Advances in Differential Equations

Stability of cnoidal waves

Jaime Angulo Pava, Jerry L. Bona, and Marcia Scialom

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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.

Article information

Adv. Differential Equations, Volume 11, Number 12 (2006), 1321-1374.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B10: Periodic solutions 35B35: Stability 35Q51: Soliton-like equations [See also 37K40] 37K15: Integration of completely integrable systems by inverse spectral and scattering methods 37K40: Soliton theory, asymptotic behavior of solutions 76B25: Solitary waves [See also 35C11] 76E99: None of the above, but in this section


Angulo Pava, Jaime; Bona, Jerry L.; Scialom, Marcia. Stability of cnoidal waves. Adv. Differential Equations 11 (2006), no. 12, 1321--1374.

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