Periodic travelling wave

<h2 id="history-of-research">History of research</h2>
While periodic travelling waves have been known as solutions of the <a href="/facts/Wave_equation/J2ksD2Az">wave equation</a> since the 18th century, their study in <a href="/facts/Nonlinear_system/4aYItALC">nonlinear systems</a> began in the 1970s. A key early research paper was that of <a href="/facts/Nancy_Kopell/uggMIw8V">Nancy Kopell</a> and Lou Howard<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> which proved several fundamental results on periodic travelling waves in <a href="/facts/Reaction%25E2%2580%2593diffusion_equation/deqI4VAP">reaction–diffusion equations</a>. This was followed by significant research activity during the 1970s and early 1980s. There was then a period of inactivity, before interest in periodic travelling waves was renewed by mathematical work on their generation,<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> and by their detection in <a href="/facts/Ecology/YE8upuJU">ecology</a>, in spatiotemporal data sets on cyclic populations.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> Since the mid-2000s, research on periodic travelling waves has benefitted from new computational methods for studying their <a href="/facts/Stability_theory/Su5sHnxs">stability</a> and absolute stability.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a><a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

<h2 id="families">Families</h2>
The existence of periodic travelling waves usually depends on the <a href="/facts/Parameter/zFvVFMv9">parameter</a> values in a mathematical equation. If there is a periodic travelling wave solution, then there is typically a family of such solutions, with different wave speeds. For partial differential equations, periodic travelling waves typically occur for a continuous range of wave speeds.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>

<h2 id="stability">Stability</h2>
An important question is whether a periodic travelling wave is <a href="/facts/Stability_theory/Su5sHnxs">stable</a> or <a href="/facts/Instability/V0uNz1oI">unstable</a> as a solution of the original mathematical system. For partial differential equations, it is typical that the wave family subdivides into <a href="/facts/Stability_theory/Su5sHnxs">stable</a> and <a href="/facts/Instability/V0uNz1oI">unstable</a> 
parts.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a><a class="footnote-ref" id="fnref:20" href="#fn:20">20</a><a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> 
For unstable periodic travelling waves, an important subsidiary question is whether they are 
absolutely or convectively unstable, meaning that there are or are not stationary growing linear modes.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> This issue has only been resolved for a few partial differential equations.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a><a class="footnote-ref" id="fnref:24" href="#fn:24">24</a><a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>

<h2 id="generation">Generation</h2>
A number of mechanisms of periodic travelling wave generation are now well established. These include: 

<ul><li>Heterogeneity: spatial noise in parameter values can generate a series of bands of periodic travelling waves.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> This is important in applications to <a href="/facts/Chemical_clock/4WAdyMof">oscillatory chemical reactions</a>, where impurities can cause target patterns or spiral waves, which are two-dimensional generalisations of periodic travelling waves. This process provided the motivation for much of the work on periodic travelling waves in the 1970s and early 1980s. Landscape heterogeneity has also been proposed as a cause of the periodic travelling waves seen in ecology.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a></li>
<li>Invasions, which can leave a periodic travelling wave in their wake.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a><a class="footnote-ref" id="fnref:29" href="#fn:29">29</a><a class="footnote-ref" id="fnref:30" href="#fn:30">30</a> This is important in the <a href="/facts/Taylor%25E2%2580%2593Couette_flow/EWzC6htW">Taylor–Couette</a> system in the presence of through flow,<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> in chemical systems such as the <a href="/facts/Belousov%25E2%2580%2593Zhabotinsky_reaction/RHH660Iy">Belousov–Zhabotinsky reaction</a><a class="footnote-ref" id="fnref:32" href="#fn:32">32</a><a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> and in <a href="/facts/Predation/JEGdrX3f">predator-prey</a> systems in <a href="/facts/Ecology/YE8upuJU">ecology</a>.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a><a class="footnote-ref" id="fnref:35" href="#fn:35">35</a></li>
<li>Domain boundaries with <a href="/facts/Dirichlet_boundary_condition/m9wsgG89">Dirichlet</a> or <a href="/facts/Robin_boundary_condition/tEoxCAfR">Robin</a> boundary conditions.<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a><a class="footnote-ref" id="fnref:37" href="#fn:37">37</a><a class="footnote-ref" id="fnref:38" href="#fn:38">38</a> This is potentially important in <a href="/facts/Ecology/YE8upuJU">ecology</a>, where Robin or Dirichlet conditions correspond to a boundary between habitat and a surrounding hostile environment. However definitive <a href="/facts/Empirical_evidence/JUaYSGJc">empirical evidence</a> on the cause of waves is hard to obtain for ecological systems.</li>
<li>Migration driven by pursuit and evasion.<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a> This may be significant in <a href="/facts/Ecology/YE8upuJU">ecology</a>.</li>
<li>Migration between sub-populations,<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a> which again has potential <a href="/facts/Ecology/YE8upuJU">ecological</a> significance.</li></ul>
In all of these cases, a key question is which member of the periodic travelling wave family is selected. For most mathematical systems this remains an open problem.

