties of the 5th Order Gardner Equation

In this work, we deal with the initial value problem of the 5th-order Gardner equation in R, presenting the local well-posedness result in H-2(R). As a consequence of the local result, in addition to H-2-energy conservation law, we are able to prove the globalwell-posedness result in H-2(R). Finally as a direct application, we prove that some globally defined functions, e.g. breather solutions of 5th order Gardner equation, are H-2(R) stable.