Nonlinear Schrödinger equation

Remarks

Example: for $A \in \C$ and $k \in \R^d$, the plane wave $u(t,x) = A e^{i(k \cdot x - \omega t)}$ is a solution when $\omega = \abs{k}^2 + \mu \abs{A}^{p-1}$. Indeed $i \partial_t u = \omega u$, $\Delta u = -\abs{k}^2 u$ and $\abs{u}^{p-1} u = \abs{A}^{p-1} u$.

Non-example: the linear Schrödinger equation $i \partial_t u + \Delta u = 0$ corresponds to $\mu = 0$, which the definition excludes.

Smooth solutions that decay at infinity conserve the mass $\int \abs{u}^2 \, dx$ and the energy $$E(u) = \int_{\R^d} \left( \tfrac{1}{2} \abs{\nabla u}^2 + \tfrac{\mu}{p+1} \abs{u}^{p+1} \right) dx.$$ Some authors write the equation as $i \partial_t u = -\Delta u + \mu \abs{u}^{p-1} u$, which is the same equation.