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

DOI:
https://doi.org/10.1090/S1061-0022-2014-01286-1

Published electronically:
March 12, 2014

MathSciNet review:
3114849

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Abstract | References | Similar Articles | Additional Information

Abstract: The -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.

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Additional Information

**A. M. Budylin**

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

Email:
budylin@math.nw.ru

DOI:
https://doi.org/10.1090/S1061-0022-2014-01286-1

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