**Nächste Seite:**Fujita
systems**Aufwärts:**Models
of Mathematical Physics**Vorherige Seite:**Picard'
s extended Maxwell

##
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:
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:

IMBq describes the Bq equationwith .
These equations can be used to study the so-called Alvén waves.

More recently is studied are general problem

Furthermore, boundary and initial conditions are posed:
**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.

**Nächste Seite:**Fujita
systems**Aufwärts:**Models
of Mathematical Physics**Vorherige Seite:**Picard'
s extended Maxwell
collected and worked out by Wolfgang Sprößig, TU-Bergakademie
Freiberg