How close Are Integrable and Non-integrable Models A Parametric Case Study Based on the Salerno Model Thudiyangal Mithun1Aleksandra Maluckov2Ana Man ci c3Avinash Khare4and Panayotis G. Kevrekidis1

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How close Are Integrable and Non-integrable Models:

A Parametric Case Study Based on the Salerno Model

Thudiyangal Mithun,1Aleksandra Maluckov,2Ana Manˇci´c,3Avinash Khare,4and Panayotis G. Kevrekidis1

1Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515, USA

2COHERENCE, Vinˇca Institute of Nuclear Sciences, National Institute of the Republic of Serbia,

University of Belgrade, P. O. B. 522, 11001 Belgrade, Serbia

3COHERENCE, Dept. of Physics, Faculty of Sciences and Mathematics,

University of Niˇs, P.O.B. 224, 18000 Niˇs, Serbia

4Department of Physics, Savitribai Phule Pune University, Pune 411007, India

In the present work we revisit the Salerno model as a prototypical system that interpolates between

a well-known integrable system (the Ablowitz-Ladik lattice) and an experimentally tractable non-

integrable one (the discrete nonlinear Schr¨odinger model). The question we ask is: for “generic”

initial data, how close are the integrable to the non-integrable models? Our more precise formulation

of this question is: how well is the constancy of formerly conserved quantities preserved in the non-

integrable case? Upon examining this, we ﬁnd that even slight deviations from integrability can be

sensitively felt by measuring these formerly conserved quantities in the case of the Salerno model.

However, given that the knowledge of these quantities requires a deep physical and mathematical

analysis of the system, we seek a more “generic” diagnostic towards a manifestation of integrability

breaking. We argue, based on our Salerno model computations, that the full spectrum of Lyapunov

exponents could be a sensitive diagnostic to that eﬀect.

I. INTRODUCTION

The topic of nonlinear dynamical lattices and energy

localization in them has been prevalent in a large ar-

ray of studies over the past few decades [1, 2]. Indeed,

since the proposal of intrinsic localized modes in anhar-

monic crystals [3, 4], there has been an ever expanding

range of disciplines where relevant states and their im-

plications are being identiﬁed, explored and dynamically

exploited [5]. Among the numerous associated examples,

one can list arrays of waveguides in nonlinear optics [6],

Bose-Einstein condensates in optical lattices [7], manip-

ulation of localization in micromechanical oscillator ar-

rays [8], granular crystals in materials science [9, 10], lat-

tices of electrical circuits [11], and many others including

layered antiferromagnetic crystals [12, 13], Josephson-

junction ladders [14, 15], or dynamical models of the

DNA double strand [16].

In many of these works, part of the emphasis has

been on localization and nonlinear wave structures [2,

5, 17, 18]. Important associated questions involve the

existence, dynamical stability and nonlinear dynamics of

the relevant waveforms. A parallel line of activity that

has also been central from early on has been that of po-

tential long-time ergodicity of the nonlinear lattice dy-

namical systems [1, 19]. In the latter, there have been

signiﬁcant developments in recent times, where compu-

tational resources have enabled far longer time simula-

tions of diﬀerent classes of such systems [20–22] and the

development of novel systems that are more straightfor-

ward to simulate over long times [23]. Interstingly, the

birth of the scientiﬁc ﬁeld examining nonlinear wave (soli-

tonic) structures has been strongly connected with such

ergodicity-related quests [24, 25].

The concept of integrability [26, 27] is one that is cen-

tral to both of the above directions of study. On the one

hand, the development of the inverse scattering trans-

form and the identiﬁcation of solitonic structures for a

number of these equations has been a key development

in nonlinear wave dynamics [26, 27], while on the other

hand, the inﬁnite conservation laws and associated con-

straints that such systems impose on the dynamics have

signiﬁcant bearings on the ability of the system to ex-

plore its phase space. Moreover, often integrability has

been a “helpful hand” towards trying to understand the

dynamics of weakly non-integrable systems through ap-

proaches involving perturbation theory [28]. Here, often

an eﬀective adiabaticity assumption is implied, i.e., that

the structures of the integrable (or analytically tractable)

limit are preserved but their features (e.g., amplitude,

width, speed, etc.) are modiﬁed and dynamically driven

by the non-integrable perturbations imposed. Indeed,

this proximity has been recently also of substantial math-

ematical interest through, e.g., the works of [29, 30].

In the present work, it is our intention to return to

the exploration of this topic of the eﬀective proximity of

integrable and weakly non-integrable systems. Indeed,

we leverage here a diﬀerent perspective from those of

works such as [29, 30] which focus on the (small) am-

plitude of the solution to gauge the relevant proxim-

ity. Rather, we deploy a comparison on the basis of

conservation laws of the original integrable system (see

also the work of [31]). Our aim is to explore more

broadly the phase space of the lattice dynamical sys-

tem and its constraints as we depart from the inte-

grable limit. As our platform of choice, we will uti-

lize the well-known so-called Salerno model [32], given

its natural interpolation between the well-established in-

tegrable variant of the nonlinear Schr¨odinger equation

(the so-called Ablowitz-Ladik (AL) limit) [33, 34] and

the non-integrable so-called DNLS (discrete nonlinear

Schr¨odinger) equation [18]. The advantage of this system

arXiv:2210.00851v1 [nlin.PS] 30 Sep 2022

is the availability of a homotopic parameter interpolat-

ing between these models and allowing us to explore the

departure from the integrable limit.

Our tool of choice will be the usage of conservation

laws of the AL limit initially. We will explore how “sen-

sitive” these are as probes of the breaking of integrability.

