Orbital stability for NLS
Let , and take . Some of the material we will present for general when it doesn’t simplify our work to use .
The (defocusing) nonlinear Schrödinger equation is
For a function on , the orbit of the function under the symmetries of NLS is
We say that is orbitally stable if initial data being near it implies that the solution of NLS is near it always.
The ground state equation is
The ground state equation comes from the solution of NLS. It is a fact that there is a positive bounded solution of the ground state equation, which we call a ground state.
The ground state is orbitally stable: for any there is a such that if
then for all
We define the energy functional by
so is a function of time but not of space.
It is a fact that for each there are and such that
Let ; so .
Let . We have
We shall express as a Taylor expansion about . We compute the first variation as follows:
where we used the fact that is real valued, integration by parts, and the fact that is a solution of the ground state equation. So the first variation of at is .
We now compute the second variation of .
Write . Then we have
And we assert that the remainder term of the Taylor series is , because is bounded. Therefore
We can bound using the Gagliardo-Nirenberg inequality, which gives us (for )
for some that doesn’t depend on . Therefore
so there is some such that