On the evolution of scattering data under perturbations of the Toda lattice

Bilman, Deniz; Nenciu, Irina

doi:10.1016/j.physd.2016.03.017

Cited by 6 publications

(12 citation statements)

References 36 publications

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“…Proof. For ψ, φ ∈ ℓ 2 (Z) we have lim n→±∞ W n (ψ, φ) = 0, where W n (·, ·) is the Wronskian defined in (17). Using this together with Green's formula (16) from Exercise 2 implies that…”

Section: Exercise 1 Show That If the Lax Equationmentioning

confidence: 96%

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Discrete Integrable Systems, Darboux Transformations, and Yang–Baxter Maps

Bilman¹,

Konstantinou-Rizos²

2017

Symmetries and Integrability of Difference Equations

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ABSTRACT. These lecture notes are devoted to the integrability of discrete systems and their relation to the theory of Yang-Baxter (YB) maps. Lax pairs play a significant role in the integrability of discrete systems. We introduce the notion of Lax pair by considering the well-celebrated doubly-infinite Toda lattice. In particular, we present solution of the Cauchy initial value problem via the method of the inverse scattering transform, provide a review of scattering theory of Jacobi matrices, and give the Riemann-Hilbert formulation of the inverse scattering transform. On the other hand, the LaxDarboux scheme constitutes an important tool in the theory of integrable systems, as it relates several concepts of integrability. We explain the role of Darboux and Bäcklund trasformations in the theory of integrable systems, and we show how they can be used to construct discrete integrable systems via the Lax-Darboux scheme. Moreover, we give an introduction to the theory of Yang-Baxter maps and we show its relation to discrete integrable systems. Finally, we demonstrate the construction of YangBaxter maps via Darboux transformations, using the nonlinear Schrödinger equation as illustrative example.

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Section: Exercise 1 Show That If the Lax Equationmentioning

confidence: 96%

“…respectively, so that we have Ψ(z; n) = Φ(z; n)S(z) for all n ∈ Z. Differentiate both sides of (2.3.3) and obtain the time evolution S(z; t) (see the Appendix in [17]).…”

Section: Now We Define a New Quantity T(z) Bymentioning

confidence: 99%

Discrete Integrable Systems, Darboux Transformations, and Yang–Baxter Maps

Bilman¹,

Konstantinou-Rizos²

2017

Symmetries and Integrability of Difference Equations

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“…Here the square root √ λ 2 − 1 is defined to be positive for λ > 1 and σ ac (L) is the branch cut. Under this transformation, the a.c.-spectrum, [−1, 1], is mapped to the unit circle T and the eigenvalues λ j are mapped to ζ ±1 j , with ζ j ∈ (−1, 0) ∪ (0, 1) via (5) λ j = 1 2 ζ j + ζ −1 j , for j = 1, 2, . .…”

Section: 2mentioning

confidence: 99%

“…j is equivalent to computing the 2 (Z) eigenvalues of the doubly-infinite Jacobi matrix L that is defined in (3). We approximate the eigenvalues of L by computing eigenvalues of a L K that are outside the interval [−1, 1] for a large value of K [5]. This method might fail to capture eigenvalues of L that are close to its continuous spectrum.…”

Section: Numerical Computation Of the Scattering Datamentioning

confidence: 99%

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Numerical Inverse Scattering for the Toda Lattice

Bilman

Trogdon

2017

Commun. Math. Phys.

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We present a method to compute the inverse scattering transform (IST) for the famed Toda lattice by solving the associated Riemann-Hilbert (RH) problem numerically. Deformations for the RH problem are incorporated so that the IST can be evaluated in O(1) operations for arbitrary points in the (n, t)-domain, including short-and long-time regimes. No time-stepping is required to compute the solution because (n, t) appear as parameters in the associated RH problem. The solution of the Toda lattice is computed in long-time asymptotic regions where the asymptotics are not known rigorously.The authors wish to thank Percy Deift, Peter Miller, and Irina Nenciu for useful discussions and suggestions. The authors also thank the anonymous referees for their suggestions that greatly improved the readability of our paper. DB gratefully acknowledges the hospitality of Courant Institute of Mathematical Sciences, where the majority of this work was done. The authors acknowledge the partial support of the National Science Foundation through the NSF grants DMS-1150427 (DB) and DMS-1303018 (TT). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources. 1 We omit subscripts to refer to the functions defined on Z. 2 Many results for the Toda lattice hold with less restrictive choices for σ(n) [34]. For example, the inverse scattering transform method described below can be applied for data in the so-called Marchenko class (i.e., σ(n) = 1 + |n|). We impose exponential decay for the convenience of the numerical implementation.

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On the Asymptotic Stability of $${N}$$ N -Soliton Solutions of the Defocusing Nonlinear Schrödinger Equation

Cuccagna

Jenkins

2016

Commun. Math. Phys.

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We consider the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation for finite density type initial data. Using the ∂ generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of NLS for large times in the solitonic region of space-time, |x| < 2t, and we provide bounds for the error which decay as t → ∞ for a general class of initial data whose difference from the non vanishing background possesses a fixed number of finite moments and derivatives. Using properties of the scattering map of NLS we derive, as a corollary, an asymptotic stability result for initial data which are sufficiently close to the N -dark soliton solutions of NLS.

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On the evolution of scattering data under perturbations of the Toda lattice

Cited by 6 publications

References 36 publications

Discrete Integrable Systems, Darboux Transformations, and Yang–Baxter Maps

Discrete Integrable Systems, Darboux Transformations, and Yang–Baxter Maps

Numerical Inverse Scattering for the Toda Lattice

On the Asymptotic Stability of $${N}$$ N -Soliton Solutions of the Defocusing Nonlinear Schrödinger Equation

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