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.
History of research.
While periodic travelling waves have been know 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: 
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
("r"=("u"2+"v"2)1/2) and the complex Ginzburg–Landau equation.
("A" is complex-valued). Note that these systems are the same if λ("r")=1-"r"2, ω("r")=-"c" "r"2 and "b"=0. 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 invasions 
and by zero Dirichlet boundary conditions. 
In the latter case, for the complex Ginzburg–Landau equation, the overall solution is a stationary Nozaki-Bekki hole.
Much of the work on periodic travelling waves in the complex Ginzburg–Landau equation is in the physics literature, where they are usually known as plane waves.
Numerical computation of periodic travelling waves and their stability.
For most mathematical equations, analytical calculation of periodic travelling wave solutions is not possible, and therefore it is necessary to perform numerical computations. For partial differential equations, denote by "x" and "t" the (one-dimensional) space and time variables, respectively. Then periodic travelling waves are functions of the travelling wave variable "z"="x"-"c" "t". Substituting this solution form into the partial differential equations gives a system of ordinary differential equations known as the travelling wave equations. 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. One first performs a continuation of a steady state to locate a Hopf bifurcation point. This is the starting point for a branch (family) of periodic travelling wave solutions, which one can follow by numerical continuation. In some (unusual) cases both end points of a branch (family) of periodic travelling wave solutions are homoclinic solutions, in which case one must use an external starting point, such as a numerical solution of the partial differential equations.
Periodic travelling wave stability can also be calculated numerically, by computing the spectrum. This is made easier by the fact that the spectrum of periodic travelling wave solutions of partial differential equations consists entirely of essential spectrum. 
Possible numerical approaches include Hill's method and numerical continuation of the spectrum. One advantage of the latter approach is that it can be extended to calculate boundaries in parameter space between stable and unstable waves
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. 
Using numerical continuation, Wavetrain is able to calculate the form and stability of periodic travelling wave solutions of partial differential equations, and the regions of parameter space in which waves exist and in which they are stable.
Applications.
Examples of phenomena resembling periodic travelling waves that have been found empirically include the following.