## Long-time asymptotics for the Toda lattice in the soliton region.(English)Zbl 1198.37104

The authors compute the long-time asymptotics of solutions of the initial value problem for the doubly infinite Toda lattice with decaying initial data. The asymptotic formulas are given in the soliton region $$|n/t| \geq 1 + C/t \log(t)^2$$ for some $$C>0$$. In the remaining region, the authors point out how to reduce the problem to the known case without solitons. The method is based on the nonlinear steepest descent analysis for oscillatory Riemann-Hilbert problems of [P. Deift and X. Zhou, Ann. Math. (2) 137, No. 2, 295–368 (1993; Zbl 0771.35042)]. In the present case with solitons the associated Riemann-Hilbert factorization problem posed two new difficulties: the nonuniqueness issue for the Riemann-Hilbert problem became crucial and a uniform bound for the inverse of the singular integral equation associated with a one-soliton solution was needed.

### MSC:

 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems 35Q15 Riemann-Hilbert problems in context of PDEs 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

### Keywords:

Riemann-Hilbert problem; Toda lattice; Solitons

Zbl 0771.35042
Full Text:

### References:

 [1] Beals R., Coifman R.: Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math. 37, 39–90 (1984) · Zbl 0523.34020 [2] Beals, R., Deift, P., Tomei, C.: Direct and inverse scattering on the real line. Math. Surv. Mon., vol. 28, Am. Math. Soc., Rhode Island (1988) · Zbl 0679.34018 [3] Deift, P.: Orthogonal polynomials and random matrices: a Riemann–Hilbert approach. Courant Lecture Notes, vol. 3, Am. Math. Soc., Rhode Island (1998) · Zbl 0997.47033 [4] Deift P., Zhou X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Ann. Math. (2) 137, 295–368 (1993) · Zbl 0771.35042 [5] Deift P., Kamvissis S., Kriecherbauer T., Zhou X.: The Toda rarefaction problem. Comm. Pure Appl. Math. 49(1), 35–83 (1996) · Zbl 0857.34025 [6] Egorova, I., Michor, J., Teschl, G.: Soliton solutions of the Toda hierarchy on quasi-periodic background revisited. Math. Nach. (2008, to appear) · Zbl 1166.37028 [7] Kamvissis S.: On the long time behavior of the doubly infinite Toda lattice under initial data decaying at infinity. Comm. Math. Phys. 153(3), 479–519 (1993) · Zbl 0773.35074 [8] Kamvissis S., Teschl G.: Stability of periodic soliton equations under short range perturbations. Phys. Lett. A 364(6), 480–483 (2007) · Zbl 1203.35226 [9] Kamvissis, S., Teschl, G.: Stability of the periodic Toda lattice under short range perturbations. arXiv: 0705.0346 · Zbl 1301.37055 [10] Krüger, H., Teschl, G.: Stability of the periodic Toda lattice in the soliton region. arXiv:0807.0244 · Zbl 1185.37168 [11] Muskhelishvili N.I.: Singular Integral Equations. P. Noordhoff Ltd., Groningen (1953) · Zbl 0051.33203 [12] Novokshenov V.Yu., Habibullin, I.T.: Nonlinear differential-difference schemes integrable by the method of the inverse scattering problem. Asymptotics of the solution for t. Sov. Math. Doklady 23/2, 304–307 (1981) [13] Prössdorf S.: Some Classes of Singular Equations. North-Holland, Amsterdam (1978) [14] Teschl G.: Inverse scattering transform for the toda hierarchy. Math. Nach. 202, 163–171 (1999) · Zbl 1120.37315 [15] Teschl, G.: On the initial value problem of the Toda and Kac-van Moerbeke hierarchies. In: Weikard, R., Weinstein, G. (eds.) Differential Equations and Mathematical Physics. 375–384, AMS/IP Studies in Advanced Mathematics, vol. 16, Am. Math. Soc., Providence (2000) · Zbl 1056.37083 [16] Teschl, G.: Jacobi Operators and completely integrable nonlinear lattices. Math. Surv. and Mon. 72, Am. Math. Soc., Rhode Island (2000) · Zbl 1056.39029 [17] Teschl G.: Almost everything you always wanted to know about the Toda equation. Jahresber. Deutsch. Math.-Verein. 103(4), 149–162 (2001) · Zbl 1039.37058 [18] Toda M.: Theory of Nonlinear Lattices. 2nd enl. edn. Springer, Berlin (1989) · Zbl 0694.70001 [19] Zhou X.: The Riemann–Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20(4), 966–986 (1989) · Zbl 0685.34021
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.