Conserved energies for the cubic NLS in 1-d

TL;DR

This paper establishes the existence of conserved energies equivalent to Sobolev norms for the cubic NLS and mKdV equations in one dimension, extending the understanding of their invariants across various regularity levels.

Contribution

It proves the existence of new conserved energies for cubic NLS and mKdV that match the $H^s$ norms for specified ranges of s, broadening the class of conserved quantities.

Findings

Conserved energies exist for the cubic NLS for all s > -1/2.

Conserved energies exist for the mKdV for all s ≥ -1.

These energies are equivalent to the $H^s$ norms of solutions.

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 $s\ge -1$.