Q2. What is the spectral parameter for the KdV hierarchy?
Since K2 is computed for the same set of eigenfunctions as K1, the authors may think of K2 as being parameterized by the Lax spectral parameter ζ in the same way that K1 is.
Q3. What is the integrand signature of the imaginary axis?
The semiinfinite component of σL gives rise to a single covering of the imaginary axis, while an additional double covering of a symmetric interval on the imaginary axis around the origin corresponds to ζ ∈ [2k2 − 1, k2].
Q4. How did Benjamin prove that the cnoidal waves were indeed stable?
By using the integrable structure associated with KdV, the results of [6] on the nature of the spectrum for the linearized operator, and the Lyapunov function construction ideas presented in [20], the authors are able to show that these energetically unstable waves are indeed orbitally stable.
Q5. What is the spectral stability of the cnoidal wave?
A separation of variables v(y, τ) = eλτV (y) yields the spectral problemλV = ∂yLV, L = −∂2y + c− U. (2.5)Spectral stability of the cnoidal wave (2.3) with respect to perturbations that are bounded on the whole line is established by demonstrating that the spectrum σ(∂yL) of the operator ∂yL does not intersect the right-half complex λ plane.
Q6. What is the sign of K1 for all bounded V?
The Krein signature is defined as the sign ofK1 = 〈V,LV 〉n := ∫ nK(k) −nK(k) V ∗LV dy, (3.2)where n ∈ N and V is a solution of the spectral problem (2.5).
Q7. What is the spectral stability of the cnoidal waves?
The spectral and linear stability of the cnoidal waves were the subject of [6], and their orbital stability with respect to harmonic (same period) perturbations was covered in [21] (using integrability) and [3] (not using integrability).
Q8. What is the reason why the authors can construct a Lyapunov functional when it is clear?
The fact that the authors are able to construct a Lyapunov functional when it is clear the energy cannot play this role, is a consequence of the integrability of the KdV equation, which provides us with an infinite number of candidate Lyapunov functionals.
Q9. What is the reason why the Krein signature is not needed to be repeated?
since the eigenfunctions of the linear stability problems for the τ and τ2 flows are identical, the Krein signature calculation necessary to invoke Theorem 1 of [14] does not need to be repeated as it remains the same.
Q10. What is the cnoidal wave solution of KdV?
For instance,H2(u) = ∫ nK(k) −nK(k) ( 1 2 u2yy − 5 6 uu2y + 5 72 u4 + c21 ( 1 2 u2y − 1 6 u3 ) + c20 1 2 u2 ) dy. (3.10)Since all the flows in the KdV hierarchy commute, the cnoidal wave solution (2.3) of KdV (2.2) is a stationary (with respect to τ2) solution of the second KdV equation for a suitable choice of the constants c21 and c20.
Q11. Why was the question of the orbital stability of the cnoidal waves open until now?
Up until now the question of the orbital stability of these waves with respect to subharmonic perturbations was open, primarily because in this case the wave is not a local minimizer of a constrained energy.
Q12. How can the authors determine the spectrum of the Lax problem?
By restricting the Lax pair (2.6-2.7) to the stationary solution u(y, τ) = U(y), the spectrum σL of the Lax problem (2.6) can be determined explicitly.
Q13. What is the Krein signature for the cnoidal waves?
This implies that all perturbations that are not mere shifts of the cnoidal wave in L2per([−nK(k), nK(k)]; R) give rise to an increase in the value of H2(u).