linear stability

The differential equation

$$\backslash frac\{dx\}\{dt\}\; =\; x\; -\; x^2$$

has two stationary (time-independent) solutions: x = 0 and x = 1.

The linearization at x = 0 has the form

$\backslash frac\{dx\}\{dt\}=x$. The linearized operator is A₀ = 1. The only eigenvalue is $\backslash lambda=1$. The solutions to this equation grow exponentially;

the stationary point x = 0 is linearly unstable.

To derive the linearization at {{math|1=x = 1}}, one writes

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

The nonlinear Schrödinger equation

i\frac{\partial u}{\partial t}

= -\frac{\partial^2 u}{\partial x^2} - |u|^{2k} u, where {{math|u(x,t) ∈ C}} and {{math|k > 0}}, has solitary wave solutions of the form $\backslash phi(x)\; e^\{-i\backslash omega\; t\}$.{{ 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

}}

To derive the linearization at a solitary wave, one considers the solution in the form

$u(x,t)\; =\; (\backslash phi(x)+r(x,t))\; e^\{-i\backslash omega\; t\}$. The linearized equation on $r(x,t)$ is given by

\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},

where

with $$L\_0\; =\; -\backslash frac\{\backslash partial\}\{\backslash partial\; x^2\}\; -\; k\backslash phi^2-\backslash omega$$

and $$L\_1\; =\; -\backslash frac\{\backslash partial\}\{\backslash partial\; x^2\}\; -\; (2k+1)\; \backslash phi^2-\backslash omega$$

the differential operators.

According to Vakhitov–Kolokolov stability criterion,{{ 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

}}

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.{{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

}}

• Asymptotic stability
• Linearization (stability analysis)
• Lyapunov stability
• Orbital stability
• Stability theory
• Vakhitov–Kolokolov stability criterion

Category:Stability theory

Category:Solitons