Linear stability

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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 \frac{dr}{dt}=A r, where A is a linear operator whose spectrum contains eigenvalues with positive real part. If there are no such eigenvalues, the solution is called linearly stable. Other names for linear stability include spectral, or exponential, stability.[citation needed]

Example 1: ODE[edit]

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

Example 2: NLS[edit]

The nonlinear Schrödinger equation


i\frac{\partial u}{\partial t}
=-\frac{\partial^2 u}{\partial x^2}-|u|^{2k} u
, where u(x,t) ∈ ℂ and k > 0,

has solitary wave solutions of the form \phi(x)e^{-i\omega t} .[1] 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}\,u\\ \text{Im} \,u\end{bmatrix}=
A
\begin{bmatrix}\text{Re}\,u\\ \text{Im} \,u\end{bmatrix},

where

A=\begin{bmatrix}0&L_0\\-L_1&0\end{bmatrix},

with

L_0=-\frac{\partial}{\partial x^2}-k|u|^2-\omega

and

L_1=-\frac{\partial}{\partial x^2}-(2k+1)|u|^2-\omega

the differential operators. According to Vakhitov–Kolokolov stability criterion ,[2] 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.[3]

See also[edit]

References[edit]

  1. ^ H. Berestycki and P.-L. Lions (1983). "Nonlinear scalar field equations. I. Existence of a ground state". Arch. Rational Mech. Anal. 82: 313–345. Bibcode:1983ArRMA..82..313B. doi:10.1007/BF00250555. 
  2. ^ N.G. Vakhitov and A.A. Kolokolov (1973). "Stationary solutions of the wave equation in the medium with nonlinearity saturation". Radiophys. Quantum Electron. 16: 783–789. Bibcode:1973R&QE...16..783V. doi:10.1007/BF01031343. 
  3. ^ Manoussos Grillakis, Jalal Shatah, and Walter Strauss (1987). "Stability theory of solitary waves in the presence of symmetry. I". J. Funct. Anal. 74: 160–197. doi:10.1016/0022-1236(87)90044-9.