Periodic travelling wave

Consequently, it is a special type of spatiotemporal oscillation that is a periodic function of both space and time.

[4] 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.

A key early research paper was that of Nancy Kopell and Lou Howard[1] which proved several fundamental results on periodic travelling waves in reaction–diffusion equations.

[15][16] The existence of periodic travelling waves usually depends on the parameter values in a mathematical equation.

[1] 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.

These include: In all of these cases, a key question is which member of the periodic travelling wave family is selected.

[11][27] Thus the solution involves a spatiotemporal transition to chaos via the periodic travelling wave.

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.

These exact solutions for the periodic travelling wave families enable a great deal of further analytical study.

[15][16] Also exact solutions have been obtained for the selection problem for waves generated by invasions[22][33] and by zero Dirichlet boundary conditions.

For most mathematical equations, analytical calculation of periodic travelling wave solutions is not possible, and therefore it is necessary to perform numerical computations.

Periodic travelling waves correspond to limit cycles of these equations, and this provides the basis for numerical computations.

The standard computational approach is numerical continuation of the travelling wave equations.

This is the starting point for a branch (family) of periodic travelling wave solutions, which one can follow by numerical continuation.

Periodic travelling wave stability can also be calculated numerically, by computing the spectrum.

[15] One advantage of the latter approach is that it can be extended to calculate boundaries in parameter space between stable and unstable waves[40] Software: The free, open-source software package Wavetrain http://www.ma.hw.ac.uk/wavetrain is designed for the numerical study of periodic travelling waves.

Examples of phenomena resembling periodic travelling waves that have been found empirically include the following.