## 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.

### MSC:

 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
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### References:

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