Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Quasiclassical asymptotics for solutions of the matrix conjugation problem with rapid oscillation of off-diagonal entries

Author: A. M. Budylin
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 25 (2013), nomer 2.
Journal: St. Petersburg Math. J. 25 (2014), 205-222
MSC (2010): Primary 35Q15; Secondary 45E10, 45E99
Published electronically: March 12, 2014
MathSciNet review: 3114849
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The $ (2\times 2)$-matrix conjugation problem (Riemann-Hilbert problem) with rapidly oscillating off-diagonal entries is considered, along with its applications to nonlinear problems of mathematical physics. The phase function that determines oscillation is assumed to have finitely many simple stationary points and to admit power-like growth at infinity. Quasiclassical asymptotics are constructed for solutions of such a problem in the class of Hölder functions, under appropriate restrictions on the entries of the conjugation matrix. It is proved that, after separation of a certain background, the stationary points of the phase function contribute to the asymptotics additively. Along with the M. G. Kreĭn theory, the justification of the resulting asymptotic solutions employs the stationary phase method and the Schwarz alternating method.

References [Enhancements On Off] (What's this?)

  • 1. A. M. Budylin and V. S. Buslaev, Quasiclassical asymptotics of the solutions of matrix Riemann-Hilbert problems with quadratic oscillation of non-diagonal elements, Funktsional. Anal. i Priložen., 2013 (Russian) (to appear).
  • 2. A. M. Budylin and V. S. Buslaev, Quasiclassical asymptotics of the resolvent of an integral convolution operator with a sine kernel on a finite interval, Algebra i Analiz 7 (1995), no. 6, 79–103 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 6, 925–942. MR 1381979
  • 3. I. Ts. Gokhberg and N. Ya. Krupnik, \cyr Vvedenie v teoriyu odnomernykh singulyarnykh integral′nykh operatorov., Izdat. “Štiinca”, Kishinev, 1973 (Russian). MR 0405177
  • 4. P. A. Deift, A. R. It⋅s, and X. Zhou, Long-time asymptotics for integrable nonlinear wave equations, Important developments in soliton theory, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1993, pp. 181–204. MR 1280475
  • 5. P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), no. 2, 295–368. MR 1207209,
  • 6. -, Long-time behavior of the non-focusing nonlinear Schrödinger equation -- a case study, Univ. of Tokyo, Tokyo, 1994.
  • 7. G. G. Varzugin, Asymptotics of oscillatory Riemann-Hilbert problems, J. Math. Phys. 37 (1996), no. 11, 5869–5892. MR 1417182,
  • 8. Yen Do, A nonlinear stationary phase method for oscillatory Riemann-Hilbert problems, Int. Math. Res. Not. IMRN 12 (2011), 2650–2765. MR 2806592,

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35Q15, 45E10, 45E99

Retrieve articles in all journals with MSC (2010): 35Q15, 45E10, 45E99

Additional Information

A. M. Budylin
Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 185504, Russia

Keywords: Matrix conjugation problem, quasiclassical asymptotics, singular integral equations, nonlinear equations of mathematical physics
Received by editor(s): October 26, 2012
Published electronically: March 12, 2014
Additional Notes: Supported by RFBR (grant no. 11-01-00458) and by Ministry of Education and Science of RF, grant nos. 8501 as of 07.09.2012, and 2012-1.5-12-000-1003-016.
Dedicated: To blessed memory of my Teacher Vladimir Savel’evich Buslaev
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society