In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly or exponentially unstable if the linearization of the equation at this solution has the form , where A is a linear operator whose spectrum contains points with positive real part. If there are no such eigenvalues, the solution is called linearly, or spectrally, stable.
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 . The solutions to this equation grow exponentially; the stationary point x = 0 is linearly unstable.
To derive the linearizaton at x = 1, one writes , where r = x − 1. The linearized equation is then ; the linearized operator is A = −1, the only eigenvalue is , hence this stationary point is linearly stable.
Example 2: NLS
- , where u(x,t) ∈ ℂ and k > 0,
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 unstable.
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.
- Asymptotic stability
- Linearization (stability analysis)
- Lyapunov stability
- Orbital stability
- Stability theory
- Vakhitov–Kolokolov stability criterion
- 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.
- 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%26QE...16..783V. doi:10.1007/BF01031343.
- 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.