Nonstationary inverse scattering problems for a system of first-order hyperbolic equations on the semi-axis. (Q2748476)

In this monograph the author presents some results (partly new) concerning direct and inverse scattering for first-order hyperbolic systems on the half-axis \(\mathbb{R}_+\). The systems under consideration are of the form NEWLINE\[NEWLINE\xi_i\partial_t\psi_i- \partial_x\psi_i= \sum_{1\leq j\leq n} u_{ij}(x,t) \psi_j,\quad x\in\mathbb{R}_+,\quad t\in \mathbb{R},\quad i\leq i\leq n,\tag{1}NEWLINE\]NEWLINE where \(n\geq 3\), \(u_{ii}= 0\), \(u_{ij}\) is a measurable complex valued function verifying \(|u_{ij}(x,t)|\leq C(1+|x|)^{-1-\varepsilon}(1+|t|)^{-1-\varepsilon}\), \(C,\varepsilon> 0\). As for \(\xi_i\), there exists \(k\in \{1,2,\dots, n\}\) such that \(\xi_1> \xi_2>\cdots> \xi_k> 0> \xi_{k+1}>\cdots> \xi_n\) (there are \(k\) incident waves and \(n-k\) scattered waves).NEWLINENEWLINENEWLINEThe most general results are obtained in the case \(k= n-1\) (or \(k=1\)). In the direct problem, one is looking for the scattering operator \(S\), defined as follows. Let us consider \(N-1\) problems: for \(j\in \{1,2,\dots, n-1\}\) and \(a_1,\dots, a_{n-1}\in L^2(\mathbb{R})\) (defining the incident waves), prove that there exists an unique system \(\{\psi^j_i(x,t)\}_{1\leq i\leq n}\) satisfying (1), the boundary condition \(\psi^j_n(0,1)= \psi^j_j(0,t)\) and the asymptotic conditions \(\psi^j_i(m,t)= a_i(t+ \xi_ix)+ O(1)\), \(x\to\infty\), \(1\leq i\leq n-1\). Then \(\psi^j_n(x,t)= b_j(t+ \xi_nx)+ O(1)\), \(x\to\infty\), where \(b_j\in L^2(\mathbb{R})\), and the scattering operator \(S\) is well defined by \(S(a_1,\dots, a_{n-1})= (b_1,\dots, b_{n-1})\). In the inverse problem, one is looking for the coefficients \(u_{ij}\) given the scattering operator \(S\). For the solution, one makes use of an inverse problem on \(\mathbb{R}\) and of a certain decomposition of the scattering operator.NEWLINENEWLINENEWLINEThe two problems are also studied in the following cases: \(n= 4\) and \(k=2\), \(n=3\) and \(\xi_1= \xi_2> \xi_3\), and for Dirac systems. In the cases \(n=2m\) and \(k=m\), \(n= 5\) and \(k=2\), the direct problem only is studied.