Continuous spectra and numerical eigenvalues

1. On numerical spectra for the linearized Burgers’ equation The stability of a traveling wave depends on the spectrum of a differential operator L obtained by linearization about the wave profile. As a simple example, consider Burgers’ equation ut = uxx − 1 2 (u)x, x ∈ R, t ≥ 0, with stationary solution U(x) = − tanh x 2 . Linearization about U(x) leads to the spectral problem Lu ≡ uxx − (Uu)x = su where L : H2(R) → L2(R). (1) In this case, the operator L has the simple eigenvalue s0 = 0 with corresponding eigenfunction u0(x) = U (x). Also, since U(x) → ±1 as x → ∓∞, the operator L has the same continuous spectrum as the operators L+u = uxx + ux and L−u = uxx − ux. Therefore, the continuous spectrum of L is the parabolic line σcont = {s ∈ C | s = −k2 + ik, k ∈ R} (2) obtained by applying L± to u(x) = eikx. Note that L+ and L− both have the same continuous spectrum, σcont , given in (2). Thus, for the operator L in (1) the line (2) should be thought of as double. In the next section we modify Burgers’ equation to break the double line into two distinct parabolas. Doing this in two different ways, will more clearly illustrate the main point of the paper. ∗ Corresponding author. E-mail addresses: [email protected] (O. Guba), [email protected] (J. Lorenz). 1 SandiaNational Laboratories is amulti-program laboratory operated by Sandia Corporation, awholly owned subsidiary of LockheedMartin Corporation, for the US Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. 0895-7177/$ – see front matter© 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.06.037