CONSERVED ENERGIES FOR THE CUBIC NONLINEAR SCHRODINGER EQUATION IN ONE DIMENSION

摘要

We consider the cubic nonlinear SchrOdinger (NLS) equation as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension. We prove that for each s -1/2 there exists a conserved energy which is equivalent to the H-s-norm of the solution. For the Korteweg-de Vries (KdV) equation, there is a similar conserved energy for every s = -1.