## Singular matrix conjugacy problem with rapidly oscillating off-diagonal entries. Asymptotics of the solution in the case when a diagonal entry vanishes at a stationary point.(English. Russian original)Zbl 1483.35144

St. Petersbg. Math. J. 32, No. 5, 847-864 (2021); translation from Algebra Anal. 32, No. 5, 37-61 (2020).
In this paper, the author considers a $$2 \times 2$$ matrix conjugation problem (the Riemann-Hilbert factorization problem) with rapidly oscillating off-diagonal inputs and a quadratic phase function, in particular when one of the diagonal inputs vanishes at a stationary point. The main result here is the determination of the leading term with respect to time asymptotics of the solution of this problem.

### MSC:

 35Q15 Riemann-Hilbert problems in context of PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs 45E99 Singular integral equations
Full Text:

### References:

 [1] DIZ P. A. Deift, A. R. Its, and X. Zhou, A Riemann-Hilbert approach to asymptotic problems arising in theory of random matrix models, and also in the theory of integrable statistical mechanics, Ann. of Math. (2) 146 (1997), no. 1, 149-235. 1469319 · Zbl 0936.47028 [2] DZ1 P. A. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), no. 2, 295-368. 1207209 · Zbl 0771.35042 [3] DZ2 \bysame , Long-time behavior of the non-focusing nonlinear Schr\"odinger equation – a case study, Lectures in Math. Sci., vol. 5, Tokyo Univ., Tokyo, 1994. [4] DVZ P. A. Deift, S. Venakides, and X. Zhou, The collisionless shock region for the long-time behavior of solutions of the KdV equation, Comm. Pure Appl. Math. 47 (1994), no. 2, 199-206. 1263128 · Zbl 0797.35143 [5] BBlast A. M. Budylin and V. S. Buslaev, Semiclassical asymptotics of solutions of the matrix conjugation problem with quadratic oscillation of off-diagonal elements, Funktsional. Anal. i Prilozhen. 48 (2014), no. 1, 1-18; English transl., Funct. Anal. Appl. 48 (2014), no. 1, 1-14. 3204674 · Zbl 1307.35179 [6] BE A. Erdelyi, W. Magnus, F. Oberhettinger, and F.  G. Tricomi, Tables of integral transforms. Vol. II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. 0065685 · Zbl 0055.36401 [7] BBAA2000 A. M. Budylin and V. S. Buslaev, The Gel’fand-Levitan-Marchenko equation and the asymptotic behavior of solutions of the nonlinear Schrodinger equation for large time values, Algebra i Analiz 12 (2000), no. 5, 64-105; English transl., St. Petersburg Math. J. 12 (2001), no. 5, 761-789. 1812942 · Zbl 0998.35050 [8] F M. V. Fedoryuk, Asymptotic analysis. Linear ordinary differential equations, Nauka, Moscow, 1983; English transl., Springer-Verlag, Berlin, 1993. 1295032 · Zbl 0538.34001
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.