Computing the total energy of Nonlinear Schrödinger (NLS) equation NLS: > $$ i\, u_t + \frac 12 u_{xx} \pm \lVert u\rVert^2u=0 $$ > > Show that the following energy of the nonlinear Schrödinger (NLS) equation is constant $$ E=\int\limits_{-\infty}^\infty \left( \frac 12 \lVert u_x \rVert^2\mp \frac 12 \lVert u \rVert^4 \right) \, dx = \mbox{const} $$ I don't know how I can show this.

Here is an outline of a non-rigorous answer, which I suspect is all you want.

Your equation is of the form $i u_t = Hu$. Multiply it by $\bar u_t$ to get $i \|u_t\|^2 = \bar u_t Hu$. Now take the real part this to get $$ 0 = \bar u_t Hu + u_t H \bar{u_t} .$$ Now integrate both sides with respect to $x$ from $-\infty$ to $\infty$. You will have to do an integration by parts to show $$ \int_{-\infty}^\infty \bar u_t u_{xx} \, dx = - \int_{-\infty}^\infty \bar u_{tx} u_{x} \, dx $$ and another equality which is simply the complex conjugate of this. (You will need to suppose that $u \to 0$ as $x \to \pm \infty$ to delete the cross terms.) Then use formulas like $$ \partial_t \|u_{x}\|^2 = \bar u_{xt} u_x + u_{xt} \bar u_x .$$ Then pull the derivatives outside of the integrals.

Making it all rigorous is quite a bit harder, and involves Hilbert spaces and Sobolev spaces, etc.