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Boussinesq equation

The French scientist Joseph Boussinesq (1842-1929) described in the 1870s model equations for the propagation of long waves on the surface of water with a small amplitude. Today one has to distinguish two essentially different Boussinesq (Bq) equations. These equations read as follows:
\begin{eqnarray*}u_{tt}-u_{xx}\pm u_{xxxx}=(u^2)_{xx}\end{eqnarray*}
The equation with the minus sign is called ''bad'' Bq equation and that with plus sign is just the ''good'' Bq equation. The last one describes the two-dimensional irrotational flow of an inviscid liquid in a uniform rectancular channel. There are known results due to local well-posedness, global existence and blow-up of some solutions. The ''bad'' Bq equation is used to describe two-dimensional flow of shallow-water waves having small amplitudes. There is a dense connection to the so-called Fermi-Pasta-Ulam (FPU) problem. Only some soliton-type solutions are known.
 

Bq-type equations can be considered as the first model for nonl-linear, dispersive wave propagation. V. G. Makhankov showed in his paper Dynamics of classical solitons, Physic Reports A review Section of Physics Letters (Section C) 35C(1), 1-128 in 1978 that the ''bad'' Bq equation can be approached by an so-called Improved (Improved Modified) Bq equation IBq (IMBq)) which reads as follows:

\begin{eqnarray*}u_{tt}-u_{xx}-u_{xxtt}=(u^k)_{xx} \quad\mbox{where}\quad(k=2,3)\end{eqnarray*}
IMBq describes the Bq equationwith $k=3$. These equations can be used to study the so-called Alvén waves.
 

More recently is studied are general problem

\begin{eqnarray*}u_{tt}-u_{xx}-u_{xxtt}=(\sigma (u))_{xx} \quad\mbox{for} \quad(0<x<1,\;t>0).\end{eqnarray*}
Furthermore, boundary and initial conditions are posed:
\begin{eqnarray*}&&u(0,t)=u(1,t)=0\quad t>0\\&&u(x,0=u_0(x), u_t(x,0)=u_1(x)\quad (0\le x\le1)\end{eqnarray*}
A late reference: Yang Zhijian (1998). Existence and Non-existence of Global Solutions to a Generalized Modification of the Improved Boussinesq Equation, MMAS 21, 1467-1477.
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collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg