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| In mathematics, in the theory of [[differential equations]] and [[dynamical systems]], a particular [[stationary state|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 <math>\frac{dr}{dt}=A r</math>, where ''A'' is a linear [[Operator (mathematics)|operator]] whose [[Spectrum (functional analysis)|spectrum]] contains points with positive real part. If there are no such eigenvalues, the solution is called '''linearly''', or '''spectrally''', '''stable'''.
| | See a list of screenshots here |
| | | https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang&ilshowall=1 |
| ==Example 1: ODE==
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| The differential equation
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| :<math>\frac{dx}{dt}=x-x^2</math> | |
| has two stationary (time-independent) solutions: ''x'' = 0 and ''x'' = 1.
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| The linearization at ''x'' = 0 has the form
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| <math>\frac{dx}{dt}=x</math>. The solutions to this equation grow exponentially;
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| the stationary point ''x'' = 0 is linearly unstable.
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| To derive the linearizaton at ''x'' = 1, one writes
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| <math>\frac{dr}{dt}=(1+r)-(1+r)^2=-r-r^2</math>, where ''r'' = ''x'' − 1. The linearized equation is then <math>\frac{dr}{dt}=-r</math>; the linearized operator is ''A'' = −1, the only eigenvalue is <math>\lambda=-1</math>, hence this stationary point is linearly stable.
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| ==Example 2: NLS== | |
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| The [[nonlinear Schrödinger equation]]
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| : <math>
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| i\frac{\partial u}{\partial t}
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| =-\frac{\partial^2 u}{\partial x^2}-|u|^{2k} u
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| </math>, where ''u''(''x'',''t'') ∈ ℂ and ''k'' > 0,
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| has [[soliton|solitary wave solutions]] of the form <math>\phi(x)e^{-i\omega t}</math>
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| .<ref>{{ cite journal | |
| |author=H. Berestycki and P.-L. Lions
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| |title=Nonlinear scalar field equations. I. Existence of a ground state
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| |journal=Arch. Rational Mech. Anal.
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| |volume=82
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| |year=1983
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| |pages=313–345
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| |doi=10.1007/BF00250555
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| |bibcode=1983ArRMA..82..313B}}</ref>
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| To derive the linearization at a solitary wave, one considers the solution in the form
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| <math>u(x,t)=(\phi(x)+r(x,t))e^{-i\omega t}</math>. The linearized equation on <math>r(x,t)</math>
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| is given by
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| :<math>
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| \frac{\partial}{\partial t}\begin{bmatrix}\text{Re}\,u\\ \text{Im} \,u\end{bmatrix}=
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| A
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| \begin{bmatrix}\text{Re}\,u\\ \text{Im} \,u\end{bmatrix},
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| </math>
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| where
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| : <math>A=\begin{bmatrix}0&L_0\\-L_1&0\end{bmatrix},</math>
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| with
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| : <math>L_0=-\frac{\partial}{\partial x^2}-k|u|^2-\omega</math>
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| and
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| : <math>L_1=-\frac{\partial}{\partial x^2}-(2k+1)|u|^2-\omega</math>
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| the [[differential operators]].
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| According to [[Vakhitov–Kolokolov stability criterion]]
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| ,<ref>{{ cite journal
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| |author=N.G. Vakhitov and A.A. Kolokolov
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| |title=Stationary solutions of the wave equation in the medium with nonlinearity saturation
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| |journal=Radiophys. Quantum Electron.
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| |volume=16
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| |year=1973
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| |pages=783–789
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| |doi=10.1007/BF01031343
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| |bibcode=1973R%26QE...16..783V }}</ref>
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| 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.
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| It should be mentioned that linear stability does not automatically imply stability;
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| 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 [[Orbital stability|orbitally stable]].<ref>{{cite journal
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| |author=Manoussos Grillakis, Jalal Shatah, and Walter Strauss
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| |title=Stability theory of solitary waves in the presence of symmetry. I
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| |journal=J. Funct. Anal.
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| |volume=74
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| |year=1987
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| |pages=160–197
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| |doi=10.1016/0022-1236(87)90044-9}}</ref>
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| ==See also==
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| *[[Asymptotic stability]]
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| *[[Linearization#Stability_analysis|Linearization (stability analysis)]]
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| *[[Lyapunov stability]]
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| *[[Orbital stability]]
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| *[[Stability theory]]
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| *[[Vakhitov–Kolokolov stability criterion]]
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| ==References==
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| <references />
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| [[Category:Stability theory]]
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| [[Category:Solitons]]
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