Introduction to nonlinear discrete systems: Theory and modeling

B. A. Umarov; E. N. Tsoy

REVIEW 2 major objections 4 minor 40 references

Introduction to nonlinear discrete systems: Theory and modeling

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proposes a self-contained, code-backed introduction to nonlinear discrete systems built on the DNLS equation, and argues that the same analytical and numerical sequence transfers to other lattice models.

desk verdict A useful teaching tutorial on the DNLS equation, but Eq. (2) contains a real sign error that makes Problem 3 ask students to prove a false invariant. read the letter →

arxiv 1908.01497 v1 pith:MHIWZMJJ submitted 2019-08-05 nlin.PS

E. N. Tsoy , B. A. Umarov This is my paper · ORCID

classification nlin.PS

keywords discretenonlinearSchrödingerequationsolitonsmodulationalinstabilityanti-continuumlimitlinearstabilityanalysiswaveguidearraysBose-EinsteincondensatesMathematicateachingcode

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that the discrete nonlinear Schrödinger (DNLS) equation is a good universal model for teaching nonlinear discrete systems. It walks through the full analysis pipeline: plane-wave solutions and their dispersion relation, modulational instability, approximate and numerically exact discrete solitons, and linear stability. Every step is implemented in a Mathematica program, so a student can reproduce the figures and then modify the code for other systems. The intended payoff is that the basic toolkit of nonlinear lattice physics becomes accessible without needing specialist monographs.

What carries the argument

The load-bearing object is the discrete nonlinear Schrödinger (DNLS) equation, a nearest-neighbor lattice model with cubic onsite nonlinearity. The argument is carried by a small set of ansätze: the plane wave $u_k=a\,e^{i(qk-\omega t)}$ for dispersion, the modulated plane wave $u_k=(a+b_k(t))e^{i(qk-\omega t)}$ for modulational instability, the stationary mode $u_k=U_k e^{-i\eta t}$ for solitons, and the perturbed stationary mode $u_k=(U_k+w_k(t))e^{-i\eta t}$ for stability. Substituting each ansatz and linearizing in the small quantities reduces the problem to algebraic or eigenvalue equations, and the anti-continuum limit provides the localized seed that the Newton-Raphson solver (Mathematica's FindRoot) needs. The stability verdict is read from the complex frequencies of the final eigenvalue problem: any nonzero imaginary part means instability.

What would settle it

Run the supplied Mathematica code with $\beta=0.5$, $\gamma=1$, and $\eta=-2$: the on-site soliton should stay stationary with maximum imaginary part of the modulation eigenvalues around $10^{-7}$, the inter-site soliton should be unstable, and a noise-modulated plane wave with $\epsilon=10^{-3}$ should break into soliton-like pulses; failure of any of these outputs would show the tutorial does not deliver its central claim.

Watch

Extended reading notes

Core claim

The authors claim that the main ideas and methods of nonlinear discrete-system theory can be presented through the DNLS equation $i\,du_k/dt + \beta(u_{k+1}+u_{k-1}) + \gamma|u_k|^2u_k=0$, and that the analysis sequence is generic. They derive the plane-wave dispersion relation $\omega = -(2\beta\cos q + \gamma a^2)$, the modulational-instability condition $(\Omega - 2\beta\sin Q\sin q)^2 = 8\beta\sin^2(Q/2)\cos q\,[2\beta\sin^2(Q/2)\cos q - \gamma a^2]$, the stationary soliton equation $-\beta(U_{k+1}+U_{k-1}) - \gamma|U_k|^2U_k = \eta U_k$, and a linear-stability eigenvalue problem for small perturbations. The paper also uses the continuous limit and the anti-continuum limit $\beta=0$ to construct soliton approximations, and supplies Mathematica code covering every one of these steps.

Load-bearing premise

The load-bearing premise is that the toolkit demonstrated on the DNLS equation is generic enough to transfer to other discrete systems, which the paper asserts in Section 4 but does not demonstrate in detail.

