# Conserved energies for the cubic NLS in 1-d

Research paper by Herbert Koch, Daniel Tataru

Indexed on: 08 Jul '16Published on: 08 Jul '16Published in: Mathematics - Analysis of PDEs

#### Abstract

We consider the cubic Nonlinear Schr\"odinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension. We prove that for each $s>-\frac12$ 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 $s\ge -1$.