Mass conservation for the nonlinear Schrödinger equation

For the nonlinear Schrödinger equationwith real-valued $F$, a sufficiently regular, decaying solution with finite L2 norm conservesDifferentiating gives $M^{\prime }(t)=2\mathrm{Re}\int u_t\overline{u}$. Substitute $u_t=i\Delta u+iF(|u{|}^2)u$. Integration by parts makes $\int \Delta u\overline{u}=-\int |\nabla u{|}^2$ real, and $\int F(|u{|}^2)|u{|}^2$ is real as well, so multiplication by $i$ gives zero real part.

The real coefficient is essential to this calculation; phase covariance alone does not imply mass conservation. For an $H^1$ solution obtained by a well-posed approximation theory, the invariant extends from smooth solutions by continuity in the L2 norm.^[ref]