Periodic travelling wave

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, excitable systems and reaction–diffusion–advection systems. 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, integrodifference equations, coupled map lattices and cellular automata 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. 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, and by their detection in ecology, in spatiotemporal data sets on cyclic populations. Since the mid-2000s, research on periodic travelling waves has benefitted from new computational methods for studying their stability and absolute stability. The existence of periodic travelling waves usually depends on the parameter values in a mathematical equation.

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