Nonlinear Schrödinger Equation
The nonlinear Schrödinger equation
where u(x,t) ∈ C and k > 0, has solitary wave solutions of the form .[4]
To derive the linearization at a solitary wave, one considers the solution in the form
. The linearized equation on is given by
where
with
and
the differential operators.
According to Vakhitov–Kolokolov stability criterion,[5]
when k > 2, the spectrum of A has positive point eigenvalues, so that the linearized equation is linearly (exponentially) unstable; for 0 < k ≤ 2, the spectrum of A is purely imaginary, so that the corresponding solitary waves are linearly stable.
It should be mentioned that linear stability does not automatically imply stability;
in particular, when k = 2, the solitary waves are unstable. On the other hand, for 0 < k < 2, the solitary waves are not only linearly stable but also orbitally stable.[6]