This chapter describes the Gelfand-Dickey systems and inelastic solitons. The Boussinesq equation is known and called N-soliton solutions—for example, solutions that exhibit asymptotically N solitary waves of the typical sech2-form. The chapter mainly interested in the Boussinesq equation in imaginary time. This equation renders “inelastic solitons”—for example, solitary waves of the sech2-form which may stick together after interaction thus forming a new sech2-wave. These inelastic solitons can be obtained via an auto-Bäcklund transformation for the Gelfand-Dickey system associated with the Boussinesq-type equation. In general the solutions constructed by the method described in the chapter may have singularities because the Wronskians used may have zeros. © 1993, Academic Press Inc.