Editorial extensions

If this is right

A student who follows the paper and runs the Mathematica code can reproduce the dispersion relation, the modulational-instability growth, and the soliton stability spectra for the stated parameters.
The same five-step sequence (plane wave, dispersion, modulational instability, stationary soliton, linear stability) can be applied to other discrete systems, as the paper claims in Section 4 and demonstrates via the suggested problems.
The code is written for nearest-neighbor equations and can be extended to more general couplings, different nonlinearities, and higher-dimensional arrays.
Modulational instability of a nonlinear plane wave leads to break-up into soliton-like pulses, which is how the paper motivates the relevance of discrete solitons.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The Mathematica code could be refactored into a reusable template: changing the coupling term and nonlinearity in MC.1 and re-running MC.2–MC.10 would give a working stability analysis for other lattice models without re-deriving the linear algebra.
The eigenvalue formulation suggests a natural extension to two- and three-dimensional arrays by altering the neighbor indexing in the code, a step the paper mentions but does not carry out.
The paper's implicit pedagogical claim is testable: comparing students who work through this DNLS pipeline against students who only see linear lattice theory would show whether the nonlinear toolkit transfers as claimed.
Because the anti-continuum limit provides the numerical seed, the method is especially well suited to strongly localized modes, which is where discrete effects differ most from continuum solitons.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a teaching paper, not a research contribution, and the main derivations are mostly standard and correct. The one serious problem is Eq. (2): the Hamiltonian as written, with the +γ/2 term, is not conserved by Eq. (1). I checked this by direct differentiation on a simple four-site example with β=γ=1 and initial data (1,2i,0,0); dH/dt comes out to 24, not zero. The sign of the nonlinear term is flipped. That makes Problem 3, which asks students to prove P and H are invariants, a false statement. For a pedagogical paper, this is not cosmetic.

What the paper does well: it walks through plane waves, the modulational instability condition, approximate solitons, the anti-continuum limit, and the linear stability analysis, all with a working Mathematica script that reproduces the figures. The derivations are self-contained, the cited literature is appropriate, and the numerical methods (Newton–Raphson, linearized eigenvalues) are standard. For a lecturer who wants a ready-made module on nonlinear discrete systems, this is a solid starting point.

The other soft spots are minor. The statement that "there is no exact DS solution" is a bit loose; there are no simple closed-form solutions, but the paper itself computes exact stationary solutions numerically. The claim that the approach transfers to other discrete systems is plausible but not demonstrated, which is acceptable for a tutorial.

The dispersion relation, MI condition, and stability equations check out against the literature. The citation pattern is normal; there are no self-citation issues.

Bottom line: with the sign in Eq. (2) fixed (the γ/2 should be −γ/2, or equivalently the β term should be −β), this would be a decent teaching resource. As it stands, I would not want students to work through Problem 3 without rewriting it. Because the pedagogical framing is good and the flaw is easy to repair, it deserves a serious referee rather than a desk rejection.

Referee Report

2 major / 4 minor

Summary. The manuscript proposes a pedagogical introduction to nonlinear discrete systems, using the discrete nonlinear Schrödinger (DNLS) equation as a universal model. It covers plane-wave dispersion, modulational instability, discrete bright solitons, the anti-continuum limit, numerical construction of stationary solitons, and linear stability analysis, and it supplies a Mathematica script together with twelve problems. The paper is aimed at university courses and claims to provide a self-contained analytical and numerical toolkit that can be extended to other discrete systems.

Significance. If corrected, this would be a useful teaching resource for Eur. J. Phys.: the analytical derivations are standard, the Mathematica code is provided as a working supplement, and the problems extend the methods to impurity modes, surface modes, cubic-quintic, and Ablowitz-Ladik/Salerno models. The modulational-instability condition and the linear stability equations are consistent with the literature, and the code appears to reproduce known results. The paper does not claim new research results; its value lies in accessibility and reproducibility, which are genuine strengths.

major comments (2)

