Stability of the periodic Toda lattice under short range perturbations
📝 Abstract
We consider the stability of the periodic Toda lattice (and slightly more generally of the algebro-geometric finite-gap lattice) under a short range perturbation. We prove that the perturbed lattice asymptotically approaches a modulated lattice. More precisely, let $g$ be the genus of the hyperelliptic curve associated with the unperturbed solution. We show that, apart from the phenomenon of the solitons travelling on the quasi-periodic background, the $n/t $-pane contains $g+2$ areas where the perturbed solution is close to a finite-gap solution in the same isospectral torus. In between there are $g+1$ regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free lattice ( $g=0 $) the isospectral torus consists of just one point and we recover the known result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve. Our method relies on the equivalence of the inverse spectral problem to a matrix Riemann–Hilbert problem defined on the hyperelliptic curve and generalizes the so-called nonlinear stationary phase/steepest descent method for Riemann–Hilbert problem deformations to Riemann surfaces.
💡 Analysis
We consider the stability of the periodic Toda lattice (and slightly more generally of the algebro-geometric finite-gap lattice) under a short range perturbation. We prove that the perturbed lattice asymptotically approaches a modulated lattice. More precisely, let $g$ be the genus of the hyperelliptic curve associated with the unperturbed solution. We show that, apart from the phenomenon of the solitons travelling on the quasi-periodic background, the $n/t $-pane contains $g+2$ areas where the perturbed solution is close to a finite-gap solution in the same isospectral torus. In between there are $g+1$ regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free lattice ( $g=0 $) the isospectral torus consists of just one point and we recover the known result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve. Our method relies on the equivalence of the inverse spectral problem to a matrix Riemann–Hilbert problem defined on the hyperelliptic curve and generalizes the so-called nonlinear stationary phase/steepest descent method for Riemann–Hilbert problem deformations to Riemann surfaces.
📄 Content
A classical result going back to Zabusky and Kruskal [46] states that a decaying (fast enough) perturbation of the constant solution of a soliton equation eventually splits into a number of “solitons”: localized travelling waves that preserve their shape and velocity after interaction, plus a decaying radiation part. This is the motivation for the result presented here. Our aim is to investigate the case where the constant background solution is replaced by a periodic one. We provide the detailed analysis in the case of the Toda lattice though it is clear that our methods apply to other soliton equations as well.
In the case of the Korteweg-de Vries equation the asymptotic result was first shown by Šabat [37] and by Tanaka [40]. Precise asymptotics for the radiation part were first formally derived by Zakharov and Manakov [45] and by Ablowitz and Segur [1], [38] with further extensions by Buslaev and Sukhanov [5]. A detailed rigorous justification not requiring any a priori information on the asymptotic form of the solution was first given by Deift and Zhou [6] for the case of the modified Korteweg-de Vries equation, inspired by earlier work of Manakov [31] and Its [19] (see also [20], [21], [22]). For further information on the history of this problem we refer to the survey by Deift, Its, and Zhou [8].
A naive guess would be that the perturbed periodic lattice approaches the unperturbed one in the uniform norm. However, as pointed out in [25] this is wrong: In Figure 1 the two observed lines express the variables a(n, t) of the Toda lattice (see (1.1) below) at a frozen time t. In areas where the lines seem to be continuous this is due to the fact that we have plotted a huge number of particles and also due to the 2-periodicity in space. So one can think of the two lines as the evenand odd-numbered particles of the lattice. We first note the single soliton which separates two regions of apparent periodicity on the left. Also, after the soliton, we observe three different areas with apparently periodic solutions of period two. Finally there are some transitional regions in between which interpolate between the different period two regions. It is the purpose of this paper to give a rigorous and complete mathematical explanation of this picture. This will be done by formulating the inverse spectral problem as a vector Riemann-Hilbert problem on the underlying hyperelliptic curve and extending the nonlinear steepest descent method to this new setting. While Riemann-Hilbert problem on Riemann surfaces have been considered in detail before, see for example the monograph by Rodin [36], we extend this theory as well (see e.g. our novel solution formula for scalar Riemann-Hilbert problems in Theorem 4.3).
Consider the doubly infinite Toda lattice in Flaschka’s variables (see e.g. [15], [41], [42], or [44]) (1.1) ḃ(n, t) = 2(a(n, t) 2 -a(n -1, t) 2 ), ȧ(n, t) = a(n, t)(b(n + 1, t) -b(n, t)),
(n, t) ∈ Z × R, where the dot denotes differentiation with respect to time.
In case of a constant background the long-time asymptotics were first computed by Novokshenov and Habibullin [34] and later made rigorous by Kamvissis [23] under the additional assumption that no solitons are present. The full case (with solitons) was only recently presented by Krüger and Teschl in [28] (for a review see also [29]).
Here we will consider a quasi-periodic algebro-geometric background solution (a q , b q ), to be described in the next section, plus a short-range perturbation (a, b) satisfying (1.2) n∈Z n 6 (|a(n, t) -a q (n, t)| + |b(n, t) -b q (n, t)|) < ∞ for t = 0 and hence for all (see e.g. [11]) t ∈ R. The perturbed solution can be computed via the inverse scattering transform. The case where (a q , b q ) is constant is classical (see again [15], [41] or [44]), while the more general case we want here was solved only recently in [11] (see also [32]).
To fix our background solution, consider a hyperelliptic Riemann surface of genus g with real moduli E 0 , E 1 , …., E 2g+1 . Choose a Dirichlet divisor D μ and introduce
where A p0 (α p0 ) is Abel’s map (for divisors) and Ξ p0 , U 0 are some constants defined in Section 2. Then our background solution is given in terms of Riemann theta functions (defined in (2.14)) by
where ã, b ∈ R are again some constants.
We can of course view this hyperelliptic Riemann surface as formed by cutting and pasting two copies of the complex plane along bands. Having this picture in mind, we denote the standard projection to the complex plane by π.
Assume for simplicity that the Jacobi operator (1.5) H(t)f (n) = a(n, t)f (n + 1) + a(n -1, t)f (n -1) + b(n, t)f (n), f ∈ ℓ 2 (Z), corresponding to the perturbed problem (1.1) has no eigenvalues. In this paper we prove that for long times the perturbed Toda lattice is asymptotically close to the following limiting lattice defined by (1.6) ∞ j=n ( a l (j, t) a q (j, t)
where R is the associated reflection coefficient, ζ ℓ is a canonical basis of hol
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