Parabolic PDEs.

  For (2.3) the program can : 

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Trace out branches of spatially homogeneous solutions. This amounts to a bifurcation analysis of the algebraic system (2.1). However, AUTO uses a related system instead, in order to enable the detection of bifurcations to wave train solutions of given wave speed. More precisely, bifurcations to wave trains are detected as Hopf bifurcations along fixed point branches of the related ODE

 

where z = x - ct , with the wave speed c specified by the user.
(Demo wav; Run 2.)

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Trace out branches of periodic wave solutions to (2.3) that emanate from a Hopf bifurcation point of Equation 2.4. The wave speed c is fixed along such a branch, but the wave length L, i.e., the period of periodic solutions to (2.4), will normally vary. If the wave length L becomes large, i.e., if a homoclinic orbit of Equation 2.4 is approached, then the wave tends to a solitary wave solution of (2.3).
(Demo wav; Run 3.)

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Trace out branches of waves of fixed wave length L in two parameters. The wave speed c may be chosen as one of these parameters. If L is large then such a continuation gives a branch of approximate solitary wave solutions to (2.3).
(Demo wav; Run 4.)

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Do time evolution calculations for (2.3), given periodic initial data on the interval [0,L]. The initial data must be specified on [0,1] and L must be set separately because of internal scaling. The initial data may be given analytically or obtained from a previous computation of wave trains, solitary waves, or from a previous evolution calculation. Conversely, if an evolution calculation results in a stationary wave then this wave can be used as starting data for a wave continuation calculation.
(Demo wav; Run 5.)

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Do time evolution calculations for (2.3) subject to user-specified boundary conditions. As above, the initial data must be specified on [0,1] and the space interval length L must be specified separately. Time evolution computations of (2.3) are adaptive in space and in time. Discretization in time is not very accurate : only implicit Euler. Indeed, time integration of (2.3) has only been included as a convenience and it is not very efficient. (Demos pd1, pd2.)

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Compute curves of stationary solutions to (2.3) subject to user-specified boundary conditions. The initial data may be given analytically, obtained from a previous stationary solution computation, or from a time evolution calculation.
(Demos pd1, pd2.)

In connection with periodic waves, note that (2.4) is just a special case of (9.1) and that its fixed point analysis is a special case of (2.1). One advantage of the built-in capacity of AUTO to deal with problem (2.3) is that the user need only specify f, D, and c. Another advantage is the compatibility of output data for restart purposes. This allows switching back and forth between evolution calculations and wave computations.