Existence and orbital stability of cnoidal waves for a 1D Boussinesq equation
The authors of this very interesting paper study existence as well as nonlinear orbital stability of periodic travelling-wave solutions of the nonlinear one-dimensional Boussinesq-type equation written in the form \[ \Phi_{tt}-\Phi_{xx}+a\Phi_{xxxx}-b\Phi_{xxtt}+\Phi_{t}\Phi_{xx}+2\Phi_{x}\Phi_{xt}=0, \] where \(a\), \(b\) are positive numbers. This equation is a rescaled version of the one-dimensional Benney-Luke equation \[ \Phi_{tt}-\Phi_{xx}+\mu (a\Phi_{xxxx}-b\Phi_{xxtt})+\varepsilon (\Phi_{t}\Phi_{xx}+2\Phi_{x}\Phi_{xt})=0, \] which is derived from evolution of two-dimensional long water waves with surfaces tension. The function \(\Phi (x,t)\) is the nondimensional velocity potential at the bottom fluid boundary, \(\mu \) represents the long-wave parameter (dispersion coefficient), \(\varepsilon \) represents the amplitude parameter, and \(a-b+1/3=\sigma \) is the Bond number associated with the surface tension. Periodic travelling-wave solutions with an arbitrary fundamental period \(T_0\) are obtained by using Jacobian elliptic functions. It turns out that there exists stable (in orbital sense) solutions, that is a consequence of the general stability criteria established by \textit{M. Grillakis}, \textit{J. Shatah}, and \textit{W. Strauss} [J. Funct. Anal. 74, 160--197 (1987; Zbl 0656.35122)]. The periodic eigenvalue problem associated to the Lamé equation is studied as well.
scientific article; zbMATH DE number 5282464