Long-time asymptotics for Toda shock waves in the modulation region
DOI:
https://doi.org/10.15407/mag19.02.396Ключові слова:
Рiвняння Тоди, задача Рiмана–Гiльберта, тип сходинки, хвиля стисненняАнотація
Ми показуємо, що хвиля стиснення для ланцюжка Тоди є асимптотично наближеною до модульованого скiнченнозонного розв’язку у правому модуляцiйному регiонi. Ранiше нами було виведено формули для головних членiв асимптотичного розвинення цiєї хвилi стиснення в усiх п’ятьох принципових регiонах, а також було припущено, що у двох модуляцiйних регiонах наступний член є порядку $O(t^{-1})$. У данiй роботi ми доводимо цей факт i дослiджуємо, як дискретний спектр та резонанси впливають на провiдну асимптотику. Основним внеском є розв’язання локальних задач Рiмана–Гiльберта (задач параметрiксу) i строге обґрунтування заключного асимптотичного аналiзу. Зокрема, це включає в себе побудову належного матричного розв’язку модельної задачi Рiмана–Гiльберта.
Mathematical Subject Classification 2020: 37K40, 35Q53, 37K45, 35Q15
Посилання
K. Andreiev, I. Egorova, T.L. Lange, and G. Teschl, Rarefaction waves of the Korteweg-de Vries equation via nonlinear steepest descent, J. Differential Equations 261 (2016), 5371--5410. https://doi.org/10.1016/j.jde.2016.08.009
M. Bleher, Lectures on Random Matrix Models: The Riemann-Hilbert Approach, Random Matrices, Random Processes and Integrable Systems, (Ed. J. Harnad), CRM Series in Mathematical Physics, Springer, New York, 2001,251--349. https://doi.org/10.1007/978-1-4419-9514-8_4
A.M. Bloch and Y. Kodama, The Whitham equation and shocks in the Toda lattice, Proceedings of the NATO Advanced Study Workshop on Singular Limits of Dispersive Waves held in Lyons, July 1991, Plenum Press, New York, 1994. https://doi.org/10.1007/978-1-4615-2474-8_1
A.M. Bloch and Y. Kodama, Dispersive regularization of the Whitham equation for the Toda lattice, SIAM J. Appl. Math. 52 (1992), 909--928. https://doi.org/10.1137/0152052
A. Boutet de Monvel, I. Egorova, and E. Khruslov, Soliton asymptotics of the Cauchy problem solution for the Toda lattice, Inverse Problems 13 (1997), 223--237. https://doi.org/10.1088/0266-5611/13/2/003
A. Boutet de Monvel and I. Egorova, The Toda lattice with step-like initial data. Soliton asymptotics, Inverse Problems 16 (2000), 955--977. https://doi.org/10.1088/0266-5611/16/4/306
P. Deift, Some open problems in random matrix theory and the theory of integrable systems. II, SIGMA 13 (2017), 016. https://doi.org/10.3842/SIGMA.2017.016
P. Deift, S. Kamvissis, T. Kriecherbauer, and X. Zhou, The Toda rarefaction problem, Comm. Pure Appl. Math. 49 (1996), 35--83. https://doi.org/10.1002/(SICI)1097-0312(199601)49:1%3C35::AID-CPA2%3E3.0.CO;2-8
P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335--1425. https://doi.org/10.1002/(SICI)1097-0312(199911)52:11%3C1335::AID-CPA1%3E3.0.CO;2-1
P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, Ann. of Math. (2) 137 (1993), 295--368. https://doi.org/10.2307/2946540
P. Deift, S. Venakides, and X. Zhou, The collisionless shock region for the long time behavior of solutions of the KdV equation, Comm. Pure and Appl. Math. 47 (1994), 199--206. https://doi.org/10.1002/cpa.3160470204
B.A. Dubrovin, Theta functions and nonlinear equations, Russian Math. Surveys 36 (1981), 11--92. https://doi.org/10.1070/RM1981v036n02ABEH002596
I. Egorova and Z. Gladka, V. Kotlyarov, and G. Teschl, Long-time asymptotics for the Korteweg-de Vries equation with steplike initial data, Nonlinearity 26 (2013), 1839--1864. https://doi.org/10.1088/0951-7715/26/7/1839
I. Egorova and J. Michor, How discrete spectrum and resonances influence the asymptotics of the Toda shock wave, SIGMA 17 (2021), 045. https://doi.org/10.3842/SIGMA.2021.045
I. Egorova, J. Michor, and G. Teschl, Scattering theory for Jacobi operators with general steplike quasi-periodic background, J. Math. Phys. Anal. Geom. 4 (2008), 33--62.
