Geodesic
Quasi-Classical Asymptotics of Solutions to the Matrix Factorization Problem with Quadratically Oscillating Off-Diagonal Elements
Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 1, pp. 1-18

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The paper investigates the asymptotic behavior of solutions to the $2\times 2$ matrix factorization (Riemann–Hilbert) problem with rapidly oscillating off-diagonal elements and quadratic phase function. A new approach to study such problems based on the ideas of the stationary phase method and M. G. Krein's theory is proposed. The problem is model for investigating the asymptotic behavior of solutions to factorization problems with several turning points. Power-order complete asymptotic expansions for solutions to the problem under consideration are found. These asymptotics are used to construct asymptotics for solutions to the Cauchy problem for the nonlinear Schrödinger equation at large times.
Keywords: oscillatory Riemann–Hilbert problems, semiclassical asymptotics, singular integral equations, nonlinear equations of mathematical physics. 
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     author = {A. M. Budylin and V. S. Buslaev},
     title = {Quasi-Classical {Asymptotics} of {Solutions} to the {Matrix} {Factorization} {Problem} with {Quadratically} {Oscillating} {Off-Diagonal} {Elements}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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     number = {1},
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A. M. Budylin; V. S. Buslaev. Quasi-Classical Asymptotics of Solutions to the Matrix Factorization Problem with Quadratically Oscillating Off-Diagonal Elements. Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 1, pp. 1-18. http://geodesic.mathdoc.fr/item/FAA_2014_48_1_a0/