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.
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, 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.
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. 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. 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. 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. Invasions, which can leave a periodic travelling wave in their wake. This is important in the Taylor–Couette system in the presence of through flow, in chemical systems such as the Belousov–Zhabotinsky reaction and in predator-prey systems in ecology. Domain boundaries with Dirichlet or Robin boundary conditions. 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. This may be significant in ecology. Migration between sub-populations, 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
∂ 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}
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