I. Egorova, J. Michor, and G. Teschl, Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds, J. Math. Phys. 50 (2009), 103522. https://doi.org/10.1063/1.3239507
I. Egorova, J. Michor, and G. Teschl, Scattering theory with finite-gap backgrounds: transformation operators and characteristic properties of scattering data, Math. Phys. Anal. Geom. 16 (2013), 111--136. https://doi.org/10.1007/s11040-012-9121-y
I. Egorova, J. Michor, and G. Teschl, Long-time asymptotics for the Toda shock problem: non-overlapping spectra, J. Math. Phys. Anal. Geom. 14 (2018), 406--451. https://doi.org/10.15407/mag14.04.406
I. Egorova, M. Piorkowski, and G. Teschl, On Vector and Matrix Riemann-Hilbert problems for KdV shock waves, preprint, https://arxiv.org/abs/1907.09792.
M. Girotti, T. Grava, R. Jenkins, and K.D.T.-R. McLaughlin, Rigorous asymptotics of a KdV soliton gas, Comm. Math. Phys. 384 (2021), 733--784. https://doi.org/10.1007/s00220-021-03942-1
F. Gesztesy and G. Teschl, Commutation methods for Jacobi operators , J. Differential Equations 128 (1996), 252--299. https://doi.org/10.1006/jdeq.1996.0095
K. Grunert and G. Teschl, Long-time asymptotics for the Korteweg--de Vries equation via nonlinear steepest descent, Math. Phys. Anal. Geom. 12 (2009), 287--324. https://doi.org/10.1007/s11040-009-9062-2
A. Its, Large N-asymptotics in random matrices. Random Matrices, Random Processes and Integrable Systems, CRM Series in Mathematical Physics, Springer, New York, 2011. https://doi.org/10.1007/978-1-4419-9514-8_5
V.P. Kotlyarov, A.M. Minakov, Riemann-Hilbert problem to the modified Korteweg-de Vries equation: Long-time dynamics of the step-like initial data, J. Math. Phys. 51 (2010), 093506. https://doi.org/10.1063/1.3470505
S. Kamvissis, On the Toda shock problem, Phys. D, 65 (1993), 242--256. https://doi.org/10.1016/0167-2789(93)90161-S
H. Krüger and G. Teschl, Long-time asymptotics for the Toda lattice in the soliton region, Math. Z. 262 (2009), 585--602. https://doi.org/10.1007/s00209-008-0391-9
H. Krüger and G. Teschl, Long-time asymptotics of the Toda lattice for decaying initial data revisited, Rev. Math. Phys. 21 (2009), 61--109. https://doi.org/10.1142/S0129055X0900358X
J. Lenells, Matrix Riemann-Hilbert problems with jumps across Carleson contours, Monatsh. Math. 186 (2018), 111--152. https://doi.org/10.1007/s00605-017-1019-0
J. Michor, Wave phenomena of the Toda lattice with steplike initial data , Phys. Lett. A 380 (2016), 1110--1116. https://doi.org/10.1016/j.physleta.2016.01.033
A. Minakov, Riemann-Hilbert problem for Camassa-Holm equation with step-like initial data, J. Math. Anal. Appl. 429 } (2015), 81--104. https://doi.org/10.1016/j.jmaa.2015.03.059
A. Minakov, On the solution of the Zakharov-Shabat system, which arises in the analysis of the largest real eigenvalue in the real Ginibre ensemble, preprint, https://arxiv.org/abs/1905.03369.
N.I. Muskhelishvili, Singular Integral Equations, P. Noordhoff Ltd., Groningen, 1953.
M. Piorkowski, Parametrix problem for the Korteweg-de Vries equation with steplike initial data, preprint, https://arxiv.org/abs/1908.11340.
G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon., 72 , Amer. Math. Soc., Rhode Island, 2000. https://doi.org/10.1090/surv/072
M. Toda, Theory of Nonlinear Lattices, Springer, Berlin, 1989. https://doi.org/10.1007/978-3-642-83219-2
S. Venakides, P. Deift, and R. Oba, The Toda shock problem, Comm. Pure Appl. Math. 44 (1991), 1171--1242. https://doi.org/10.1002/cpa.3160440823