[Eq. (2), §3 Problem 3] The Hamiltonian H in Eq. (2) is not an invariant of Eq. (1). With β0=0, differentiating H = Σ[β|u_k-u_{k-1}|² + (γ/2)|u_k|⁴] along the flow gives a generally nonzero result. For a periodic N=4 chain with β=γ=1 and initial condition (1,2i,0,0), direct differentiation using Eq. (1) gives dH/dt=24. The correct conserved quantity is H' = Σ[β0|u_k|² + β|u_k-u_{k-1}|² - (γ/2)|u_k|⁴]. Because Problem 3 explicitly asks students to prove that P and H in Eq. (2) are invariants, this is a load-bearing error that must be corrected.
[Sec. 2.2, Eq. (8)] The continuum limit is misstated by a factor of two. Using the intended phase transformation u_k = w(x_k,t) e^{2iβt} and expanding u_{k+1}+u_{k-1} ≈ 2u_k + h²∂²u/∂x² in Eq. (1) gives i w_t + βh² w_xx + γ|w|²w = 0, not 2βh². Equation (9) is consistent with the coefficient β (for h=1), so Eq. (8) should be corrected, and the missing imaginary unit in the transformation w ≈ u_k exp(−2βt) should also be fixed.

minor comments (4)

[Abstract and §1] The phrase 'nonlinear distributed systems' should read 'nonlinear discrete systems'; the paper treats discrete lattices, not distributed (continuum) systems.
[Sec. 2.2, first sentence] The statement 'There is no exact DS solutions of the DNLS equation' is too absolute and should be qualified, for example as 'no exact closed-form bright-soliton solutions in the general nonintegrable case', since integrable variants such as the Ablowitz-Ladik equation in Problem 11 do possess exact soliton solutions.
[Problem 5, §3] Typographical error: 'staggered solions' should be 'staggered solitons'.
[Eq. (9), Sec. 2.2] The soliton width in Eq. (9) is written without the lattice spacing h; if h is retained, the width should be ν = h√(2β)/(A√γ).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all derivations are self-contained and no prediction reduces to a fitted input.

full rationale

The paper's analytical chain is carried out explicitly inside the manuscript: the dispersion relation (4) follows from substituting the plane-wave ansatz (3) into Eq. (1); the modulational-instability condition (7) follows from linearizing the perturbed solution (5); the stationary soliton equation (11) follows from the substitution uk(t)=Uk exp(-i eta t); and the stability analysis follows from linearizing around the numerically found soliton profile. No parameter is fitted to a subset of data and then used to predict a closely related quantity, and no 'prediction' is defined in terms of the model input. Citations to the DNLS literature are contextual or problem-suggesting rather than load-bearing derivations. The only self-citation by an author of this paper is reference [23] in Problem 10, and it is used only as a pointer for study, not as a justification of the paper's central claim. The suspected sign error in the Hamiltonian H in Eq. (2) is a correctness concern, not a circularity concern: Problem 3 asks students to verify the invariant, so the invariant is not being used as an input to derive the result. Because the derivations are self-contained, code-reproducible, and compared against known textbook results, no circular step is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new particles, forces, or conserved quantities. Its content is standard DNLS theory; the only additions are numerical illustrations and a teaching code.

assumptions (3)

domain assumption The DNLS equation (1) is a valid model for the physical systems listed: waveguide arrays, BEC in optical lattices, Josephson junction arrays, and others.
The introduction asserts these applications based on citations [1,4,6,12,13] rather than deriving them in this paper.
domain assumption Any initial field distribution with finite power P eventually splits into spreading waves and discrete solitons, justifying the focus on these modes.
Stated in Section 1 citing [2,5]; used to motivate the entire analysis but not proven in the paper.
standard math Newton-Raphson iteration converges to a physical soliton profile for the chosen initial conditions.
Standard numerical method, used in MC.3; the paper gives no convergence proof, which is normal for a tutorial.

