Similarity solutions of the generalized Korteweg-de Vries equation

Numerical simulations and group invariance considerations point to the exis tence of similarity solutions of the form U(x, t) = 1/(t(*)-t)2/3p psi(x(*)-x-c(t(*) -t)(1/3)/(t(*) -t)(1/3)) (*) of the generalized Korteweg-de Vries equation U-i + (UUx)-U-p + U-xxx = 0 (**) Here, x(*), t(*) and c are real parameters, x and t are real variables with t not equal t(*), p is a positive integer and interest is focussed on the case where p greater than or equal to 4 for which solutions of the initial- value problem for (**) are not known to be always globally defined. It is s hown that smooth solutions of (**) of the form appearing in (*) do indeed e xist. Some detailed properties of the function psi appearing in (*) are als o obtained.