Riemann-Hilbert factorization of matrices invariant under inversion in a circle

Yamane, Hideshi

doi:10.1090/proc/14398

Riemann-Hilbert factorization of matrices invariant under inversion in a circle
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Proc. Amer. Math. Soc. 147 (2019), 2147-2157 Request permission

Abstract:

We consider matrix functions with certain invariance under inversion in the unit circle. If such a function satisfies a positivity assumption on the unit circle, then only zero partial indices appear in its Riemann-Hilbert (Wiener-Hopf) factorization. It implies the unique solvability of a certain class of Riemann-Hilbert boundary value problems. It includes the ones associated with the inverse scattering transform of the focusing/defocusing integrable discrete nonlinear Schrödinger equations.

References

M. J. Ablowitz, B. Prinari, and A. D. Trubatch, Discrete and continuous nonlinear Schrödinger systems, London Mathematical Society Lecture Note Series, vol. 302, Cambridge University Press, Cambridge, 2004. MR 2040621
R. Beals and R. R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math. 37 (1984), no. 1, 39–90. MR 728266, DOI 10.1002/cpa.3160370105
Kevin F. Clancey and Israel Gohberg, Factorization of matrix functions and singular integral operators, Operator Theory: Advances and Applications, vol. 3, Birkhäuser Verlag, Basel-Boston, Mass., 1981. MR 657762
Percy Deift and Xin Zhou, Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space, Comm. Pure Appl. Math. 56 (2003), no. 8, 1029–1077. Dedicated to the memory of Jürgen K. Moser. MR 1989226, DOI 10.1002/cpa.3034
Baoqiang Xia and A. S. Fokas, Initial-boundary value problems associated with the Ablowitz-Ladik system, Phys. D 364 (2018), 27–61. MR 3737868, DOI 10.1016/j.physd.2017.10.004
Georgi Semenovich Litvinchuk and Ilya Matveyevich Spitkovskiĭ, Factorization of measurable matrix functions, Mathematical Research, vol. 37, Akademie-Verlag, Berlin, 1987. With a foreword by Bernd Silbermann. MR 925825, DOI 10.1007/978-3-0348-6266-0
Hideshi Yamane, Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation, J. Math. Soc. Japan 66 (2014), no. 3, 765–803. MR 3238317, DOI 10.2969/jmsj/06630765
Hideshi Yamane, Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation II, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 020, 17. MR 3322338, DOI 10.3842/SIGMA.2015.020
H. Yamane, Long-time asymptotics for the integrable discrete nonlinear Schrödinger equation: the focusing case, arXiv 1512.01760 [math-ph], to appear in Funk. Ekvac.
Xin Zhou, The Riemann-Hilbert problem and inverse scattering, SIAM J. Math. Anal. 20 (1989), no. 4, 966–986. MR 1000732, DOI 10.1137/0520065
Xin Zhou, Direct and inverse scattering transforms with arbitrary spectral singularities, Comm. Pure Appl. Math. 42 (1989), no. 7, 895–938. MR 1008796, DOI 10.1002/cpa.3160420702
Xin Zhou, $L^2$-Sobolev space bijectivity of the scattering and inverse scattering transforms, Comm. Pure Appl. Math. 51 (1998), no. 7, 697–731. MR 1617249, DOI 10.1002/(SICI)1097-0312(199807)51:7<697::AID-CPA1>3.0.CO;2-1