Scattering and inverse scattering for first order systems

Abstract

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.

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

“…We can write (2.6) as where the off-diagonal entries can be expressed in terms of L(A) or R(A) by using 10) which is immediate from (2.33).…”

Complete Integrability of Shock Clustering and Burgers Turbulence

“…Such solutions are called wavefunctions (we use the terminology of [45]). The scattering theory for this equation when A is a matrix that vanishes on the diagonal was considered by Zakharov et al [48] and by Beals and Coifman [13,14]. In our work, A ∈ m. As a consequence, even though A is completely determined by its off-diagonal elements, its diagonal entries do not vanish.…”

“…This theory does not directly apply to (1.21) since m is not semi-simple. We modify [13,14] to obtain a scattering and inverse scattering theory for (1.21). Among other results, we obtain a hierarchy of integrable flows and global well-posedness theorems for (1.21) including the probabilistically natural case.…”

“…In our work, A ∈ m. As a consequence, even though A is completely determined by its off-diagonal elements, its diagonal entries do not vanish. Thus, the scattering theory of [13,14] does not immediately apply to our model and we have to rederive some results. For the most part, this is straightforward.…”

Inverse Scattering on the Half-Line for ZS-AKNS Systems with Integrable Potentials

“…Recently, Demontis and van der Mee in [17][18][19] studied the problem of reconstructing non-self-adjoint 326 R. O. Hryniv and S. S. Manko IEOT matrix Zakharov-Shabat and AKNS systems. An interesting approach to inverse scattering analysis of more general systems was suggested by Beals and Coifman [8][9][10] and Beals, Deift, and Tomei [11]. The motivation for the present work was two-fold.…”