PDE 5: Boussinesq equation

\[u_{tt}+\alpha u_{xx}+\beta (u^2)_{xx}+\gamma u_{xxxx} = 0.\]

One possibility is to set \(\alpha =0 , \beta=\frac{1}{2}\), and \(\gamma=1 \), another choice is \(\alpha=-1 , \beta=-1 , \gamma=\pm 1 \). A pure initial value problem is possible, i.e. where \(u(x,0)\) and \(u_t(x,0)\) are given for all \(t>0\), but boundary value problems are also relevant. If possible try and be guided by possible applications!

Google: Boussinesq equation, water waves, shallow water, nonlinear waves, dispersive waves.

Challenges: Nonlinearity, high order derivatives, physical relevance.