# Linear stability

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{{Short description|State of linear equations}}
In mathematics, in the theory of [differential equations](/source/differential_equations) and [dynamical systems](/source/dynamical_systems), a particular [stationary or quasistationary solution](/source/stationary_or_quasistationary_solution) to a nonlinear system is called '''linearly unstable '''if the [linearization](/source/linearization) of the equation at this solution has the form <math>dr/dt = A r</math>, where ''r'' is the perturbation to the steady state, ''A'' is a linear [operator](/source/Operator_(mathematics)) whose [spectrum](/source/Spectrum_(functional_analysis)) 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'''.<ref>[V. I. Arnold](/source/V._I._Arnold), ''[Ordinary Differential Equations](/source/Ordinary_Differential_Equations_(textbook))''. MIT Press, Cambridge, MA (1973)</ref><ref>P. Glendinning, Stability, instability and chaos: an introduction to the theory of nonlinear differential equations. Cambridge university press, 1994.</ref> If there exists 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".<ref>V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations", Princeton Univ. Press (1960)</ref>

==Examples==
===Ordinary differential equation===
The differential equation 
<math display="block">\frac{dx}{dt} = x - x^2</math>
has two stationary (time-independent) solutions: ''x''&nbsp;=&nbsp;0 and ''x''&nbsp;=&nbsp;1.
The linearization at ''x''&nbsp;=&nbsp;0 has the form
<math>\frac{dx}{dt}=x</math>. The linearized operator is ''A''<sub>0</sub>&nbsp;=&nbsp;1. The only eigenvalue is <math>\lambda=1</math>. The solutions to this equation grow exponentially; 
the stationary point ''x''&nbsp;=&nbsp;0 is linearly unstable.

To derive the linearization at {{math|1=''x'' = 1}}, one writes
<math>\frac{dr}{dt} = (1+r)-(1+r)^2 = -r-r^2</math>, where {{math|1=''r'' = ''x'' − 1}}. The linearized equation is then <math>\frac{dr}{dt} = -r</math>; the linearized operator is {{math|1=''A''<sub>1</sub> = −1}}, the only eigenvalue is <math>\lambda=-1</math>, hence this stationary point is linearly stable.

===Nonlinear Schrödinger Equation===
The [nonlinear Schrödinger equation](/source/nonlinear_Schr%C3%B6dinger_equation)
<math display="block">
i\frac{\partial u}{\partial t}
= -\frac{\partial^2 u}{\partial x^2} - |u|^{2k} u,</math> where {{math|''u''(''x'',''t'') &isin; '''C'''}} and {{math|''k'' > 0}}, has [solitary wave solutions](/source/soliton) of the form <math>\phi(x) e^{-i\omega t}</math>.<ref>{{ cite journal
|author=H. Berestycki and P.-L. Lions
|title=Nonlinear scalar field equations. I. Existence of a ground state
|journal=Arch. Rational Mech. Anal.
|volume=82
|issue=4
|year=1983
|pages=313–345
|doi=10.1007/BF00250555
|bibcode=1983ArRMA..82..313B|s2cid=123081616
}}</ref>
To derive the linearization at a solitary wave, one considers the solution in the form
<math>u(x,t) = (\phi(x)+r(x,t)) e^{-i\omega t}</math>. The linearized equation on <math>r(x,t)</math> is given by
<math display="block">
\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},
</math>
where <math display="block">A = \begin{bmatrix} 0 &L_0 \\ -L_1 & 0 \end{bmatrix},</math>
with <math display="block">L_0 = -\frac{\partial}{\partial x^2} - k\phi^2-\omega</math>
and <math display="block">L_1 = -\frac{\partial}{\partial x^2} - (2k+1) \phi^2-\omega</math>
the [differential operators](/source/differential_operators).
According to [Vakhitov–Kolokolov stability criterion](/source/Vakhitov%E2%80%93Kolokolov_stability_criterion),<ref>{{ cite journal
|author=N.G. Vakhitov and A.A. Kolokolov
|title=Stationary solutions of the wave equation in the medium with nonlinearity saturation
|journal=Radiophys. Quantum Electron.
|volume=16
|issue=7
|year=1973
|pages=783–789
|doi=10.1007/BF01031343
|bibcode=1973R&QE...16..783V |s2cid=123386885
}}</ref>
when {{math|''k'' > 2}}, the spectrum of ''A'' has positive point eigenvalues, so that the linearized equation is linearly (exponentially) unstable; for {{math|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 {{math|1=''k'' = 2}}, the solitary waves are unstable.  On the other hand, for {{math|0 < ''k'' < 2}}, the solitary waves are not only linearly stable but also [orbitally stable](/source/Orbital_stability).<ref>{{cite journal
|author=Manoussos Grillakis, Jalal Shatah, and Walter Strauss
|title=Stability theory of solitary waves in the presence of symmetry. I
|journal=J. Funct. Anal.
|volume=74
|year=1987
|pages=160–197
|doi=10.1016/0022-1236(87)90044-9|doi-access=free
}}</ref>

==See also==
*[Asymptotic stability](/source/Asymptotic_stability)
*[Linearization (stability analysis)](/source/Linearization)
*[Lyapunov stability](/source/Lyapunov_stability)
*[Orbital stability](/source/Orbital_stability)
*[Stability theory](/source/Stability_theory)
*[Vakhitov–Kolokolov stability criterion](/source/Vakhitov%E2%80%93Kolokolov_stability_criterion)

==References==
<references />

Category:Stability theory
Category:Solitons

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