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Cite this review

Pith. "Pith review of Introduction to nonlinear discrete systems: Theory and modeling." pith.science (2026). https://pith.science/paper/MHIWZMJJ

@misc{pith2026190801497,
  author       = {Pith},
  title        = {Pith review of: Introduction to nonlinear discrete systems: Theory and modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MHIWZMJJ}},
  note         = {Machine review of arXiv:1908.01497}
}

read the original abstract

An analysis of discrete systems is important for understanding of various physical processes, such as excitations in crystal lattices and molecular chains, the light propagation in waveguide arrays, and the dynamics of Bose-condensate droplets. In basic physical courses, usually linear properties of discrete systems are studied. In this paper we propose a pedagogical introduction to the theory of nonlinear distributed systems. The main ideas and methods are illustrated using a universal model for different physical applications, the discrete nonlinear Schr\"{o}dinger (DNLS) equation. We consider solutions of the DNLS equation and analyze their linear stability. The notions of nonlinear plane waves, modulational instability, discrete solitons and the anti-continuum limit are introduced and thoroughly discussed. A Mathematica program is provided for better comprehension of results and further exploration. Also, few problems, extending the topic of the paper, for independent solution are given.

Figures

Figures reproduced from arXiv: 1908.01497 by the authors.

**Figure 1.** Examples of systems, described by the DNLS equation. (a) An array of optical rib waveguides. Light in a waveguide (a spot at k = 0) is coupled with nearest neighbor waveguides. (b) Clouds of Bose-Einstein condensate, distributed in minima (k = −1, 0 and 1) of the periodic optical potential. a BEC in a periodic potential can be found in Ref. [13]. A BEC in optical lattices, as an ensemble of coherent objects, is acti… view at source ↗

**Figure 2.** The dynamics of a plane wave with random modulations. The plane wave breaks up due to MI into a set of soliton-like pulses [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Dynamics of (a) on-site and (b) inter-site discrete solitons, obtained from numerical simulations of Eq. (1). The system parameters are β = 0.5, γ = 1, and η = −2. We can check numerically that the solution found is indeed the eigenmode. For this purpose, we integrate numerically Eq. (1) taking the solution of Eqs. (11) as the initial condition, see MC.4. We expect a stationary evolution at least for small t. Such n… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: (a) shows that numerical accuracy of eigenvalue calculations is of order 10−7 . -10-7 0 10-7 -3 -2 -1 0 1 2 3 Im Re (a) -2 -1 0 1 2 -3 -2 -1 0 1 2 3 Im Re (b) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

[1]

Each site is characterized by some varia bles, so that at a given time these variables specify a state of the system

Introduction A discrete system is a system that consists of several (or an inﬁnit e number of) well- separated points (sites). Each site is characterized by some varia bles, so that at a given time these variables specify a state of the system. A change of var iables on each site may depend on values on other sites. Many important physical systems, such a...

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Plane waves For discrete systems, it is instructive to start the analysis from a p lane wave solution

Analysis of waves in discrete systems 2.1. Plane waves For discrete systems, it is instructive to start the analysis from a p lane wave solution. We look for a solution in the following form: uk(t) = a exp[i(qk − ωt)], (3) where a, q, and ω are the amplitude, the wave number and the frequency of the plane wave. A substitution of Eq. (3) into Eq. (1) resul...

work page
[3]

The aim of the problems is t o help a deeper understanding of the methods reviewed in this paper, and a lso to develop skills in working with Mathematica program in Appendix A

Questions and Problems In this Section, we suggest few problems. The aim of the problems is t o help a deeper understanding of the methods reviewed in this paper, and a lso to develop skills in working with Mathematica program in Appendix A. Problems 1-7 c onsider the fundamental properties of plane waves and solitons. These pr operties are common for var...

work page
[4]

Conclusion The basic steps in the study of discrete systems have been presen ted. These steps include the construction of plane wave solutions and soliton solutions of the discrete system, the derivation of equations for small modulations, and ana lysis of stability. The corresponding code is implemented in Mathematica. It is demonst rated that theoretica...

work page
[5]

}; eqn= I*u[k]’[t] + β*(u[k+1][t]+u[k-1][t]) + γ*(u[k][t])∧ 2*Conjugate[u[k][t]]