<h2 id="spatiotemporal-chaos">Spatiotemporal chaos</h2>

It is common that for some <a href="/facts/Parameter/zFvVFMv9">parameter</a> values, the periodic travelling waves arising from a wave generation mechanism are unstable. In such cases the solution usually evolves to spatiotemporal <a href="/facts/Chaos_theory/LTC17vsF">chaos</a>.<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a><a class="footnote-ref" id="fnref:42" href="#fn:42">42</a> Thus the solution involves a spatiotemporal transition to chaos via the periodic travelling wave.

<h2 id="lambdaomega-systems-and-the-complex-ginzburglandau-equation">Lambda–omega systems and the complex Ginzburg–Landau equation</h2>
There are two particular mathematical systems that serve as prototypes for periodic travelling waves, and which have been fundamental to the development of mathematical understanding and theory. These are the "lambda-omega" class of <a href="/facts/Reaction%25E2%2580%2593diffusion_system/deqI4VAP">reaction–diffusion equations</a><a class="footnote-ref" id="fnref:43" href="#fn:43">43</a>

∂
              u
            
            
              ∂
              t
            
          
        
        =
        
          
            
              
                ∂
                
                  2
                
              
              u
            
            
              ∂
              
                x
                
                  2
                
              
            
          
        
        +
        λ
        (
        r
        )
        u
        −
        ω
        (
        r
        )
        v
      
    
    {\displaystyle {\frac {\partial u}{\partial t}}={\frac {\partial ^{2}u}{\partial x^{2}}}+\lambda (r)u-\omega (r)v}

∂
              v
            
            
              ∂
              t
            
          
        
        =
        
          
            
              
                ∂
                
                  2
                
              
              v
            
            
              ∂
              
                x
                
                  2
                
              
            
          
        
        +
        ω
        (
        r
        )
        u
        +
        λ
        (
        r
        )
        v
      
    
    {\displaystyle {\frac {\partial v}{\partial t}}={\frac {\partial ^{2}v}{\partial x^{2}}}+\omega (r)u+\lambda (r)v}

(
 
 
 
 r
 =
 
 
 
 u
 
 2
 
 
 +
 
 v
 
 2
 
 
 
 
 
 
 {\textstyle r={\sqrt {u^{2}+v^{2}}}}
 
) and the complex <a href="/facts/Ginzburg%25E2%2580%2593Landau_theory/VAHPgSzY">Ginzburg–Landau</a> equation.<a class="footnote-ref" id="fnref:44" href="#fn:44">44</a>

 
 
 
 
 
 
 ∂
 A
 
 
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 =
 A
 +
 (
 1
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 i
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 )
 
 
 
 
 ∂
 
 2
 
 
 A
 
 
 ∂
 
 x
 
 2
 
 
 
 
 
 −
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 c
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 |
 
 A
 
 
 |
 
 
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 A
 
 
 {\displaystyle {\frac {\partial A}{\partial t}}=A+(1+ib){\frac {\partial ^{2}A}{\partial x^{2}}}-(1+ic)|A|^{2}A}

