It is well-known that a number of important nonlinear evolution equations are associated to spectral problems for ordinary differential operators (see [l], [4]). The initial value problem for the evolution equation can, in principle, be solved by solving an inverse scattering problem. Schematically, the unknown function u( * , t ) (possibly vector-valued) is identified with or transformed into the coefficients q( -, t ) of a differential operator L,. A spectral problem is associated to L, which carries (at least formally) some asymptotic information called the scattering data u( * , t ) . The original nonlinear evolution of y or equivalently of q, corresponds to a trivially solvaole linear evolution of the scattering data u.The analytical theory of scattering and inverse scattering in various cases has been treated, for example, in [l], [6], [lo], and other papers of these authors. It should be noted, though, that in much of the literature the expression "solvable by the inverse scattering method" designates evolutions associated to spectral problems for which certain purely formal scattering data would evolve linearly if it existed. The proposed scattering data may exist only for compactly supported or exponentially vanishing q, and the support condition or the vanishing condition may be destroyed by the evolution itself. In short, problems may have been termed "solvable" when neither the scattering map q* o nor the inverse map u-q has been seriously investigated. (For such problems one has recipes to produce special solutions, such as soliton or multi-soliton solutions, but the general initial value problem may be untouched.)A satisfactory analytical treatment of scattering and inverse scattering for a given spectral problem should aim for the following:(i) to formulate a notion of scattering data u which is meaningful for (essentially) all reasonable coefficients q, such as q E L';(ii) to show that q + u is injective; (iii) to characterize scattering data by determining all the algebraic or topo-(iv) to show that for (essentially) each set of data satisfying the constraints, (v) to discuss the relationship of such analytic properties of q as smoothness logical constraints such data satisfy; there is a corresponding q ; or decay at 00 with corresponding properties of u.

dedicated to gian-carlo rotaClassical results of Stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the Camassa Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon antipeakon pairs, and the details of the collisions are analyzed using results from the moment problem. A sharp result on the steepening of the slope at the time of collision is given. Asymptotic formulas are given, and the scattering shifts are calculated explicitly. Academic Press

The theory of valuations, O. F. G. Schilling 5 The kernel function and conformal mapping, S. Bergman 6 Introduction to the theory of algebraic functions of one variable, C. C. Chevalley 7.1 The algebraic theory of semigroups, Volume I, A. H. Clifford and G. B. Preston 7.2 The algebraic theory of semigroups, Volume II, A. H. Clifford and G. B. Preston 8 Discontinuous groups and automorphic functions, J. Lehner 9 Linear approximation, Arthur Sard 10 An introduction to the analytic theory of numbers, R. Ayoub 11 Fixed points and topological degree in nonlinear analysis, J. Cronin 12 Uniform spaces, J. R. Isbell

A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts.

No abstract

The acoustic scattering operator on the real line is mapped to a Schro dinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transformation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals. Academic Press

No abstract

No abstract