Rogue wave modes for the long wave-short wave resonance model

The long wave-short wave resonance model arises physically when the phase velocity of a long wave matches the group velocity of a short wave. It is a system of nonlinear evolution equations solvable by the Hirota bilinear method and also possesses a Lax pair formulation. "Rogue wave" modes, algebraically localized entities in both space and time, are constructed from the breathers by a singular limit involving a "coalescence" of wavenumbers in the long wave regime. In contrast with the extensively studied nonlinear Schrödinger case, the frequency of the breather cannot be real and must satisfy a cubic equation with complex coefficients. The same limiting procedure applied to the finite wavenumber regime will yield mixed exponential-algebraic solitary waves, similar to the classical "double pole" solutions of other evolution systems.