Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra
DOI:
https://doi.org/10.15407/mag14.04.406Анотація
Застосовуючи нелінійний аналіз найшвидшого спуску для осциляторної задачі факторизації Рімана–Гільберта, ми виводимо асимптотики при великих значеннях часу для хвилі стиску ланцюжка Тоди. Ми демонструємо, що півплощина просторової/часової змінних розпадається на п'ять основних областей. У двох зовнішніх розв'язок є асимптотично наближеним до відповідних вільних тонів. У середньому регіоні він є наближеним до двозонного розв'язку ланцюжка Тоди. У двох регіонах, що залишилися, розв'язок є асимптотично наближеним до повільно модульованої еліптичної хвилі. Зокрема, форма розв'язку в цих областях підтверджує гіпотезу Вернакідеса, Дейфта та Оба від 1991 р.
Mathematics Subject Classification: 37K40, 37K10, 37K60, 35Q15.
Ключові слова:
ланцюжок Тоди, проблема Рімана-Гільберта, хвиля стискуПосилання
A. Boutet de Monvel, I. Egorova, and E. Khruslov, Soliton asymptotics of the Cauchy problem solution for the Toda lattice, Inverse Problems 13 (1997), No. 2, 223–237.
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
P.F. Byrd and M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer, Berlin, 1954. https://doi.org/10.1007/978-3-642-52803-3
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<35::AID-CPA2
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
P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems, Ann. of Math. 137 (1993), 295–368. https://doi.org/10.2307/2946540
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), No. 11, 1335–1425.
I. Egorova, The scattering problem for step-like Jacobi operator, Mat. Fiz. Anal. Geom. 9 (2002), No. 2, 188–205.
I. Egorova, 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, J. Michor, and G. Teschl, Scattering theory for Jacobi operators with general steplike quasi-periodic background, Zh. Mat. Fiz. Anal. Geom. 4 (2008), No. 1, 33–62.
I. Egorova, J. Michor, and G. Teschl, Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds, J. Math. Physics 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, Rarefaction waves for the Toda equation via nonlinear steepest descent, Discrete Contin. Dyn. Syst. 38 (2018), 2007–2028. https://doi.org/10.3934/dcds.2018081
I. Egorova and L. Pastur, On the asymptotic properties of polynomials orthogonal with respect to varying weights and related problems of spectral theory, Algebra i Analiz 25 (2013), No. 2, 101–124 (Russian). Engl. transl.: St. Petersburg Math. J. 25 (2014), No. 2, 223–240. https://doi.org/10.1090/S1061-0022-2014-01287-3
H. Farkas and I. Kra, Riemann Surfaces, GTM 71, Springer, New York, 1980. https://doi.org/10.1007/978-1-4684-9930-8
B.L. Holian, H. Flaschka, and D.W. McLaughlin, Shock waves in the Toda lattice: Analysis, Phys. Rev. A 24 (1981), 2595–2623. https://doi.org/10.1103/PhysRevA.24.2595
B.L. Holian and G.K. Straub, Molecular dynamics of shock waves in one-dimensional chains, Phys. Rev. B 18 (1978), 1593–1608. https://doi.org/10.1103/PhysRevB.18.1593
A.R. Its, Asymptotics of solutions of the nonlinear Schrödinger equation and isomonodromic deformations of systems of linear differential equations, Soviet Math. Dokl. 24 (1981), 452–456.
S. Kamvissis, On the Toda shock problem, Phys. D 65 (1993), 242–256. https://doi.org/10.1016/0167-2789(93)90161-S
S. Kamvissis, On the long time behavior of the doubly infinite Toda lattice under initial data decaying at infinity, Comm. Math. Phys. 153 (1993), No. 3, 479–519.
S. Kamvissis and G. Teschl, Stability of periodic soliton equations under short range perturbations, Phys. Lett. A 364 (2007), 480–483. https://doi.org/10.1016/j.physleta.2006.12.032
S. Kamvissis and G. Teschl, Long-time asymptotics of the periodic Toda lattice under short-range perturbations, J. Math. Phys. 53 (2012), 073706. https://doi.org/10.1063/1.4731768
V.P. Kotlyarov and 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
V.P. Kotlyarov and A.M. Minakov, Step-initial function to the mKdV equation: Hyper-elliptic long-time asymptotics of the solution, J. Math. Phys. Anal. Geom. 8 (2012), 38–62.
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
H. Krüger and G. Teschl, Stability of the periodic Toda lattice in the soliton region, Int. Math. Res. Not. 2009 (2009), No. 21, 3996–4031.
S.V. Manakov, Nonlinear Frauenhofer diffraction, Sov. Phys. JETP 38 (1974), No. 4, 693–696.
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. Mikikits-Leitner and G. Teschl, Long-time asymptotics of perturbed finite-gap Korteweg–de Vries solutions, J. d’Analyse Math. 116 (2012), 163–218. https://doi.org/10.1007/s11854-012-0005-7
A.M. Minakov, Asymptotics of rarefaction wave solution to the mKdV equation, J. Math. Phys. Anal. Geom. 7 (2011), 59–86.
N.I. Muskhelishvili, Singular Integral Equations, P. Noordhoff Ltd., Groningen, 1953.
Yu. Rodin, The Riemann Boundary Problem on Riemann Surfaces, Mathematics and its Applications (Soviet Series) 16, D. Reidel Publishing Co., Dordrecht, 1988. https://doi.org/10.1007/978-94-009-2885-5
G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon. 72, Amer. Math. Soc., Rhode Island, 2000.
G. Teschl, Algebro-geometric constraints on solitons with respect to quasi-periodic backgrounds, Bull. London Math. Soc. 39 (2007), No. 4, 677–684.
G. Teschl, On the spatial asymptotics of solutions of the Toda lattice, Discrete Contin. Dyn. Syst. 27 (2010), 1233–1239. https://doi.org/10.3934/dcds.2010.27.1233
S. Venakides, P. Deift, and R. Oba, The Toda shock problem, Comm. Pure Appl. Math. 44 (1991), No. 8–9, 1171–1242.
