Differential and Integral Equations

Existence of traveling-wave solutions to Boussinesq systems

Min Chen, Nghiem V. Nguyen, and Shu-Ming Sun

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In this manuscript, the existence of traveling-wave solutions to Boussinesq systems \begin{equation*} \left\{ \begin{matrix} \eta_t +u_x+(\eta u)_x +au_{xxx}-b \eta_{xxt}=0, &\\ u_t + \eta_x +uu_x +c\eta_{xxx} - d u_{xxt}=0,&\\ \end{matrix} \right. \end{equation*} is established. We prove that all the systems with $ a<0,$ $ c<0$ and $ b=d$ exhibit traveling-wave solutions with small propagation speeds. The result complements our earlier work [6] on a restricted family of the systems where both existence and stability of traveling-wave solutions were established in the presence of large surface tension, namely when $a+b+c+d<0$.

Article information

Differential Integral Equations, Volume 24, Number 9/10 (2011), 895-908.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35A15: Variational methods 35B35: Stability 35Q35: PDEs in connection with fluid mechanics


Chen, Min; Nguyen, Nghiem V.; Sun, Shu-Ming. Existence of traveling-wave solutions to Boussinesq systems. Differential Integral Equations 24 (2011), no. 9/10, 895--908. https://projecteuclid.org/euclid.die/1356012891

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