Front (physics)

<h2 id="fronts-connecting-stable---unstable-homogeneous-states">Fronts connecting stable - unstable homogeneous states</h2>
<p>The most simple example of front solution connecting a homogeneous stable state with a homogeneous unstable state can be shown in the one-dimensional <a href="/facts/Fisher%25E2%2580%2593Kolmogorov_equation/v112v7zY">Fisher–Kolmogorov equation</a>:
</p>

N
          
            t
          
        
        =
        D
        
          N
          
            x
            x
          
        
        +
        r
        N
        (
        
          N
          
            0
          
        
        −
        N
        )
      
    
    {\displaystyle N_{t}=DN_{xx}+rN(N_{0}-N)}

<p>that describes a simple model for the density 
  
    
      
        N
        (
        x
        ,
        t
        )
      
    
    {\displaystyle N(x,t)}
  
 of population. This equation has two steady states, 
  
    
      
        N
        =
        0
      
    
    {\displaystyle N=0}
  
, and 
  
    
      
        N
        =
        
          N
          
            0
          
        
      
    
    {\displaystyle N=N_{0}}
  
. This solution corresponds to extinction and saturation of population. Observe that this model is spatially-extended, because it includes a diffusion term given by the second derivative. The state 
  
    
      
        N
        ≡
        
          N
          
            0
          
        
      
    
    {\displaystyle N\equiv N_{0}}
  
 is stable as a simple linear analysis can show and the state 
  
    
      
        N
        =
        0
      
    
    {\displaystyle N=0}
  
 is unstable. There exist a family of front solutions connecting 
  
    
      
        N
        =
        
          N
          
            0
          
        
      
    
    {\displaystyle N=N_{0}}
  
 with 
  
    
      
        N
        =
        0
      
    
    {\displaystyle N=0}
  
, and such solution are propagative. Particularly, there exist one solution of the form 
  
    
      
        N
        (
        t
        ,
        x
        )
        =
        N
        (
        x
        −
        v
        t
        )
      
    
    {\displaystyle N(t,x)=N(x-vt)}
  
, with 
  
    
      
        v
      
    
    {\displaystyle v}
  
 is a velocity that only depends on 
  
    
      
        D
      
    
    {\displaystyle D}
  
 and 
  
    
      
        r
      
    
    {\displaystyle r}
  
<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>

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

<ol>
<li id="fn:1"><p>Pismen, L. M. (2006). Patterns and interfaces in dissipative dynamics. Berlin: Springer. ISBN 978-3-540-30430-2. <a href="978-3-540-30430-2" target="_blank">978-3-540-30430-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Horsthemke, Vicenç Mendéz, Sergei Fedotov, Werner (2010). Reaction-transport systems : mesoscopic foundations, fronts, and spatial instabilities. Heidelberg: Springer. ISBN 978-3642114427.{{cite book}}:  CS1 maint: multiple names: authors list (link) <a href="978-3642114427" target="_blank">978-3642114427</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Pomeau, Y. (1986). "Front motion, metastability and subcritical bifurcations in hydrodynamics". Physica D: Nonlinear Phenomena. 23 (1–3): 3–11. Bibcode:1986PhyD...23....3P. doi:10.1016/0167-2789(86)90104-1. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Alvarez-Socorro, A. J.; Clerc, M.G.; González-Cortés, G; Wilson, M. (2017). "Nonvariational mechanism of front propagation: Theory and experiments". Physical Review E. 95 (1): 010202. Bibcode:2017PhRvE..95a0202A. doi:10.1103/PhysRevE.95.010202. hdl:10533/232239. PMID 28208393. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Uchiyama, Kohei (1977). "The behavior of solutions of the equation of Kolmogorov–Petrovsky–Piskunov". Proceedings of the Japan Academy, Series A, Mathematical Sciences. 53 (7): 225–228. doi:10.3792/pjaa.53.225. <a href="https://www.researchgate.net/publication/38389672" target="_blank">https://www.researchgate.net/publication/38389672</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

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