COMMUTATION METHODS APPLIED TO THE mKdV-EQUATION

An explicit construction of solutions of the modified Korteweg-de Vries equation given a solution of the (ordinary) Korteweg-de Vries equation is provided. Our theory is based on commutation methods (i.e., N = 1 supersymmetry) underlying Miura's transformation that links solutions of the two evolution equations. In connection with the extensively studied Korteweg-de Vries (KdV-) equation its cousins, the modified Korteweg-de Vries (mKdV,-) equations have also been investigated. In fact, (1.2+) has been treated in some detail in the literature: See, e.g., [12, 51, 67, 68, 69, 74, 11 1] for existence and uniqueness questions, [58, 105, 107, 1 13, 1 141 for the derivation of the N-soliton solutions and [94, 1081 for the general approach to (1.2+) via inverse scattering techniques. The Lax pair for (1.2 f) has been given in [I071 and finally (1.2 f ) have been found to be subordinate to the AKNS-ZS-theory [4, 931 (cf. [34]). For more recent work on (1.2+) see, e.g., [11, 13, 70, 1181. Surprisingly enough, apart from existence and uniqueness questions of solutions of (1.2-) in [12, 51, 68, 69, 74, l l l], no detailed study of (1.2-) seems to have appeared until 1984 when Grosse [54] (see also [55]) derived the N-soliton solutions of (1.2-) . The present paper is devoted to a detailed investigation of real-valued solutions of the mKdV--equation (1.2-) (from now on simply denoted by mKdV). Received by the editors January 15, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 35620, 76B25.