Spectral Stability of Nonlinear Waves in KdV‐Type Evolution Equations

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 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.

A Hamiltonian–Krein (Instability) Index Theory for Solitary Waves to KdV‐Like Eigenvalue Problems

“…REMARK 1. After the completion of this paper we became aware of the result of Pelinovsky [20]. He proves the same result as that given in this paper (Theorem 10); however, his proof is quite different than that presented herein.…”

“…It was recently shown by Frank and Lenzmann [9] that if 0 < p < p max (s), then (20) has an unique (up to translation) ground state solution Q which is positive and bell-shaped, i.e., even and decreasing in (0, ∞). In addition, the solution Q has decay as ξ → +∞; in fact, |Q(ξ )| ≤ C|ξ | −1 (see [9], lemma 5.6]).…”

“…We now discuss abstract conditions that would imply Comparing these results with the corresponding results of [16], theorem 1], we see that regarding the requirements on the multiplier α(ξ ), they are slightly more general than the ones proposed by Lin (see also Proposition 17). Now, using the existence Equation (20) we have that Q p+1 , Q = Q, Q + |∂ ξ | s Q, Q = Q, Q + |∂ ξ | s/2 Q, |∂ ξ | s/2 Q ; thus, we can rewrite the above to say that In particular, the wave is spectrally stable if 1 ≤ s < 2 and 0 < p ≤ 4. If the inequality is reversed the linearized eigenvalue problem has precisely one positive real eigenvalue, and the rest of the spectrum is purely imaginary.…”

New variational characterization of periodic waves in the fractional Korteweg–de Vries equation

“…The existence region of the periodic waves with the zero mean for α near α 0 is unfolded in the new variational characterization. Moreover, spectral stability of periodic waves with respect to perturbations of the same period is obtained from the sharp criterion of monotonicity of the map from the wave speed to the wave momentum similarly to the stability criterion for solitary waves, see [9,26,29,36] for review.…”

Barotropic instability of shear flows

“…In Refs. and , the index formulas were studied for Korteweg‐de Vries (KdV)‐type equations in the whole line, where does not have bounded inverse. Lin and Zeng generalized these results in Ref.…”