What is relative equilibria and flow of PDE?

Question

Picture below is from Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials. The NLS is nonlinear Schrodinger equation $$ ih\frac{\partial \psi}{\partial t} = -\frac{h^2}{2}\Delta \psi + V\psi-|\psi|^{p-1}\psi $$ What is the flow of a equation ? And what is relative equilibria ?

Answer 1

The key to understanding a relative equilibrium is the group action. Since you have a group action, you can consider the orbits that its action generates. A relative equilibrium is then a solution to the equations that evolves through such an orbit. Since you didn't include in your image the bound states constructed in the text, I can't specifically point to how this is done in your text. If I had to guess, though, I would say that the bound states change in time precisely through the $S^1$ action, i.e. the time dependence is just phase change: something like $\phi(x,t) = \psi(x) e^{i \omega t}$.

Here "the flow of NLS" is just referring to the time evolution of the solution. The notation / terminology is borrowed from ODE / dynamical systems. In that case for linear problems $\dot{x} = A(t) x$ we have a fundamental matrix (solution operator) $\Phi(t)$, and the solution is given by $x(t) = \Phi(t) x_0$. The "flow" of problem is then the map $t \mapsto x(t) = \Phi(t) x_0$, which tells us how the solutions "flows in time." For linear evolution PDEs we can often do similar things, writing $t \mapsto u(\cdot,t) = S(t) u_0$ for the flow, with $S(t)$ the "solution operator." For example, for the linear heat or Schrodinger equations (without potential), the operator $S(t)$ can be explicitly written out via convolution . This is what the author of the text means here. The question is whether the bound states are stable in the sense that if we start with some data $\phi_0$ near (in some sense) the bound state, does the solution $\phi(\cdot,t)$ stay near (in some sense) the evolution of the bound state for all $t > 0$? This is just the idea of stability from ODE ported to the PDE setting.