(A is complex-valued). Note that these systems are the same if λ(r) = 1 − r2, ω(r) = −c r2 and b = 0. Both systems can be simplified by rewriting the equations in terms of the amplitude (r or |A|) and the phase (arctan(v/u) or arg A). Once the equations have been rewritten in this way, it is easy to see that solutions with constant amplitude are periodic travelling waves, with the phase being a linear function of <a href="/facts/Euclidean_space/R2UbzmzM">space</a> and time. Therefore, u and v, or Re(A) and Im(A), are <a href="/facts/Sine_wave/mD1CB09M">sinusoidal</a> functions of space and time.
These exact solutions for the periodic travelling wave families enable a great deal of further analytical study. Exact conditions for the <a href="/facts/Stability_theory/Su5sHnxs">stability</a> of the periodic travelling waves can be found,<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a><a class="footnote-ref" id="fnref:46" href="#fn:46">46</a> and the condition for absolute stability can be reduced to the solution of a simple <a href="/facts/Polynomial/Lzak8VVx">polynomial</a>.<a class="footnote-ref" id="fnref:47" href="#fn:47">47</a><a class="footnote-ref" id="fnref:48" href="#fn:48">48</a> Also exact solutions have been obtained for the selection problem for waves generated by invasions<a class="footnote-ref" id="fnref:49" href="#fn:49">49</a><a class="footnote-ref" id="fnref:50" href="#fn:50">50</a> 
and by zero Dirichlet boundary conditions.<a class="footnote-ref" id="fnref:51" href="#fn:51">51</a><a class="footnote-ref" id="fnref:52" href="#fn:52">52</a> 
In the latter case, for the complex Ginzburg–Landau equation, the overall solution is a stationary Nozaki-Bekki hole.<a class="footnote-ref" id="fnref:53" href="#fn:53">53</a><a class="footnote-ref" id="fnref:54" href="#fn:54">54</a>
Much of the work on periodic travelling waves in the complex Ginzburg–Landau equation is in the <a href="/facts/Physics/a7aH2y08">physics</a> literature, where they are usually known as <a href="/facts/Plane_wave/0aDUbNhQ">plane waves</a>.

<h2 id="numerical-computation-of-periodic-travelling-waves-and-their-stability">Numerical computation of periodic travelling waves and their stability</h2>
For most mathematical equations, <a href="/facts/Mathematical_analysis/XXeR26Ih">analytical</a> calculation of periodic travelling wave solutions is not possible, and therefore it is necessary to perform <a href="/facts/Numerical_analysis/QT9cYa9z">numerical computations</a>. For <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a>, denote by x and t the (one-dimensional) space and time variables, respectively. Then periodic travelling waves are functions of the travelling wave variable z=x-c t. Substituting this solution form into the partial differential equations gives a system of <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equations</a> known as the travelling wave equations. Periodic travelling waves correspond to <a href="/facts/Limit_cycle/LmM9EVoA">limit cycles</a> of these equations, and this provides the basis for <a href="/facts/Numerical_analysis/QT9cYa9z">numerical computations</a>. The standard computational approach is <a href="/facts/Numerical_continuation/2ypyzVVk">numerical continuation</a> of the travelling wave equations. One first performs a continuation of a <a href="/facts/Steady_state/qgrL4BNs">steady state</a> to locate a <a href="/facts/Hopf_bifurcation/WuFmnr96">Hopf bifurcation</a> point. This is the starting point for a branch (family) of periodic travelling wave solutions, which one can follow by numerical continuation. In some (unusual) cases both end points of a branch (family) of periodic travelling wave solutions are <a href="/facts/Homoclinic_orbit/4Cuw16Il">homoclinic</a> solutions,<a class="footnote-ref" id="fnref:55" href="#fn:55">55</a> in which case one must use an external starting point, such as a numerical solution of the partial differential equations.
Periodic travelling wave <a href="/facts/Stability_theory/Su5sHnxs">stability</a> can also be calculated numerically, by computing the <a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">spectrum</a>. This is made easier by the fact that the spectrum of periodic travelling wave solutions of partial differential equations consists entirely of <a href="/facts/Essential_spectrum/EHN47PNM">essential spectrum</a>.<a class="footnote-ref" id="fnref:56" href="#fn:56">56</a> 
Possible numerical approaches include Hill's method<a class="footnote-ref" id="fnref:57" href="#fn:57">57</a> and numerical continuation of the spectrum.<a class="footnote-ref" id="fnref:58" href="#fn:58">58</a> One advantage of the latter approach is that it can be extended to calculate boundaries in <a href="/facts/Parameter/zFvVFMv9">parameter</a> space between stable and unstable waves<a class="footnote-ref" id="fnref:59" href="#fn:59">59</a>
Software: The free, <a href="/facts/Open-source_software/HzTaB8E9">open-source software</a> package Wavetrain <a href="http://www.ma.hw.ac.uk/wavetrain">http://www.ma.hw.ac.uk/wavetrain</a> is designed for the numerical study of periodic travelling waves.<a class="footnote-ref" id="fnref:60" href="#fn:60">60</a> 
Using <a href="/facts/Numerical_continuation/2ypyzVVk">numerical continuation</a>, Wavetrain is able to calculate the form and stability of periodic travelling wave solutions of partial differential equations, and the regions of parameter space in which waves exist and in which they are stable.

