Many nonlinear PDEs have solutions that become singular in finite time. Examples are the inviscid Burgers equation,
u_t + u u_x = 0, (1)
and the nonlinear heat equation,
u_t - u_xx - u^2 = 0. (2)
Eq. (1) exhibits shocks, eq. (2) a pole-type blow-up. A less familiar example that also displays blow-up is
u_t + (H(u)u)_x - 0.1 u_xx = 0, (3)
where H represents the Hilbert transform (the coefficient of the diffusion term was chosen arbitrarily).
In the last decade or two, it has been recognized that the analytic structure of the solutions u(x+iy,t) in the complex plane may aid in the understanding of the formation of singularities.
This paper proposes a two-step method for the computation of the singularities of PDEs such as (1)-(3). The first step is the numerical solution of the PDE up to a point near breakdown. Here we assume periodic solutions in x, so a Fourier spectral method is used. The second step is to continue the numerical solution analytically into the complex plane. For this step the epsilon algorithm is used to sum the Fourier series, which is just an efficient way of computing a certain Pade approximation.
We have tested this strategy on the model equations (1)-(3). Explicit solutions of equations (1) and (3) were available, which we could use for comparison. We know of no explicit solution of (2) that is periodic in x.
Starting with equation (3), the following animation 10frame (1.1MB)/ 20frame (2.3MB) shows the mechanism of blow-up. The actual solution is shown on the left. In the middle figure this solution is shown in the complex plane (upper half). The red dot represents the location of a pole as it moves down towards the real axis. Owing to symmetry there is a conjugate pole in the lower half-plane (not shown here). When these two poles coalesce on the real axis, the blow-up occurs. Both the first and second figures in this animation were computed using the theoretical solution that was at our disposal. The figure on the right, however, has been numerically computed using the procedures suggested in this paper. It approximates the middle figure, and evidently the two figures are hardly distinguishable.
Next we look at equation (1). With a certain (complex) initial condition the solution to (1) can be computed explicitly. Using this solution one finds that |u(x,t)| displays a cusp-like singularity, as shown in the left figure of the following animation 10frame (1.0MB)/ 20frame (2.1MB). Here the mechanism of blow-up is not poles moving onto the real axis, but rather a branch point singularity. This can be seen in the middle figure of the animation, which shows the theoretical solution analytically continued into the lower half-plane. (In the upper half-plane the solution is analytic and not interesting, and therefore not displayed here.) The needle-like structure at x=pi/2 represents the branch cut that moves upwards towards the real axis. When it hits, the cusp singularity is formed. The third figure in the animation shows the numerical approximation to the middle figure, as computed by the procedures suggested in this paper. The branch cut is approximated by a string of poles, which is typical behavior for Pade approximants.
Finally we turn to equation (2). Like equation (3) it exhibits blow-up in finite time, except that the peak points down (rather than up) and the rate of blow-up is more violent. This can be seen in the numerical solution shown on the left in the following animation 10frame (0.7MB)/ 20frame (1.5MB). The figure on the right is the analytical continuation of this numerical solution, as computed by the procedures suggested in this paper. The animation suggests that as in the case of equation (3), the mechanism of blow-up is a conjugate pair of poles that coalesce on the real axis. The speed of approach to the real axis is, however, not constant. Initially the poles are at infinity (the initial condition is an entire function). They then move in very quickly from infinity to z = pi*i and z = -pi*i, where they hover for a while before zooming onto the real axis at great speed. Here is a graph (169KB) showing the imaginary part of the computed pole as a function of time. We do not know if this peculiar behavior of the poles of equation (2) has been noted before.