A Coupling Problem for Entire Functions and Its Application to the Long-time Asymptotics of Integrable Wave Equations

We propose a novel technique for analyzing the long-time asymptotics of integrable wave equations in the case where 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 approach, which does not work for these kind of problems. As a prototypical example we investigate the long-time asymptotics of the dispersionless Camassa–Holm equation improving the currently known results.