  In 1963 a partial differential equation with a convolution non-linearity was
introduced in connection with a quantum mechanical many-body problem, namely
the gas of bosonic particles. This equation is mathematically interesting for
several reasons. (1) Although the equation was expected to be valid only for
small values of the parameters, further investigation showed that predictions
based on the equation agree well over the {\it entire range} of parameters with
what is expected to be true for the solution of the true many-body problem. (2)
The novel nonlinearity is easy to state but seems to have almost no literature
up to now. (3) The earlier work did not prove existence and uniqueness of a
solution, which we provide here along with properties of the solution such as
decay at infinity.
