Stability of travelling waves to Korteweg--de Vries type equations with fractional dispersion
Kaito Kokubu
TL;DR
The paper studies the stability of travelling waves for Korteweg–de Vries type equations with fractional dispersion $D_x^{\sigma}$ ($1\leq \sigma\leq 2$) and double-power nonlinearities, focusing on ground-state travelling waves arising from stationary problems $D_x^{\sigma}\phi + c\phi - a\phi^{p} - \phi^{q}=0$ with $2\le p<q$ and $a\in\{\pm1\}$. It builds a variational-analytic framework using energy $E$, mass $M$, and action $S_c=E+cM$, and characterizes ground states as minimizers on the Nehari constraint $K_c(v)=0$ with existence depending on parity of $p,q$ and the dispersion strength; ground states can be chosen even and, in most cases, strictly positive. Stability is addressed via the Grillakis–Shatah–Strauss theory: a spectral coercivity condition $\langle S_c''(\phi_c)v,v\rangle \ge C_1\|v\|^2$ on the orthogonal subspace (to $\phi_c$ and its translation) ensures orbital stability, and the paper proves this condition in regimes specific to the sign pattern (cases I, II-1, II-2) and large/small wave speeds $c$. Additionally, the work proves convergence results: scaled ground states converge to limiting fractional or classical ground states $\psi_{1,p}$, $\psi_{1,q}$ or $\chi_{1,q}$ as $c\to0$ or $\infty$, providing a coherent picture of the stability landscape and a classification of travelling-wave stability phenomena for these nonlocal Gardner-type equations.
Abstract
We study stability of travelling wave solutions to Korteweg--de Vries type equations which has the fractional dispersion and integer-indices double power nonlinearities. It may depend on parity combinations of the two indices and the strength of dispersion whether these equations have a ground state solution. Therefore, we observe the stability phenomena on travelling wave solutions from the perspective of the parities and the dispersion, and we give the classification of phenomena on travelling wave solutions. In this paper, we focus on stable travelling wave solutions.