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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.)

Citations:

Zbl 0771.35042
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References:

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