In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form
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Example 1: ODE
The differential equation
has two stationary (time-independent) solutions: x = 0 and x = 1. The linearization at x = 0 has the form
To derive the linearizaton at x = 1, one writes
Example 2: NLS
The nonlinear Schrödinger equation
has solitary wave solutions of the form
where
with
and
the differential operators. According to Vakhitov–Kolokolov stability criterion , 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.