Numerical simulations and group invariance considerations point to the existence
of similarity solutions of the form
formula here
of the generalized Korteweg–de Vries equation
formula here
Here, x∗, t∗ and c are real parameters,
x and t are real variables with t≠t∗, p is a
positive integer and interest is focussed on the case where p[ges ]4 for which solutions
of the initial-value problem for (**) are not known to be always globally defined. It
is shown that smooth solutions of (**) of the form appearing in (*) do indeed exist.
Some detailed properties of the function ψ appearing in (*) are also obtained.