The inverse scattering problems for the hyperbolic equations and their application to nonlinear integrable systems (Q1822697)

The inverse scattering problem is studied for a system of two first order differential equations of the form \[ \psi_{1x}(x,y)+u_ 1(x,y)\psi_ 2(x,y)=0,\quad \psi_{2y}(x,y)+u_ 2(x,y)\psi_ 1(x,y)=0, \] (x,y)\(\in R^ 2\), \(u_ 1,u_ 2\in L^ 2(R^ 2).\) The scattering data are the operators \(F_{12}\), \(Y_{21}\) which are entries of the \(2\times 2\) operator matrices defined by the formulas \(S=I+F\), \(S^{-1}=I+Y\), where S is the S-operator transforming the incident wave into the scattered wave: \(b=Sa\), \(a_ 1(y)=\psi_ 1(- \infty,y)\), \(a_ 2(x)=\psi_ 2(x,-\infty)\), \(b_ 1(y)=\psi_ 1(+\infty,y)\), \(b_ 2(x)=\psi_ 2(x,+\infty)\). Factorization of S plays an important role in the author's argument. The results on inverse scattering problem are applied to integration of some nonlinear evolution equations. Global existence of the solution to a Cauchy problem is studied.