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2006 Stability of cnoidal waves
Jaime Angulo Pava, Jerry L. Bona, Marcia Scialom
Adv. Differential Equations 11(12): 1321-1374 (2006).


This paper is concerned with the stability of periodic travelling-wave solutions of the Korteweg-de Vries equation $$ u_t+uu_x+u_{xxx}=0. $$ Here, $u$ is a real-valued function of the two variables $x,t\in{\mathbb R}$ and subscripts connote partial differentiation. These special solutions were termed cnoidal waves by Korteweg and de Vries. They also appear in earlier work of Boussinesq. It is shown that these solutions are stable to small, periodic perturbations in the context of the initial-value problem. The approach is that of the modern theory of stability of solitary waves, but adapted to the periodic context. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.


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Jaime Angulo Pava. Jerry L. Bona. Marcia Scialom. "Stability of cnoidal waves." Adv. Differential Equations 11 (12) 1321 - 1374, 2006.


Published: 2006
First available in Project Euclid: 18 December 2012

zbMATH: 1147.35079
MathSciNet: MR2276856

Primary: 35Q53
Secondary: 35B10, 35B35, 35Q51, 37K15, 37K40, 76B25, 76E99

Rights: Copyright © 2006 Khayyam Publishing, Inc.


Vol.11 • No. 12 • 2006
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