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 ${\displaystyle dr/dt=Ar}$, where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part. If all the eigenvalues have negative real part, then the solution is called linearly stable. Other names for linear stability include exponential stability or stability in terms of first approximation.^[1][2] If there exist an eigenvalue with zero real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem".^[3]

Examples

Ordinary differential equation

The differential equation

$${\displaystyle {\frac {dx}{dt))=x-x^{2))$$

${\displaystyle {\frac {dx}{dt))=x}$

${\displaystyle \lambda =1}$

To derive the linearization at x = 1, one writes ${\displaystyle {\frac {dr}{dt))=(1+r)-(1+r)^{2}=-r-r^{2))$, where r = x − 1. The linearized equation is then ${\displaystyle {\frac {dr}{dt))=-r}$; the linearized operator is A₁ = −1, the only eigenvalue is ${\displaystyle \lambda =-1}$, hence this stationary point is linearly stable.

Nonlinear Schrödinger Equation

The nonlinear Schrödinger equation

$${\displaystyle i{\frac {\partial u}{\partial t))=-{\frac {\partial ^{2}u}{\partial x^{2))}-|u|^{2k}u,}$$

${\displaystyle \phi (x)e^{-i\omega t))$

${\displaystyle u(x,t)=(\phi (x)+r(x,t))e^{-i\omega t))$

${\displaystyle r(x,t)}$

$${\displaystyle {\frac {\partial }{\partial t)){\begin{bmatrix}{\text{Re))\,r\\{\text{Im))\,r\end{bmatrix))=A{\begin{bmatrix}{\text{Re))\,r\\{\text{Im))\,r\end{bmatrix)),}$$

$${\displaystyle A={\begin{bmatrix}0&L_{0}\\-L_{1}&0\end{bmatrix)),}$$

$${\displaystyle L_{0}=-{\frac {\partial }{\partial x^{2))}-k\phi ^{2}-\omega }$$

$${\displaystyle L_{1}=-{\frac {\partial }{\partial x^{2))}-(2k+1)\phi ^{2}-\omega }$$

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]