Continuity and minimization of spectrum related with the two-component Novikov equation

Work of Guoliang Shi and Jun Yan

joint with Lianyuan Duan

In this paper, we study the spectral properties of a third-order measure differential equation associated with the two-component Novikov (Geng–Xue) system. The coefficients are functions of bounded variation, and the spectral problem is non-self-adjoint. Our main contributions include: establishing the continuous dependence of eigenvalues on the coefficients under weak* and total variation topologies; proving the Fréchet differentiability of algebraically simple eigenvalues; and deriving conditions under which the spectrum is nonnegative and algebraically simple using oscillatory kernel theory. Furthermore, we solve extremal problems for the lowest nonzero eigenvalue, obtaining sharp lower bounds and identifying minimizers in terms of Dirac measures.These results provide a theoretical foundation for inverse spectral problems and the study of peakon solutions.

J. Differential Equations

https://doi.org/10.1016/j.jde.2024.08.025