Spectral Stability of Nonlinear Waves in KdV‐Type Evolution Equations

This chapter focuses on the spectral stability of nonlinear waves in Korteweg‐de Vries (KdV) type evolution equations. The relevant eigenvalue problem is defined by the composition of an unbounded self‐adjoint operator with a finite number of negative eigenvalues and an unbounded non‐invertible operator ∂x. The instability index theorem is proven under a generic assumption on the self‐adjoint operator both in the case of solitary waves and periodic waves. This result is reviewed in the context of recent results on spectral stability of nonlinear waves in KdV‐type evolution equations. The chapter contains historical notes devoted to the theorem. The chapter shows how to recover the correct count of eigenvalues for the example of the focusing modified KdV equation. More direct approaches to the stability of solitary waves in Boussinesq systems can be found in recent works.