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