Exact location of the Lax spectrum for periodic traveling waves of the focusing mKdV equation

Determine the exact location of the Lax spectrum for periodic traveling waves of the focusing modified Korteweg–de Vries equation u_t + 6 u^2 u_x + u_{xxx} = 0 by characterizing all continuous spectral bands of the eigenvalue parameter lambda in the spectral problem psi_x = L(U, lambda) psi, where U(x) is an L-periodic traveling-wave profile, including the precise band endpoints and whether the bands lie on the real axis, the imaginary axis, or in the complex plane.

Background

The paper analyzes spectral stability of periodic traveling waves in the focusing mKdV equation by leveraging the relation between squared eigenfunctions of the AKNS Lax pair and eigenfunctions of the linearized stability problem. The main theorems on stability and instability (for the two waveform families generalizing dnoidal and cnoidal waves) assume specific configurations of the Lax spectrum bands and then map them to stability spectra via Lambda = ±2√P(lambda).

While the symmetry properties of the Lax spectrum are established, the authors emphasize that its exact location is not known and provide extensive numerical computations to illustrate likely configurations, especially highlighting complex bands for cnoidal-type waves. A precise analytical determination of the Lax spectrum would enable a fully rigorous characterization of stability bands and their transformations.