= Mass conservation for the nonlinear Schrödinger equation
{c}
For the <nonlinear Schrödinger equation>
$$
iu_t+\Delta u+F(|u|^2)u=0
$$
with real-valued $F$, a sufficiently regular, decaying solution with finite <L2 norm> conserves
$$
M(u(t))=\int_{\mathbb R^d}|u(t,x)|^2\,dx.
$$
Differentiating gives $M'(t)=2\operatorname{Re}\int u_t\overline u$. 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>.<Nonlinear dispersive equations: local and global analysis by Terence Tao>[Section 3.4, conservation laws and their extension through well-posedness]{ref}
Back to article page
