Spectral stability of periodic traveling waves in Caudrey-Dodd-Gibbon-Sawada-Kotera Equation
Sudhir Singh, Ashish Kumar Pandey, Nitesh Sharma
TL;DR
The paper investigates spectral stability of small-amplitude periodic traveling waves for the one-dimensional CDG–SK equation $u_t+u_{xxxxx}+15(uu_{xx}+u^3)_x=0$. It first proves the existence of such waves via Lyapunov–Schmidt reduction, obtaining explicit small-amplitude expansions $w(z)=a w_1(z)+a^2 w_2(z)+a^3 w_3(z)+O(a^4)$ and $c=c_0+c_2 a^2+O(a^4)$ with $c_0=k^4$, $w_1= ext{cos}z$, $w_2=- rac{15}{2k^2}+ rac{1}{2k^2} ext{cos}(2z)$, $w_3= rac{3}{16k^4} ext{cos}(3z)$, and $c_2=105$. It then analyzes the linearized operator via Floquet–Bloch theory, decomposing the spectrum into a finite-dimensional critical part and the remainder, and shows that for small $a$ all eigenvalues of the reduced $3 imes3$ problem lie on the imaginary axis, hence the waves are spectrally stable to both localized and co-periodic perturbations. This provides a rigorous link between the integrable structure and linear stability for periodic waves in a fifth-order KdV-type system and extends stability insights from Kawahara-type models to the CDG–SK context. The results rely on careful perturbation arguments and an explicit discriminant analysis of the reduced matrix to guarantee purely imaginary spectra in the small-amplitude regime.
Abstract
We study the spectral stability of the one-dimensional small-amplitude periodic traveling wave solutions of the (1+1)-dimensional Caudrey-Dodd-Gibbon-Sawada-Kotera equation. We show that these waves are spectrally stable with respect to co-periodic as well as square integrable perturbations.