<h2 id="applications">Applications</h2>
Examples of phenomena resembling periodic travelling waves that have been found <a href="/facts/Empirical_evidence/JUaYSGJc">empirically</a> include the following.

<ul><li>Many natural populations undergo multi-year cycles of abundance. In some cases these population cycles are spatially organised into a periodic travelling wave. This behaviour has been found in <a href="/facts/Vole/yPC6PvYk">voles</a> in Fennoscandia<a class="footnote-ref" id="fnref:61" href="#fn:61">61</a> and Northern UK,<a class="footnote-ref" id="fnref:62" href="#fn:62">62</a> <a href="/facts/Geometer_moth/ces6X6JW">geometrid moths</a> in Northern Fennoscandia,<a class="footnote-ref" id="fnref:63" href="#fn:63">63</a> larch budmoths in the European Alps<a class="footnote-ref" id="fnref:64" href="#fn:64">64</a> and <a href="/facts/Red_grouse/2qtjn2zo">red grouse</a> in Scotland.<a class="footnote-ref" id="fnref:65" href="#fn:65">65</a></li>
<li>In <a href="/facts/Semi-arid_climate/XnSYao1E">semi-deserts</a>, <a href="/facts/Vegetation/nAHr9kjk">vegetation</a> often self-organises into <a href="/facts/Patterns_in_nature/handzeJh">spatial patterns</a>.<a class="footnote-ref" id="fnref:66" href="#fn:66">66</a> On slopes, this typically consists of stripes of vegetation running parallel to the <a href="/facts/Contour_line/35Qten5r">contours</a>, separated by stripes of bare ground; this type of banded vegetation is sometimes known as <a href="/facts/Tiger_bush/xtNLbekg">Tiger bush</a>. Many observational studies have reported slow movement of the stripes in the uphill direction.<a class="footnote-ref" id="fnref:67" href="#fn:67">67</a> However, in a number of other cases the data points clearly to stationary patterns,<a class="footnote-ref" id="fnref:68" href="#fn:68">68</a> and the question of movement remains controversial. The conclusion that is most consistent with available data is that some banded vegetation patterns move while others do not.<a class="footnote-ref" id="fnref:69" href="#fn:69">69</a> Patterns in the former category have the form of periodic travelling waves.</li>
<li>Travelling bands occur in <a href="/facts/Oscillation_theory/xSxr5zC4">oscillatory</a> and <a href="/facts/Excitable_medium/LSjRJ6I5">excitable</a> chemical reactions. They were observed in the 1970s in the <a href="/facts/Belousov%25E2%2580%2593Zhabotinsky_reaction/RHH660Iy">Belousov–Zhabotinsky reaction</a><a class="footnote-ref" id="fnref:70" href="#fn:70">70</a> and they formed an important motivation for the mathematical work done on periodic travelling waves at that time. More recent research has also exploited the capacity to link the experimentally observed bands with mathematical theory of periodic travelling waves via detailed modelling.<a class="footnote-ref" id="fnref:71" href="#fn:71">71</a></li>
<li>Periodic travelling waves occur in the Sun, as part of the <a href="/facts/Solar_cycle/QrP4Egea">solar cycle</a>.<a class="footnote-ref" id="fnref:72" href="#fn:72">72</a><a class="footnote-ref" id="fnref:73" href="#fn:73">73</a> They are a consequence of the generation of the Sun's <a href="/facts/Magnetic_field/Ta8Ev588">magnetic field</a> by the <a href="/facts/Solar_dynamo/xYT7FEcm">solar dynamo</a>. As such, they are related to <a href="/facts/Sunspot/7E8cWSGl">sunspots</a>.</li>
<li>In <a href="/facts/Fluid_dynamics/y7uzSCfg">hydrodynamics</a>, <a href="/facts/Convection/E8vsCPhu">convection</a> patterns often involve periodic travelling waves. Specific instances include binary fluid convection<a class="footnote-ref" id="fnref:74" href="#fn:74">74</a> and heated wire convection.<a class="footnote-ref" id="fnref:75" href="#fn:75">75</a></li>
<li>Patterns of periodic travelling wave form occur in the "printer's instability", in which the thin gap between two rotating acentric cylinders is filled with oil.<a class="footnote-ref" id="fnref:76" href="#fn:76">76</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Plane_wave/0aDUbNhQ">Plane wave</a></li>
<li><a href="/facts/Reaction%25E2%2580%2593diffusion_system/deqI4VAP">Reaction–diffusion system</a></li>
<li><a href="/facts/Wave/fU0HH6St">Wave</a></li></ul>

