A generalized long-wave limit method with spectral perturbations

Abstract

We develop a generalized long-wave limit method for constructing higher-order (degenerate) lump solutions of $(2+1)$-dimensional integrable systems. Starting from the multi-soliton tau function, we introduce wavenumber rescalings, spectral perturbations, and higher-order phase modulations in a single small parameter, and choose their coefficients so that the lower-order Taylor coefficients vanish. Applying this framework to the Kadomtsev--Petviashvili I equation, we explicitly derive second-order two- and three-lump solutions as well as third-order three-, four-, five-, and six-lump solutions. The long-time asymptotics of the peak positions and amplitudes describe anomalous scattering of these degenerate lump solutions. Extensions to hybrid higher-order lump solutions associated with multiple distinct spectral parameters are also discussed.

arXiv commentAdded new results in Sections 4.2 and 4.3; Updated pictures