Linear stability

<h2 id="examples">Examples</h2>
<h3>Ordinary differential equation</h3>
The differential equation

d
              x
            
            
              d
              t
            
          
        
        =
        x
        −
        
          x
          
            2
          
        
      
    
    {\displaystyle {\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

d
 x
 
 
 d
 t
 
 
 
 =
 x
 
 
 {\displaystyle {\frac {dx}{dt}}=x}
 
. The linearized operator is A0 = 1. The only eigenvalue is 
 
 
 
 λ
 =
 1
 
 
 {\displaystyle \lambda =1}
 
. The solutions to this equation grow exponentially; 
the stationary point x = 0 is linearly unstable.
To derive the linearization at x = 1, one writes

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

<h3>Nonlinear Schrödinger Equation</h3>
The <a href="/facts/Nonlinear_Schr%C3%B6dinger_equation/gfGkRApW">nonlinear Schrödinger equation</a>

i
 
 
 
 ∂
 u
 
 
 ∂
 t
 
 
 
 =
 −
 
 
 
 
 ∂
 
 2
 
 
 u
 
 
 ∂
 
 x
 
 2
 
 
 
 
 
 −
 
 |
 
 u
 
 
 |
 
 
 2
 k
 
 
 u
 ,
 
 
 {\displaystyle 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 <a href="/facts/Soliton/5rC19BSU">solitary wave solutions</a> of the form 
 
 
 
 ϕ
 (
 x
 )
 
 e
 
 −
 i
 ω
 t
 
 
 
 
 {\displaystyle \phi (x)e^{-i\omega t}}
 
.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
To derive the linearization at a solitary wave, one considers the solution in the form

u
        (
        x
        ,
        t
        )
        =
        (
        ϕ
        (
        x
        )
        +
        r
        (
        x
        ,
        t
        )
        )
        
          e
          
            −
            i
            ω
            t
          
        
      
    
    {\displaystyle u(x,t)=(\phi (x)+r(x,t))e^{-i\omega t}}
  
. The linearized equation on 
  
    
      
        r
        (
        x
        ,
        t
        )
      
    
    {\displaystyle r(x,t)}
  
 is given by

∂
            
              ∂
              t
            
          
        
        
          
            [
            
              
                
                  
                    Re
                  
                  
                  r
                
              
              
                
                  
                    Im
                  
                  
                  r
                
              
            
            ]
          
        
        =
        A
        
          
            [
            
              
                
                  
                    Re
                  
                  
                  r
                
              
              
                
                  
                    Im
                  
                  
                  r
                
              
            
            ]
          
        
        ,
      
    
    {\displaystyle {\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 
  
    
      
        A
        =
        
          
            [
            
              
                
                  0
                
                
                  
                    L
                    
                      0
                    
                  
                
              
              
                
                  −
                  
                    L
                    
                      1
                    
                  
                
                
                  0
                
              
            
            ]
          
        
        ,
      
    
    {\displaystyle A={\begin{bmatrix}0&L_{0}\\-L_{1}&0\end{bmatrix}},}

with 
  
    
      
        
          L
          
            0
          
        
        =
        −
        
          
            ∂
            
              ∂
              
                x
                
                  2
                
              
            
          
        
        −
        k
        
          ϕ
          
            2
          
        
        −
        ω
      
    
    {\displaystyle L_{0}=-{\frac {\partial }{\partial x^{2}}}-k\phi ^{2}-\omega }

and 
  
    
      
        
          L
          
            1
          
        
        =
        −
        
          
            ∂
            
              ∂
              
                x
                
                  2
                
              
            
          
        
        −
        (
        2
        k
        +
        1
        )
        
          ϕ
          
            2
          
        
        −
        ω
      
    
    {\displaystyle L_{1}=-{\frac {\partial }{\partial x^{2}}}-(2k+1)\phi ^{2}-\omega }

the <a href="/facts/Differential_operators/Nr15J0IE">differential operators</a>.
According to <a href="/facts/Vakhitov%E2%80%93Kolokolov_stability_criterion/JS2l81H3">Vakhitov–Kolokolov stability criterion</a>,<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
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 <a href="/facts/Orbital_stability/YGH2P2LC">orbitally stable</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Asymptotic_stability/r3je5zaQ">Asymptotic stability</a></li>
<li><a href="/facts/Linearization/ohYSeQR3">Linearization (stability analysis)</a></li>
<li><a href="/facts/Lyapunov_stability/r3je5zaQ">Lyapunov stability</a></li>
<li><a href="/facts/Orbital_stability/YGH2P2LC">Orbital stability</a></li>
<li><a href="/facts/Stability_theory/Su5sHnxs">Stability theory</a></li>
<li><a href="/facts/Vakhitov%E2%80%93Kolokolov_stability_criterion/JS2l81H3">Vakhitov–Kolokolov stability criterion</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">V.I. Arnold, Ordinary Differential Equations. MIT Press, Cambridge, MA (1973) <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">P. Glendinning, Stability, instability and chaos: an introduction to the theory of nonlinear differential equations. Cambridge university press, 1994. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations", Princeton Univ. Press (1960) <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">H. Berestycki and P.-L. Lions (1983). "Nonlinear scalar field equations. I. Existence of a ground state". Arch. Rational Mech. Anal. 82 (4): 313–345. Bibcode:1983ArRMA..82..313B. doi:10.1007/BF00250555. S2CID 123081616. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">N.G. Vakhitov and A.A. Kolokolov (1973). "Stationary solutions of the wave equation in the medium with nonlinearity saturation". Radiophys. Quantum Electron. 16 (7): 783–789. Bibcode:1973R&QE...16..783V. doi:10.1007/BF01031343. S2CID 123386885. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">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.{{cite journal}}: CS1 maint: multiple names: authors list (link) <a href="https://doi.org/10.1016%2F0022-1236%2887%2990044-9" target="_blank">https://doi.org/10.1016%2F0022-1236%2887%2990044-9</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Linear stability open-in-new

Linear stability