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

<ol>
<li id="fn:1">N. Kopell, L.N. Howard (1973) "Plane wave solutions to reaction–diffusion equations", Stud. Appl. Math. 52: 291–328. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">I. S. Aranson, L. Kramer (2002) "The world of the complex Ginzburg–Landau equation", Rev. Mod. Phys. 74: 99–143. DOI:10.1103/RevModPhys.74.99 <a href="http://link.aps.org/doi/10.1103/RevModPhys.74.99" target="_blank">http://link.aps.org/doi/10.1103/RevModPhys.74.99</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">S. Coombes (2001) "From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves", Math. Biosci. 170: 155–172.
DOI:10.1016/S0025-5564(00)00070-5 <a href="https://dx.doi.org/10.1016/S0025-5564(00)00070-5" target="_blank">https://dx.doi.org/10.1016/S0025-5564(00)00070-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">J.A. Sherratt, G. J. Lord (2007) "Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments", Theor. Popul. Biol. 71 (2007): 1–11.
DOI:10.1016/j.tpb.2006.07.009 <a href="https://dx.doi.org/10.1016/j.tpb.2006.07.009" target="_blank">https://dx.doi.org/10.1016/j.tpb.2006.07.009</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">S. A. Gourley, N. F. Britton (1993) "Instability of traveling wave solutions of a population model with nonlocal effects", IMA J. Appl. Math. 51: 299–310.
DOI:10.1093/imamat/51.3.299 <a href="https://archive.today/20130415152916/http://imamat.oxfordjournals.org/content/51/3/299.short" target="_blank">https://archive.today/20130415152916/http://imamat.oxfordjournals.org/content/51/3/299.short</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">P. Ashwin, M. V. Bartuccelli, T. J. Bridges, S. A. Gourley (2002) "Travelling fronts for the KPP equation with spatio-temporal delay", Z. Angew. Math. Phys. 53: 103–122.
DOI:0010-2571/02/010103-20 <a href="https://link.springer.com/article/10.1007%2Fs00033-002-8145-8?LI=true" target="_blank">https://link.springer.com/article/10.1007%2Fs00033-002-8145-8?LI=true</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">M. Kot (1992) "Discrete-time travelling waves: ecological examples", J. Math. Biol. 30: 413-436. DOI:10.1007/BF00173295 <a href="https://dx.doi.org/10.1007/BF00173295" target="_blank">https://dx.doi.org/10.1007/BF00173295</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">M. D. S. Herrera, J. S. Martin (2009) "An analytical study in coupled map lattices of synchronized states and traveling waves, and of their period-doubling cascades", Chaos, Solitons & Fractals 42: 901–910.DOI:10.1016/j.chaos.2009.02.040 <a href="https://dx.doi.org/10.1016/j.chaos.2009.02.040" target="_blank">https://dx.doi.org/10.1016/j.chaos.2009.02.040</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">J. A. Sherratt (1996) "Periodic travelling waves in a family of deterministic cellular automata", Physica D 95: 319–335.
DOI:10.1016/0167-2789(96)00070-X <a href="https://dx.doi.org/10.1016/0167-2789(96)00070-X" target="_blank">https://dx.doi.org/10.1016/0167-2789(96)00070-X</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">M. Courbage (1997) "On the abundance of traveling waves in 1D infinite cellular automata", Physica D 103: 133–144.
DOI:10.1016/S0167-2789(96)00256-4 <a href="https://dx.doi.org/10.1016/S0167-2789(96)00256-4" target="_blank">https://dx.doi.org/10.1016/S0167-2789(96)00256-4</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">N. Kopell, L.N. Howard (1973) "Plane wave solutions to reaction–diffusion equations", Stud. Appl. Math. 52: 291–328. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">J. A. Sherratt (1994) "Irregular