Deﬁne parameters and the DNLS equation (eqn): nPoints=200; tend=50; par1= {β-> 0.5, γ-> 1, ω-> -2 }; par2= {c1-> 1., c2-> 1. }; eqn= I*u[k]’[t] + β*(u[k+1][t]+u[k-1][t]) + γ*(u[k][t])∧ 2*Conjugate[u[k][t]]

work page
[6]

Derive the stationary equation (eqnStat) from the DNLS equat ion : sub1= u[k ]-> Function[ {t}, U[k]*Exp[-I* ω*t]]; eqn1= Simplify[eqn /.sub1, Assumptions-> {{ω, t, U[i ]} ∈ Reals}]; eqnStat= Coefficient[eqn1, Exp[-I* ω*t]]

work page
[7]

Find eigenmode (eigenMode) from the list of stationary equations (eqnStatList): eqnStatList= Table[eqnStat, {k,1,nPoints}] /. {U[0]-> U[nPoints], U[nPoints+1]-> U[1] }; init= Table[ {U[k], (c1*KroneckerDelta[nPoints/2,k] + c2*KroneckerDelta[nPoints/2+1,k]) /.par2 }, {k,1,nPoints}]; eigenMode= FindRoot[(eqnStatList /.par1)==0, init]; init1= Table[U[k], {k,...

work page
[8]

Solve the DNLSE, using eigenmode as an initial condition: initialCond= Table[u[k][0]==init1[[k]], {k,1,nPoints}]; dnlsList= Join[Table[(eqn /.par1)==0, {k,1,nPoints}] /. {u[0][t]-> u[nPoints][t], u[nPoints+1][t]-> u[1][t] }, initialCond]; sol1= NDSolve[dnlsList, Table[u[k], {k,1,nPoints}], {t, 0, tend }, MaxSteps-> 1000000]; fig1= Evaluate[Table[Abs[u[k][...

work page

Show all 40 references

[9]

Derive an equation for the ﬁrst correction, w[i][t] : sub1= u[k ]-> Function[ {t}, (U[k] + ǫ*w[k][t]) Exp[-I* ω*t]]; res1= Coefficient[Simplify[eqn /.sub1, Assumptions-> {{ω, t, U[i ], ǫ} ∈ Reals, w[i ][t] ∈ Complexes}], Exp[-I* ω*t]]; eqnFirstCorr= Coefficient[Collect[res1, ǫ], ǫ, 1]
[10]

Split equation eqnFirstCorr into real and imaginary parts (wr, wi) : res1= Simplify[eqnFirstCorr /.w[k ] -> Function[ {t}, (wr[k][t]+I*wi[k][t])], Assumptions-> {wr[k ][t] ∈ Reals, wi[k ][t] ∈ Reals}]; eqnz1r= Simplify[ComplexExpand[Im[res1]]] eqnz1i= Simplify[ComplexExpand[Re[res1]]]
[11]

Find left-hand sides (lhs1r and lhs1i) of the equations for spatial distributions: sub2= {wr[k ]-> Function[ {t}, ar[k]*Exp[-I* Ω *t]], wi[k ]-> Function[ {t}, ai[k]*Exp[-I* Ω *t]]}; eqn1r= Expand[Coefficient[Simplify[eqnz1r /.sub2], Exp [-I*Ω *t], 1]]; eqn1i= Expand[Coefficie...
[12]

Construct a matrix of coeﬃcients in a diﬀerence form (matrF) an d a vector of unknowns (vectA): sub3= {ai[0]-> ai[nPoints], ai[nPoints+1]-> ai[1], ar[0]-> ar[nPoints], ar[nPoints+1]-> ar[1] }; matrFdiff= Join[Table[lhs1r, {k,1,nPoints}] /.sub3, Table[lhs1i, {k,1,nPoints}] /.su...
[13]

standard

Substitute eigenmode and parameters into matrFdiﬀ, and conve rt it to a “standard” form. Then ﬁnd eigenvalues of matrix F: m1= matrFdiff /.eigenMode /.par1; {bb1, matrF }= CoefficientArrays[m1, vectA]; ev1= Eigenvalues[Normal[matrF]]; Max[Im[ev1]]
[14]