We will ﬁnd that indeed “former conservation laws” will

be very sensitive to departures from the relevant limit.

However, a disadvantage of this approach is that it re-

quires a deep mathematical or physical (or both) knowl-

edge of the concrete features of the system at hand. In

that light, it is desirable to have a more general toolbox

that is somewhat “system independent” in order to (sen-

sitively) probe such departures from the integrable limit.

In that vein, we explore the maximal Lyapunov exponent

and, indeed, the full Lyapunov spectrum of the system

of interest that can be generally computed [35, 36]. We

ﬁnd that this represents a very eﬃcient tool for detect-

ing the number of available conservation laws and hence

integrability of the system, indeed one that we expect in

the future to be amenable to eﬃcient computation, e.g.,

via machine-learning techniques.

Our presentation will be structured as follows. In sec-

tion II, we present the model of interest and its associated

conservation laws that we will probe both in the inte-

grable limit and systematically as we depart from that

limit. In section III, we present our results for the corre-

sponding conservation laws and their long-time dynam-

ics. In section IV, we discuss the computation of the

Lyapunov exponent spectrum, both as regards the maxi-

mal Lyapunov exponent and as regards the full spectrum

and present associated numerical results. In section V we

summarize our ﬁndings and present our conclusions, as

well as a number of directions for future study.

II. MODEL DESCRIPTION

The equation that we will consider in the present study

involves the well-established Salerno model [32], which

interpolates between the AL and the DNLS limits. The

relevant dynamical equation reads:

idψn

dt = (1 + µ|ψn|2)(ψn+1 +ψn−1)+γ|ψn|2ψn.(1)

This system has been a natural playground for the us-

age of perturbation theory methods oﬀ of the integrable

limit [37], for the examination of the delicate issue of

mobility in lattice dynamical systems [38], for the explo-

ration of collisions [31], and for the analysis of statistical

mechanical properties of nonlinear lattices [39], among

many others.

The AL model is well-known to be integrable via the

inverse scattering transform [34]. This implies the exis-

tence of an inﬁnite number of conserved quantities con-

sidered, e.g., in the work of [40], while the non-integrable

DNLS limit is characterized solely by two integrals of mo-

tion, namely the energy and the (squared) l2norm of the

ﬁeld. Indeed, the Salerno model inherits these two con-

servation laws. More speciﬁcally, regardless of the limits,

Eq. (1) can be characterized by two conserved quantities:

the (squared) norm Aand the Hamiltonian H, i.e., the

energy of the model [37, 41] in the form:

n=1

An,An=1

µln |1 + µ|ψn|2|

n=1 h−γ

µAn+ψnψ∗

n+1 +ψ∗

nψn+1 +γ

µ|ψn|2i,

(2)

where Nis the total number of lattice nodes and periodic

boundary conditions are used. Notice that the latter will

be an important point, especially when we consider ﬁnite,

small-size lattices, as integrability of the AL model is

preserved in the case of periodic boundary conditions,

although other types of integrable boundary conditions

may also exist [42]. It is also relevant to point out that

in the DNLS limit of µ→0, application of l’Hospital

(or a Taylor expansion in µ) leads to the ﬁrst conserved

quantity turning into the (squared) l2norm.

The dynamical equations of the Salerno model in the

form of Eqs. (1) can be derived from the Hamiltonian H

according to:

dψn

dt ={H, ψn}.(3)

with respect to the canonically conjugated pairs of vari-

ables ψnand iψ∗

ndeﬁning the deformed Poisson brackets

[43]

{ψn, ψ∗

m}=i(1+µ|ψn|2)δnm,{ψn, ψm}={ψ∗

n, ψ∗

m}= 0.

(4)

Among the inﬁnite conservation laws of the AL limit,

the two that we will focus on observing here are [40]:

C1=−µX

ψ∗

nψn−1(5)

C2=−µX

ψ∗

nψn−2(1 + µ|ψn−1|2+1

2(ψ∗

nψn−1)2).

These will be our monitored quantities (that, as we will

see, will be quite informative) and which, in the following,

will be denoted as moments. In general they are complex

quantities. Therefore we considered their real and the

imaginary parts, as well as the corresponding modulus.

It is relevant to point out that C1is often thought

of as a discrete version of the momentum, yet C2does

not have an immediate interpretation at a physical level.

As an additional relevant remark, the moment C1is not

suﬃcient in order to showcase the integrable limit here,

as, for instance, it is still conserved as a quantity in the

linear limit of µ=γ= 0 (not considered in detail herein).

On the other hand, the quantity C2is strictly conserved

in the AL integrable case only and hence the combination

of these two moments should be able to provide us a

clearer signature associated with the integrable limit in

what follows.

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HowcloseAreIntegrableandNon-integrableModels:AParametricCaseStudyBasedontheSalernoModelThudiyangalMithun,1AleksandraMaluckov,2AnaMancic,3AvinashKhare,4andPanayotisG.Kevrekidis11DepartmentofMathematicsandStatistics,UniversityofMassachusetts,AmherstMA01003-4515,USA2COHERENCE,VincaInstituteofNuclear...

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How close Are Integrable and Non-integrable Models A Parametric Case Study Based on the Salerno Model Thudiyangal Mithun1Aleksandra Maluckov2Ana Man ci c3Avinash Khare4and Panayotis G. Kevrekidis1.pdf

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How close Are Integrable and Non-integrable Models A Parametric Case Study Based on the Salerno Model Thudiyangal Mithun1Aleksandra Maluckov2Ana Man ci c3Avinash Khare4and Panayotis G. Kevrekidis1

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