wakes in reaction-diffusion waves", Physica D 70: 370–382. DOI:10.1016/0167-2789(94)90072-8 <a href="https://dx.doi.org/10.1016/0167-2789(94)90072-8" target="_blank">https://dx.doi.org/10.1016/0167-2789(94)90072-8</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">S.V. Petrovskii, H. Malchow (1999) "A minimal model of pattern formation in a prey–predator system", Math. Comp. Modelling 29: 49–63. DOI:10.1016/S0895-7177(99)00070-9 <a href="https://dx.doi.org/10.1016/S0895-7177(99)00070-9" target="_blank">https://dx.doi.org/10.1016/S0895-7177(99)00070-9</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">E. Ranta, V. Kaitala (1997) "Travelling waves in vole population dynamics", Nature 390: 456. DOI:10.1038/37261 <a href="http://www.nature.com/nature/journal/v390/n6659/pdf/390456a0.pdf" target="_blank">http://www.nature.com/nature/journal/v390/n6659/pdf/390456a0.pdf</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">X. Lambin, D. A. Elston, S. J. Petty, J. L. MacKinnon (1998) "Spatial asynchrony and periodic travelling waves in cyclic populations of field voles", Proc. R. Soc. Lond. B 265: 1491–1496. DOI:10.1098/rspb.1998.0462 <a href="http://rspb.royalsocietypublishing.org/content/265/1405/1491.short" target="_blank">http://rspb.royalsocietypublishing.org/content/265/1405/1491.short</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">J.D.M. Rademacher, B. Sandstede, A. Scheel (2007) "Computing absolute and essential spectra using continuation", Physica D 229: 166–183. DOI:10.1016/j.physd.2007.03.016 <a href="/wiki/Bj%C3%B6rn_Sandstede" target="_blank">/wiki/Bj%C3%B6rn_Sandstede</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">M. J. Smith, J. D. M. Rademacher, J. A. Sherratt (2009) "Absolute stability of wavetrains can explain spatiotemporal dynamics in reaction-diffusion systems of lambda-omega type", SIAM J. Appl. Dyn. Systems 8: 1136–1159. DOI:10.1137/090747865 <a href="https://dx.doi.org/10.1137/090747865" target="_blank">https://dx.doi.org/10.1137/090747865</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">N. Kopell, L.N. Howard (1973) "Plane wave solutions to reaction–diffusion equations", Stud. Appl. Math. 52: 291–328. <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">N. Kopell, L.N. Howard (1973) "Plane wave solutions to reaction–diffusion equations", Stud. Appl. Math. 52: 291–328. <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">K. Maginu (1981) "Stability of periodic travelling wave solutions with large spatial periods in reaction-diffusion systems", J. Diff. Eqns. 39: 73–99.
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<li id="fn:21">M. J. Smith, J.A. Sherratt (2007) "The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems", Physica D 236: 90–103. DOI:10.1016/j.physd.2007.07.013 <a href="https://dx.doi.org/10.1016/j.physd.2007.07.013" target="_blank">https://dx.doi.org/10.1016/j.physd.2007.07.013</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">B. Sandstede, A. Scheel (2000) "Absolute and convective instabilities of waves on unbounded and large bounded domains", Physica D 145: 233–277.
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<li id="fn:23">I. S. Aranson, L. Kramer (2002) "The world of the complex Ginzburg–Landau equation", Rev. Mod. Phys. 74: 99–143. DOI:10.1103/RevModPhys.74.99 <a href="http://link.aps.org/doi/10.1103/RevModPhys.74.99" target="_blank">http://link.aps.org/doi/10.1103/RevModPhys.74.99</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">J.D.M. Rademacher, B. Sandstede, A. Scheel (2007) "Computing absolute and essential spectra using continuation", Physica D 229: 166–183. DOI:10.1016/j.physd.2007.03.016 <a href="/wiki/Bj%C3%B6rn_Sandstede" target="_blank">/wiki/Bj%C3%B6rn_Sandstede</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">M. J. Smith, J. D. M. Rademacher, J. A. Sherratt (2009) "Absolute stability of wavetrains can explain spatiotemporal dynamics in reaction-diffusion systems of lambda-omega type", SIAM J. Appl. Dyn. Systems 8: 1136–1159. DOI:10.1137/090747865 <a href="https://dx.doi.org/10.1137/090747865" target="_blank">https://dx.doi.org/10.1137/090747865</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">A. L. Kay, J. A. Sherratt (2000) "Spatial noise stabilizes periodic wave patterns
in oscillatory systems on finite domains", SIAM J. Appl. Math. 61: 1013–1041.
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<li id="fn:27">D. M. Johnson, O. N. Bjornstad, A.M. Liebhold (2006) "Landscape mosaic induces travelling waves of insect outbreaks", Oecologia 148: 51–60.
DOI:10.1007/s00442-005-0349-0 <a href="https://link.springer.com/article/10.1007%2Fs00442-005-0349-0?LI=true" target="_blank">https://link.springer.com/article/10.1007%2Fs00442-005-0349-0?LI=true</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">J. A. Sherratt (1994) "Irregular
