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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.〕〔name="AransonKramer2002">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 systems〔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 )〕 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. ( DOI:10.1093/imamat/51.3.299 )〕〔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 lattices〔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 )〕 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-X )〕〔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 )〕 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 on periodic travelling waves == Periodic travelling waves were first studied 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)90072-8 )〕〔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 )〕 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:10.1038/37261 )〕〔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 )〕 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.2007.03.016 )〕〔name="Smithetal2009">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 )〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Periodic travelling wave」の詳細全文を読む スポンサード リンク
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