Self-Similar Solutions of the Second Kind

In recent years there has been a surge of interest in self-similar solutions of the second kind. Such solutions are not newly discovered; they had been identified and in fact so named by Zel’dovich in 1956, in the context of a variety of problems, such as shock waves in gas dynamics [18], [34], [35], and filtration through elastoplastic materials [5], [10], [7, 8]. We cite from Zel’dovich’s Foreword to Barenblatt’s book on intermediate asymptotics [5]: We shall reserve the name solutions of the second kind for the large and ever-growing class of solutions for which the exponents are found in the process of solving the problem, analogous to the determination of eigenvalues for linear equations. For this case, conservation laws and dimensional considerations prove to be insufficient.