D.E. Pelinovsky

Spectral stability of nonlinear waves in KdV-type evolution equations

"Nonlinear Physical Systems: Spectral Analysis, Stability, and Bifurcations" (Edited by O.N. Kirillov and D.E. Pelinovsky) (Wiley-ISTE, NJ) 377-400 (2014)

Abstract:
This paper concerns spectral stability of nonlinear waves in 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 symplectic operator. 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 other recent results on spectral stability of nonlinear waves in KdV-type evolution equations.

Keywords:
KdV-type equations, solitary waves, periodic waves, spectral stability, instability index count.