S. Lafortune and D.E. Pelinovsky
Spectral instability of peakons in the b-family of the Camassa-Holm equations
SIAM J. Math. Anal. 54 (2022) 4572-4590
Abstract:
We prove spectral instability of peakons in the b-family of Camassa-Holm equations
in L2(R) that includes the integrable cases of b = 2 and b = 3. We start with a
linearized operator defined on functions in H1(R) intersecting with
W1,&infty;(R) and extend it to a linearized
operator defined on weaker functions in L2(R). For b different from 5/2, the spectrum of the linearized
operator in L2(R) is proved to cover a closed vertical strip of the complex plane. For b = 5/2,
the strip shrinks to the imaginary axis, but an additional pair of real eigenvalues exists
due to projections to the peakon and its spatial translation. The spectral instability results
agree with the linear instability results in the case of the Camassa-Holm equation for b = 2.
Keywords:
Camassa-Holm equation; peakons; spectral instability; linearized instability.