High-frequency two-dimensional asymptotic standing coastal trapped waves in nearly integrable case

Abstract

This paper continues the study of explicit asymptotic formulas for standing coastal trapped waves, focusing on the spectral properties of the operator \(\langle \nabla , D(x)\nabla \rangle \), which is the spatial component of the wave operator with a degenerating wave propagation velocity. We aim to construct spectral series—pairs of asymptotic eigenvalues and formal asymptotic eigenfunctions—corresponding to the high-frequency regime, where the eigenvalue is \(\varvec{\omega }\rightarrow \infty \). Extending earlier results, this study addresses the nearly integrable case, providing a more detailed asymptotic behavior of eigenfunctions. Depending on their domain of localization, these eigenfunctions can be expressed in terms of Airy functions and their derivatives or Bessel functions. In addition, we introduce a canonical operator with violated (imprecisely satisfied) quantization conditions.