Ordinary differential equation The differential equation \frac{dx}{dt} = x - x^2 has two stationary (time-independent) solutions:
x = 0 and
x = 1. The linearization at
x = 0 has the form \frac{dx}{dt}=x. The linearized operator is
A0 = 1. The only eigenvalue is \lambda=1. The solutions to this equation grow exponentially; the stationary point
x = 0 is linearly unstable. To derive the linearization at , one writes \frac{dr}{dt} = (1+r)-(1+r)^2 = -r-r^2, where . The linearized equation is then \frac{dr}{dt} = -r; the linearized operator is , the only eigenvalue is \lambda=-1, hence this stationary point is linearly stable.
Nonlinear Schrödinger Equation The
nonlinear Schrödinger equation i\frac{\partial u}{\partial t} = -\frac{\partial^2 u}{\partial x^2} - |u|^{2k} u, where and , has
solitary wave solutions of the form \phi(x) e^{-i\omega t}. To derive the linearization at a solitary wave, one considers the solution in the form u(x,t) = (\phi(x)+r(x,t)) e^{-i\omega t}. The linearized equation on r(x,t) is given by \frac{\partial}{\partial t}\begin{bmatrix}\text{Re}\,r\\ \text{Im} \,r\end{bmatrix}= A \begin{bmatrix}\text{Re}\,r \\ \text{Im} \,r\end{bmatrix}, where A = \begin{bmatrix} 0 &L_0 \\ -L_1 & 0 \end{bmatrix}, with L_0 = -\frac{\partial}{\partial x^2} - k\phi^2-\omega and L_1 = -\frac{\partial}{\partial x^2} - (2k+1) \phi^2-\omega the
differential operators. According to
Vakhitov–Kolokolov stability criterion, when , the spectrum of
A has positive point eigenvalues, so that the linearized equation is linearly (exponentially) unstable; for , 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 , the solitary waves are unstable. On the other hand, for , the solitary waves are not only linearly stable but also
orbitally stable. ==See also==