Quasiclassical asymptotics for solutions of the matrix conjugation problem with rapid oscillation of off-diagonal entries. (English. Russian original) Zbl 1311.35174

St. Petersbg. Math. J. 25, No. 2, 205-222 (2014); translation from Algebra Anal. 25, No. 2, 75-100 (2013).
Summary: 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.


35Q15 Riemann-Hilbert problems in context of PDEs
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45E99 Singular integral equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI


[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). · Zbl 1307.35179
[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. · Zbl 0862.35148
[3] Введение в теорию одномерных сингулярных интеграл\(^{\приме}\)ных операторов., Издат. ”Šтиинца”, Кишинев, 1973 (Руссиан).
[4] P. A. Deift, A. R. It\cdot 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. · Zbl 0926.35132
[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. · Zbl 0771.35042
[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. · Zbl 0860.35091
[8] Yen Do, A nonlinear stationary phase method for oscillatory Riemann-Hilbert problems, Int. Math. Res. Not. IMRN 12 (2011), 2650 – 2765. · Zbl 1222.35138
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.