Plot eigenvalues on the complex plane: ev1Fig= Table[ {Re[ev1[[i]]], Im[ev1[[i]]] }, {i,1,2*nPoints}]; ListPlot[ev1Fig, PlotRange-> All, PlotMarkers-> Graphics[{Blue,Thick,Circle[]}, ImageSize-> 8]]
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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Each site is characterized by some varia bles, so that at a given time these variables specify a state of the system

Introduction A discrete system is a system that consists of several (or an inﬁnit e number of) well- separated points (sites). Each site is characterized by some varia bles, so that at a given time these variables specify a state of the system. A change of var iables on each site may depend on values on other sites. Many important physical systems, such a...

work page

[2] [2]

Plane waves For discrete systems, it is instructive to start the analysis from a p lane wave solution

Analysis of waves in discrete systems 2.1. Plane waves For discrete systems, it is instructive to start the analysis from a p lane wave solution. We look for a solution in the following form: uk(t) = a exp[i(qk − ωt)], (3) where a, q, and ω are the amplitude, the wave number and the frequency of the plane wave. A substitution of Eq. (3) into Eq. (1) resul...

work page

[3] [3]

The aim of the problems is t o help a deeper understanding of the methods reviewed in this paper, and a lso to develop skills in working with Mathematica program in Appendix A

Questions and Problems In this Section, we suggest few problems. The aim of the problems is t o help a deeper understanding of the methods reviewed in this paper, and a lso to develop skills in working with Mathematica program in Appendix A. Problems 1-7 c onsider the fundamental properties of plane waves and solitons. These pr operties are common for var...

work page

[4] [4]

Conclusion The basic steps in the study of discrete systems have been presen ted. These steps include the construction of plane wave solutions and soliton solutions of the discrete system, the derivation of equations for small modulations, and ana lysis of stability. The corresponding code is implemented in Mathematica. It is demonst rated that theoretica...

work page

[5] [5]

}; eqn= I*u[k]’[t] + β*(u[k+1][t]+u[k-1][t]) + γ*(u[k][t])∧ 2*Conjugate[u[k][t]]

Deﬁne parameters and the DNLS equation (eqn): nPoints=200; tend=50; par1= {β-> 0.5, γ-> 1, ω-> -2 }; par2= {c1-> 1., c2-> 1. }; eqn= I*u[k]’[t] + β*(u[k+1][t]+u[k-1][t]) + γ*(u[k][t])∧ 2*Conjugate[u[k][t]]

work page

[6] [6]

Derive the stationary equation (eqnStat) from the DNLS equat ion : sub1= u[k ]-> Function[ {t}, U[k]*Exp[-I* ω*t]]; eqn1= Simplify[eqn /.sub1, Assumptions-> {{ω, t, U[i ]} ∈ Reals}]; eqnStat= Coefficient[eqn1, Exp[-I* ω*t]]

work page

[7] [7]

Find eigenmode (eigenMode) from the list of stationary equations (eqnStatList): eqnStatList= Table[eqnStat, {k,1,nPoints}] /. {U[0]-> U[nPoints], U[nPoints+1]-> U[1] }; init= Table[ {U[k], (c1*KroneckerDelta[nPoints/2,k] + c2*KroneckerDelta[nPoints/2+1,k]) /.par2 }, {k,1,nPoints}]; eigenMode= FindRoot[(eqnStatList /.par1)==0, init]; init1= Table[U[k], {k,...

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[8] [8]

Solve the DNLSE, using eigenmode as an initial condition: initialCond= Table[u[k][0]==init1[[k]], {k,1,nPoints}]; dnlsList= Join[Table[(eqn /.par1)==0, {k,1,nPoints}] /. {u[0][t]-> u[nPoints][t], u[nPoints+1][t]-> u[1][t] }, initialCond]; sol1= NDSolve[dnlsList, Table[u[k], {k,1,nPoints}], {t, 0, tend }, MaxSteps-> 1000000]; fig1= Evaluate[Table[Abs[u[k][...

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