The Riemann-Hilbert problem and inverse scattering. (English) Zbl 0685.34021

Zhou, Xin

SIAM J. Math. Anal. 20, No. 4, 966-986 (1989).

The inverse scattering problem of system \((d/dx)m-ad J(z)m=q(x,z)m\) for a class of J(z) and q(\(\cdot,z)\) is investigated with a more direct method based on the Fredholm theory of singular integral operators. To this end the connection between the Riemann-Hilbert factorization on self- intersecting contours and a class of singular integral equations is studied with a pair of decomposing algebras. The author also shows that the matrix functions with positive definite real parts on the real axis and Schwarz reflection invariant elsewhere only have zero partial indices. In particular, this implies the solvability for the inverse scattering problem with skew Schwarz reflection invariant system coefficients J(z) and q(\(\cdot,z)\). This includes, for instance, the system associated with the generalized sine-Gordon equation.

Reviewer: Xu Zhenyuan

MSC:

34L99	Ordinary differential operators
35Q15	Riemann-Hilbert problems in context of PDEs

Keywords:

inverse scattering problem; Riemann-Hilbert factorization; singular integral equations; sine-Gordon equation