  We propose a novel technique for analyzing the long-time asymptotics of
integrable wave equations in the case when the underlying isospectral problem
has purely discrete spectrum. To this end, we introduce a natural coupling
problem for entire functions, which serves as a replacement for the usual
Riemann-Hilbert problem, which does not apply in these cases. As a prototypical
example, we investigate the long-time asymptotics of the dispersionless
Camassa-Holm equation.
