Doubt on momentum conservation in nonlinear schrodinger

Question

I want to prove the momentum conservation of the nonlinear Schrodinger eq. $u_t=i\Delta u + i|u|^{p-1}u$. The momentum is gives by $$ Pu=2Im \int_{R^n} \overline{u}\nabla u\ dx$$ I have read that in order to prove its conservation, de misterio multiply the eq. By $\nabla \overline{u}$, and take real part. After that, I suppose we need some integration by parts. However I am stucked with this integrals, $$\int \Delta u\nabla \overline{u}$$ I guess the real part of it must be zero, but I dont see it. Any help is welcome! Thanks in advance.

Answer 1

Consider the expression $$\int \Delta u \partial_{x_j} \overline{u}\;dx$$ By integrating by parts $3$ times (twice to move the Laplacian from $u$ to $\overline{u}$, and once again to move $\partial_{x_j}$ from $\overline{u}$ to $u$), we obtain $$\int \Delta u \partial_{x_j} \overline{u}\;dx = - \int \partial_{x_j} u \Delta \overline{u}\;dx = - \overline{\int \Delta u \partial_{x_j} \overline{u}\;dx}.$$ It follows that the integral is purely imaginary, as required.