= Energy conservation for the nonlinear Schrödinger equation
{c}
For the focusing <nonlinear Schrödinger equation> on $\mathbb R^d$,
$$
iu_t+\Delta u+|u|^{p-1}u=0,\qquad p>1,
$$
the <energy> functional is
$$
E(u)=\frac12\int_{\mathbb R^d}|\nabla u|^2\,dx-\frac1{p+1}\int_{\mathbb R^d}|u|^{p+1}\,dx.
$$
For a sufficiently smooth solution with decay permitting <differentiation under the integral sign> and <integration by parts>, it is constant in time. Indeed,
$$
\frac{dE}{dt}=\operatorname{Re}\int_{\mathbb R^d}(-\Delta u-|u|^{p-1}u)\overline{u_t}\,dx
=\operatorname{Re}\int_{\mathbb R^d}i|u_t|^2\,dx=0.
$$
The cancellation is the Hamiltonian conservation mechanism: the evolution rotates the energy gradient by multiplication by $i$. The focusing sign makes the nonlinear contribution negative, so conserved <energy> alone need not bound the <L2 norm> of the <gradient>.<Low-Regularity Global Solutions to Nonlinear Dispersive Equations by Terence Tao>[Introduction, printed pp. 20–21: Hamiltonian evolution and formal invariants; low-regularity discussion distinguishes solution constructions]{ref}
For less regular solutions the displayed computation is only formal. To inherit the identity, approximate by smooth solutions and justify convergence of both energy terms. Convergence in <Sobolev space> $H^1$ controls the nonlinear term through a suitable <Sobolev embedding> only in the applicable exponent range, or with additional integrability. An arbitrary <weak solution> must not be assumed to satisfy the identity solely because the smooth equation does.
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