Classification of unstable travelling wave solutions to KdV type equations
Kaito Kokubu
TL;DR
This work classifies instability phenomena for traveling-wave solutions of KdV-type equations with fractional dispersion $D_x^{\sigma}$ ($1\le\sigma\le2$) and double-power nonlinearities by linking instability to the sign of the second λ-derivative of the action $S_c$ at scaled ground states. Using modulation theory, a virial-type functional $J_A$, and Nehari-based variational characterizations, the authors derive a sufficient condition $\partial_{\lambda}^{2} S_c(\phi_c^{\lambda})|_{\lambda=1} < 0$ that implies instability, and then analyze ground-state scaling limits to identify parameter regimes where this condition holds. They classify instability across parity/signature cases (II-1, II-2), illustrating with concrete examples and showing that instability can occur for all speeds or only above a critical speed depending on $(\sigma,p,q)$. Additionally, they prove decay estimates for ground states with polynomial decay when $1\le\sigma<2$ (and exponential decay in the classical case $\sigma=2$), using Green’s function representations and contraction mapping arguments to control higher derivatives. The results extend prior stability analyses to a broader class of Gardner-type equations and provide a comprehensive framework for predicting instability in fractional-dispersion gKdV systems.
Abstract
We study travelling wave solutions to Korteweg--de Vries type equations which have double power nonlinearities with integer indices, such as the Gardner equation, and fractional dispersion. Whether these equations have ground state solutions depends on signatures of nonlinearities and parity combinations of the two indices. The aim of this study is to give the classification of phenomena of travelling wave solutions from the perspective of the signatures and parities of the indices. In this paper, we focus on unstable travelling wave solutions.