Characterize the kernel of I + \hat f K and prove it is spanned by the translational mode

Prove that, for a smooth traveling-wave solution \hat f of the nonlinear integral equation w + u_* K w + (1/2) K(w^2) = 0 (with Kν(x) = ∫_{x-1}^{x+1} ν(y) dy), the generalized eigenspace Z_0 of the linear operator I + \hat f K acting on L^2(Ω) (Ω = (-L, L) or Ω = ℝ) is one-dimensional and spanned by the translational mode \hat f(x) \hat f'(x). That is, characterize the kernel structure of I + \hat f K and establish that no other generalized eigenfunctions exist beyond the one associated with translation invariance.

Background

Because the traveling-wave NIE is translation invariant, the operator I + \hat f K has at least one kernel element related to translations of the wave profile. The authors’ second-variation analysis suggests that the kernel should consist solely of this translational mode.

A definitive characterization of the kernel is crucial for establishing strict coercivity of the second variation away from the neutral direction and for understanding stability and uniqueness properties of the dual maximization. The authors believe the kernel is one-dimensional but currently have no proof.