wakes in reaction-diffusion waves", Physica D 70: 370–382. DOI:10.1016/0167-2789(94)90072-8 <a href="https://dx.doi.org/10.1016/0167-2789(94)90072-8" target="_blank">https://dx.doi.org/10.1016/0167-2789(94)90072-8</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">S.V. Petrovskii, H. Malchow (1999) "A minimal model of pattern formation in a prey–predator system", Math. Comp. Modelling 29: 49–63. DOI:10.1016/S0895-7177(99)00070-9 <a href="https://dx.doi.org/10.1016/S0895-7177(99)00070-9" target="_blank">https://dx.doi.org/10.1016/S0895-7177(99)00070-9</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">K. Nozaki, N. Bekki (1983) "Pattern selection and spatiotemporal transition to chaos in the Ginzburg–Landau equation", Phys. Rev. Lett. 51: 2171-2174. DOI:10.1103/PhysRevLett.51.2171 <a href="http://link.aps.org/doi/10.1103/PhysRevLett.51.2171" target="_blank">http://link.aps.org/doi/10.1103/PhysRevLett.51.2171</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">A. Tsameret, V. Steinberg (1994) "Competing states in a Couette–Taylor system with an axial flow", Phys. Rev. E 49: 4077-4086. DOI:10.1103/PhysRevE.49.4077 <a href="http://link.aps.org/doi/10.1103/PhysRevE.49.4077" target="_blank">http://link.aps.org/doi/10.1103/PhysRevE.49.4077</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">M. Ipsen, L. Kramer, P.G. Sorensen (2000) "Amplitude equations for description of chemical reaction–diffusion systems", Phys. Rep. 337: 193–235.
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<li id="fn:33">A.S. Mikhailov,
K. Showalter (2006) "Control of waves, patterns and turbulence in chemical systems", Phys. Rep. 425: 79–194. DOI:10.1016/j.physrep.2005.11.003 <a href="https://dx.doi.org/10.1016/j.physrep.2005.11.003" target="_blank">https://dx.doi.org/10.1016/j.physrep.2005.11.003</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">J.A. Sherratt, M. A. Lewis, A. C. Fowler (1995) "Ecological chaos in the wake of invasion", Proc. Natl. Acad. Sci. USA 92: 2524–2528.
10.1073/pnas.92.7.2524 <a href="http://www.pnas.org/content/92/7/2524.short" target="_blank">http://www.pnas.org/content/92/7/2524.short</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">S.V. Petrovskii, H. Malchow (2001) "Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics", Theor. Pop. Biol. 59: 157–174.
DOI:10.1006/tpbi.2000.1509 <a href="https://dx.doi.org/10.1006/tpbi.2000.1509" target="_blank">https://dx.doi.org/10.1006/tpbi.2000.1509</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">J. A. Sherratt, X. Lambin, C. J. Thomas, T. N. Sherratt (2002) "Generation of periodic waves by landscape features in cyclic
predator-prey systems" Proc. R. Soc. Lond. B 269: 327–334.
DOI:10.1098/rspb.2001.1890 <a href="http://rspb.royalsocietypublishing.org/content/269/1489/327.short" target="_blank">http://rspb.royalsocietypublishing.org/content/269/1489/327.short</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">M. Sieber, H. Malchow, S.V. Petrovskii (2010) "Noise-induced suppression of periodic travelling waves in oscillatory reaction–diffusion systems", Proc. R. Soc. Lond. A 466: 1903–1917.
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<li id="fn:38">J.A. Sherratt (2008) "A comparison of periodic travelling wave generation by Robin and Dirichlet boundary conditions in oscillatory reaction-diffusion equations". IMA J. Appl. Math. 73: 759-781.
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<li id="fn:40">M. R. Garvie, M. Golinski (2010) "Metapopulation dynamics for spatially extended predator–prey interactions", Ecological Complexity 7: 55–59.
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<li id="fn:41">J. A. Sherratt (1994) "Irregular
wakes in reaction-diffusion waves", Physica D 70: 370–382. DOI:10.1016/0167-2789(94)90072-8 <a href="https://dx.doi.org/10.1016/0167-2789(94)90072-8" target="_blank">https://dx.doi.org/10.1016/0167-2789(94)90072-8</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">S.V. Petrovskii, H. Malchow (2001) "Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics", Theor. Pop. Biol. 59: 157–174.
