Numerical Inverse Scattering for the Toda Lattice

Abstract

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.

Extreme superposition: Rogue waves of infinite order and the Painlevé-III hierarchy

“…The coefficients in the partial-fraction expansion of this rational ansatz are determined so that the jump condition produces a matrix in the interior domain that is consistent with the required analyticity and continuity at λ = ±i. It turns out that the Taylor coefficients of the entire function (16) at λ = ±i appear when these conditions are implemented, and in fact we can recognize these coefficients in the quantities F (x, t) and G (x, t) defined by (3). Thus it is possible to show the following.…”

“…The inner parametrices will have the following key properties: Explicitly substituting for the outer parametrix and using (from (179)) Im(z 0 (w)) = 1 3 √ 54 2/3 − w 2 completes the proof of the following result. 3 It is standard that for parabolic cylinder (resp., Airy) parametrices the mismatch error is proportional to the large parameter in the exponent, here T 1/3 , to the power −1/2 (resp. −1).…”

“…A plot of numerical jump contours encoding the deformations is given in Figure 15. A similar treatment for a Riemann-Hilbert problem with circular jump contours was done in [3,Sections 4.3 and 4.4], see also [18] and the references therein, in particular, [19]. The numerical routines that are used to generate the data in this work are available from the rogue-waves online repository 4 .…”

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

“…The following theorem from [121] locates the essential spectrum of L under an assumption that is weaker than (18). We present it in a version that is simplified for our setting and purposes.…”

Numerical inverse scattering for the sine-Gordon equation

“…In 2012, Trogdon, Olver and Deconinck implemented the numerical inverse scattering transform (NIST) for the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations [26]. The NIST is applied successfully to other integrable systems such as the focusing and defocusing nonlinear Schrödinger (NLS) equations [27] and the Toda lattice [9]. The NIST makes no domain approximation, does not require time-stepping and is uniformly accurate.…”