Journal of Mathematical Sciences, Vol. 136, No. 1, 2006 SOLUTION OF THE GENERAL KdV EQUATION IN THE CLASS OF STEP FUNCTIONS A. B. Khasanov ∗ and G. U. Urazboev ∗ UDC 517.9 In this work, we deduce laws of evolution of the scattering data for the Sturm–Liouville operator with a potential that is a solution of the general Korteweg-de Vries equation and general Korteweg-de Vries equation with a source in the class of step functions. Bibliography: 19 titles. 1. Introduction In 1967, Gargner, Green, Kruskal, and Miura (GGKM, [1]) had proposed the method of the inverse scattering problem for the Sturm–Liouville equation as a method for solving the Cauchy problem for the Korteveg-de Vries (KdV) equation u t − 6uu x + u xxx =0. Shortly thereafter, Lax (see [2]) noted the general character of the method. Several years later, Zakharov and Shabat solved in [3] the Cauchy problem for another important nonlinear evolutionary equation, the so-called nonlinear Schr¨ odinger equation, by a nontrivial development of the GGKM and Lax methods. Thus, a way was found for construction of several other classes of equations that can be solved by similar methods. For a detailed presentation of relations between the inverse problem and nonlinear equations of mathematical physics, see, for example, the monographs [4–8]. In [9], it was noted that the squares of eigenfunctions play an important role in eigenvalue problems for systems of first-order dofferential equations. It was shown by Newell in [7] that precisely the squares of eigenfunctions (and not eigenfunctions themselves) are essential in integrating by the method of the inverse scattering problem for the Sturm–Liouville equations. A strict proof of the latter fact is given in the monograph [8]. In the works [10–12], the KdV equations with a self-consistent source were considered; in particular, laws of evolution of the scattering data were found. It was shown that, in the general case of the KdV equation with a right-hand term, discrete eigenvalues of the problem considered are not invariant in t. A similar situation arises in problems for which the pole of the dispersion relation (see [7]) coincides with one of the values of the discrete spectrum; in this case, the corresponding solution preserves its identity, but now this identity is variable. In the work [14], the Cauchy problem for the KdV equation with initial data of “strip” type was solved by the method of the inverse scattering problem; in addition, an asymptotic solution as t →∞ was constructed in a neighborhood of the forefront. For the first time, this problem was considered in the work [14], where the Whizem approximate method was applied in construction of an asymptotic solution expressed in terms of the Jacobi elliptic functions with slowly varying parameters. In the works [15, 16], the KdV equations with a self-consistent source were integrated for a class of initial data of “step” type; in particular, laws of evolution of the scattering data were established. Denote L(t)= −D 2 + u and H = − 1 2 D 3 +2uD + u  , where u = u(x, t) and D = d dx . There exist polynomials P k (with respect to u and derivatives of u in x) such that H x P k = P  k+1 (see [17]). For example, P 0 = − 1 2 , P 1 = − 1 2 u, P 2 = 1 4 u xx − 3 4 u 2 , P 3 = − 1 8 u xxxx + 5 4 uu xx + 5 8 (u x ) 2 − 5 4 u 3 , and so on. The operator B q = q  k=0  1 2 P  k − P k d dx  (2L) q−k (1.1) ∗ Urgench State University, Uzbekistan. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 317, 2004, pp. 174–199. Original article submitted October 26, 2004. 1072-3374/06/1361-3625 c 2006 Springer Science+Business Media, Inc. 3625