DOI:10.1006/tpbi.2000.1509 <a href="https://dx.doi.org/10.1006/tpbi.2000.1509" target="_blank">https://dx.doi.org/10.1006/tpbi.2000.1509</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
<li id="fn:43">N. Kopell, L.N. Howard (1973) "Plane wave solutions to reaction–diffusion equations", Stud. Appl. Math. 52: 291–328. <a href="#fnref:43" class="footnote-back-ref">↩</a></li>
<li id="fn:44">I. S. Aranson, L. Kramer (2002) "The world of the complex Ginzburg–Landau equation", Rev. Mod. Phys. 74: 99–143. DOI:10.1103/RevModPhys.74.99 <a href="http://link.aps.org/doi/10.1103/RevModPhys.74.99" target="_blank">http://link.aps.org/doi/10.1103/RevModPhys.74.99</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></li>
<li id="fn:45">N. Kopell, L.N. Howard (1973) "Plane wave solutions to reaction–diffusion equations", Stud. Appl. Math. 52: 291–328. <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
<li id="fn:46">I. S. Aranson, L. Kramer (2002) "The world of the complex Ginzburg–Landau equation", Rev. Mod. Phys. 74: 99–143. DOI:10.1103/RevModPhys.74.99 <a href="http://link.aps.org/doi/10.1103/RevModPhys.74.99" target="_blank">http://link.aps.org/doi/10.1103/RevModPhys.74.99</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></li>
<li id="fn:47">J.D.M. Rademacher, B. Sandstede, A. Scheel (2007) "Computing absolute and essential spectra using continuation", Physica D 229: 166–183. DOI:10.1016/j.physd.2007.03.016 <a href="/wiki/Bj%C3%B6rn_Sandstede" target="_blank">/wiki/Bj%C3%B6rn_Sandstede</a> <a href="#fnref:47" class="footnote-back-ref">↩</a></li>
<li id="fn:48">M. J. Smith, J. D. M. Rademacher, J. A. Sherratt (2009) "Absolute stability of wavetrains can explain spatiotemporal dynamics in reaction-diffusion systems of lambda-omega type", SIAM J. Appl. Dyn. Systems 8: 1136–1159. DOI:10.1137/090747865 <a href="https://dx.doi.org/10.1137/090747865" target="_blank">https://dx.doi.org/10.1137/090747865</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></li>
<li id="fn:49">K. Nozaki, N. Bekki (1983) "Pattern selection and spatiotemporal transition to chaos in the Ginzburg–Landau equation", Phys. Rev. Lett. 51: 2171-2174. DOI:10.1103/PhysRevLett.51.2171 <a href="http://link.aps.org/doi/10.1103/PhysRevLett.51.2171" target="_blank">http://link.aps.org/doi/10.1103/PhysRevLett.51.2171</a> <a href="#fnref:49" class="footnote-back-ref">↩</a></li>
<li id="fn:50">J. A. Sherratt (1994) "On the evolution of periodic plane waves in reaction-diffusion equations of λ-ω type", SIAM J. Appl. Math. 54: 1374–1385. DOI: 10.1137/S0036139993243746 <a href="https://dx.doi.org/10.1137/S0036139993243746" target="_blank">https://dx.doi.org/10.1137/S0036139993243746</a> <a href="#fnref:50" class="footnote-back-ref">↩</a></li>
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<li id="fn:52">J. A. Sherratt (2003)"Periodic travelling wave selection by Dirichlet boundary conditions in oscillatory reaction-diffusion systems", SIAM J. Appl. Math. 63: 1520–1538. DOI:10.1137/S0036139902392483 <a href="https://dx.doi.org/10.1137/S0036139902392483" target="_blank">https://dx.doi.org/10.1137/S0036139902392483</a> <a href="#fnref:52" class="footnote-back-ref">↩</a></li>
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<li id="fn:58">J.D.M. Rademacher, B. Sandstede, A. Scheel (2007) "Computing absolute and essential spectra using continuation", Physica D 229: 166–183. DOI:10.1016/j.physd.2007.03.016 <a href="/wiki/Bj%C3%B6rn_Sandstede" target="_blank">/wiki/Bj%C3%B6rn_Sandstede</a> <a href="#fnref:58" class="footnote-back-ref">↩</a></li>
<li id="fn:59">J. A. Sherratt (2013) "Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential equations", Adv. Comput. Math, in press. DOI:10.1007/s10444-012-9273-0 <a href="https://link.springer.com/article/10.1007/s10444-012-9273-0" target="_blank">https://link.springer.com/article/10.1007/s10444-012-9273-0</a> <a href="#fnref:59" class="footnote-back-ref">↩</a></li>
<li id="fn:60">J.A. Sherratt (2012) "Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations",
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</ol>

Periodic travelling wave open-in-new

Periodic travelling wave