In mathematics, a periodic travelling wave (or wavetrain) is a periodic function of one-dimensional space that moves with constant speed. Consequently, it is a special type of spatiotemporal oscillation that is a periodic function of both space and time.
Periodic travelling waves play a fundamental role in many mathematical equations, including self-oscillatory systems,N. Kopell, L.N. Howard (1973) "Plane wave solutions to reaction–diffusion equations", Stud. Appl. Math. 52: 291–328.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 excitable systemsS. 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 and reaction–diffusion–advection systems.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 Equations of these types are widely used as mathematical models of biology, chemistry and physics, and many examples in phenomena resembling periodic travelling waves have been found empirically.
The mathematical theory of periodic travelling waves is most fully developed for partial differential equations, but these solutions also occur in a number of other types of mathematical system, including integrodifferential equations,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.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 integrodifference equations,M. Kot (1992) "Discrete-time travelling waves: ecological examples", J. Math. Biol. 30: 413-436. DOI:10.1007/BF00173295 coupled map latticesM. 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 and cellular automata.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-XM. 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
As well as being important in their own right, periodic travelling waves are significant as the one-dimensional equivalent of spiral waves and target patterns in two-dimensional space, and of scroll waves in three-dimensional space.
History of research
While periodic travelling waves have been known as solutions of the wave equation since the 18th century, their study in nonlinear systems began in the 1970s. A key early research paper was that of Nancy Kopell and Lou Howard which proved several fundamental results on periodic travelling waves in reaction–diffusion equations. 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,J. A. Sherratt (1994) "Irregular wakes in reaction-diffusion waves", Physica D 70: 370–382. DOI:10.1016/0167-2789(94)90072-8S.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 and by their detection in ecology, in spatiotemporal data sets on cyclic populations.E. Ranta, V. Kaitala (1997) "Travelling waves in vole population dynamics", Nature 390: 456. DOI:10.1038/37261X. 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 Since the mid-2000s, research on periodic travelling waves has benefitted from new computational methods for studying their stability and absolute stability.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.2007.03.016M. 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
Families
The existence of periodic travelling waves usually depends on the parameter 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.
Stability
An important question is whether a periodic travelling wave is stable or unstable as a solution of the original mathematical system. For partial differential equations, it is typical that the wave family subdivides into stable and unstable parts.K. Maginu (1981) "Stability of periodic travelling wave solutions with large spatial periods in reaction-diffusion systems", J. Diff. Eqns. 39: 73–99. 10.1016/0022-0396(81)90084-XM. 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 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.B. Sandstede, A. Scheel (2000) "Absolute and convective instabilities of waves on unbounded and large bounded domains", Physica D 145: 233–277. DOI:10.1016/S0167-2789(00)00114-7 This issue has only been resolved for a few partial differential equations.
Generation
A number of mechanisms of periodic travelling wave generation are now well established. These include: Heterogeneity: spatial noise in parameter values can generate a series of bands of periodic travelling waves.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. DOI:10.1137/S0036139999360696 This is important in applications to oscillatory chemical reactions, 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.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 Invasions, which can leave a periodic travelling wave in their wake.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 This is important in the Taylor–Couette system in the presence of through flow,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 in chemical systems such as the Belousov–Zhabotinsky reactionM. Ipsen, L. Kramer, P.G. Sorensen (2000) "Amplitude equations for description of chemical reaction–diffusion systems", Phys. Rep. 337: 193–235. DOI:10.1016/S0370-1573(00)00062-4A.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 and in predator-prey systems in ecology.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.92.7.2524S.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 Domain boundaries with Dirichlet or Robin boundary conditions.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.2001.1890M. 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. DOI:10.1098/rspa.2009.0611J.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. This is potentially important in ecology, where Robin or Dirichlet conditions correspond to a boundary between habitat and a surrounding hostile environment. However definitive empirical evidence on the cause of waves is hard to obtain for ecological systems. Migration driven by pursuit and evasion.V. N. Biktashev, M. A. Tsyganov (2009) "Spontaneous traveling waves in oscillatory systems with cross diffusion", Phys. Rev. E 80: art. no. 056111. DOI:10.1103/PhysRevE.80.056111 This may be significant in ecology. Migration between sub-populations,M. R. Garvie, M. Golinski (2010) "Metapopulation dynamics for spatially extended predator–prey interactions", Ecological Complexity 7: 55–59. DOI:10.1016/j.ecocom.2009.05.001 which again has potential ecological significance.
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.
Spatiotemporal chaos
It is common that for some parameter values, the periodic travelling waves arising from a wave generation mechanism are unstable. In such cases the solution usually evolves to spatiotemporal chaos. Thus the solution involves a spatiotemporal transition to chaos via the periodic travelling wave.
Lambda–omega systems and the complex Ginzburg–Landau equation
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 reaction–diffusion equations
() and the complex Ginzburg–Landau equation.
(A is complex-valued). Note that these systems are the same if , and . 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 space and time. Therefore, u and v, or Re(A) and Im(A), are sinusoidal 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 stability of the periodic travelling waves can be found, and the condition for absolute stability can be reduced to the solution of a simple polynomial. Also exact solutions have been obtained for the selection problem for waves generated by invasionsJ. 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 and by zero Dirichlet boundary conditions.N. Bekki, K. Nozaki (1985) "Formations of spatial patterns and holes in the generalized Ginzburg–Landau equation", Phys. Lett. A 110: 133–135.