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Fast Numerical Nonlinear Fourier Transforms

Sander Wahls, H. Vincent Poor

arXiv:1402.1605v3cs.ITcs.NAmath.ITmath.NAnlin.SIphysics.comp-ph

TL;DR

A fast nonlinear Fourier transform for periodic NSE signals was missing despite prior work on rapidly decaying non-periodic signals. This paper presents two fast numerical methods, with the second achieving O(D log^2 D) complexity for the defocusing NSE.

  • Problem

    A fast nonlinear Fourier transform for periodic NSE signals was missing, despite prior work on rapidly decaying non-periodic signals.

    Source: §I · p. 2

  • Method

    The paper develops two fast numerical algorithms for the nonlinear Fourier transform of periodic signals governed by the NSE.

    Source: §X · p. 16

  • Results

    The algorithms have complexities O(D^2) and O(D log^2 D) flops, respectively, and both are about an order of magnitude better than comparable algorithms.

    Source: §X · p. 16

  • Takeaways & Limitations

    Numerical examples demonstrate the feasibility of the fast transforms, and the paper discusses extensions to other cases.

    Source: §X · p. 16

    The O(D log^2 D) algorithm applies only to the defocusing NSE.

    Source: §X · p. 16

Abstract

from arXiv · show

The nonlinear Fourier transform, which is also known as the forward scattering transform, decomposes a periodic signal into nonlinearly interacting waves. In contrast to the common Fourier transform, these waves no longer have to be sinusoidal. Physically relevant waveforms are often available for the analysis instead. The details of the transform depend on the waveforms underlying the analysis, which in turn are specified through the implicit assumption that the signal is governed by a certain evolution equation. For example, water waves generated by the Korteweg-de Vries equation can be expressed in terms of cnoidal waves. Light waves in optical fiber governed by the nonlinear Schr\"odinger equation (NSE) are another example. Nonlinear analogs of classic problems such as spectral analysis and filtering arise in many applications, with information transmission in optical fiber, as proposed by Yousefi and Kschischang, being a very recent one. The nonlinear Fourier transform is eminently suited to address them -- at least from a theoretical point of view. Although numerical algorithms are available for computing the transform, a "fast" nonlinear Fourier transform that is similarly effective as the fast Fourier transform is for computing the common Fourier transform has not been available so far. The goal of this paper is to address this problem. Two fast numerical methods for computing the nonlinear Fourier transform with respect to the NSE are presented. The first method achieves a runtime of $O(D^2)$ floating point operations, where $D$ is the number of sample points. The second method applies only to the case where the NSE is defocusing, but it achieves an $O(D\log^2D)$ runtime. Extensions of the results to other evolution equations are discussed as well.

I. INTRODUCTION … A. The Finite-Band Solutions of Kotlyarov and Its

The paper develops fast nonlinear Fourier transforms for periodic NSE signals, grounding the approach in finite-band solutions whose parameters form the scattering data. It reviews how these solutions are characterized and outlines the methods and extensions developed in the paper.

I. INTRODUCTION

  • The paper targets the missing computationally efficient fast NFT for periodic NSE signals, motivated by nonlinear frequency-division multiplexing for optical communications,,.NFDM encodes information in the scattering domain, where spatial evolution is simpler, and uses the NFT at the receiver; the paper focuses on periodic boundary conditions.Source: §I · p. 2
  • The paper carries rational monodromy-matrix approximation and fast algorithms into the periodic case, introduces new coordinate transforms, and presents a defocusing alternative.It compares the algorithms with current numerical approaches, gives numerical examples, and discusses extensions to non-periodic signals and other evolution equations.Source: §I · p. 2

Notation

  • The notation section defines the number sets, complex operations, derivatives, matrix operations, and polynomial degree conventions used in the paper.These include R, C, Z, Re, Im, complex conjugation, adjoints, matrix exponentials, and matrix products.Source: Notation · p. 2

II. FINITE BAND SOLUTIONS

  • Finite-band solutions are described by finitely many parameters that serve as scattering data; any sufficiently smooth periodic NSE solution can be approximated arbitrarily well on a fixed finite time interval.The paper reviews two types of finite-band solutions, each with advantages and disadvantages.Source: §II · p. 2

A. The Finite-Band Solutions of Kotlyarov and Its

  • Kotlyarov–Its finite-band solutions are generated by coupled equations for auxiliary-spectrum modes, whose motion between main spectral points gives them wave-like behavior [24], [25].Degenerate parameters can leave the solution unchanged; the paper assumes no main-spectrum point appears more than once to simplify later exposition.Source: A. The Finite-Band Solutions of… · pp. 2–3
  • A generated function solves the NSE if and only if the specified squared-eigenfunction construction yields a polynomial of finite degree.The squared eigenfunctions have simple spatial and temporal evolution and also provide a starting point for generating NSE solutions.Source: A. The Finite-Band Solutions of… · p. 3
  • For this function, periodicity in x with period ℓ holds if and only if e^(−i(λ1 + λ2)ℓ) = 1, equivalently ℓ/[2π(λ1 + λ2)] ∈ Z.Direct substitution shows that the function solves the NSE if and only if condition (17) is satisfied.Source: A. The Finite-Band Solutions of… · p. 3

B. The Finite Band Solutions of Ma and Ablowitz

The section describes Ma and Ablowitz’s finite-band solutions and their auxiliary spectra, distinguishing a simpler reconstruction formula from more complicated spectral evolution. It also notes that no known condition guarantees that arbitrary initial parameters yield an NSE solution.

B. The Finite Band Solutions of Ma and Ablowitz

  • Ma and Ablowitz’s finite-band solutions are specified through parameters, and their auxiliary spectra evolve according to a coupled differential system (also see).The supplied passages do not include the parameter definitions or the differential equations.Source: B. The Finite Band Solutions of Ma and… · p. 4
  • The reconstruction formula is simpler, but auxiliary-spectrum evolution is more complicated; temporal evolution equations are omitted, and sheet indices change when a spectrum reaches the main spectrum [35, Eqs. (4.2)+(8.4)].The corresponding finite-band signal is given, but its formula is not included in the supplied passage.Source: B. The Finite Band Solutions of Ma and… · p. 4
  • No known condition guarantees that arbitrary initial parameters produce an NSE solution, unlike for the finite-band solutions of Kotlyarov and Its.The main spectrum is the same as in Sec. II-A.Source: B. The Finite Band Solutions of Ma and… · p. 4
  • Even for one band (N = 1), finding the Ma–Ablowitz auxiliary spectra is complicated [35, Sec. 2.4].The general form of q(x, t) is referenced, but not shown in the supplied passage.Source: B. The Finite Band Solutions of Ma and… · p. 4

C. Construction of Finite-Band Solutions for Information Transmission in Optical Fiber · III. THE NONLINEAR FOURIER TRANSFORM

The section discusses constructing periodic finite-band NSE signals with prescribed scattering data for optical-fiber transmission, then reviews the spectral framework used to compute the nonlinear Fourier transform. Periodic signals offer fixed temporal support, while their construction and computation involve distinct challenges and methods.

C. Construction of Finite-Band Solutions for Information Transmission in Optical Fiber

  • Prior prescribed-data transmission constructions,,,,,,, target vanishing boundaries, leave degrees unused, and some degrees appear sensitivity-prone [9, VIII].With vanishing boundary conditions, temporal spread is difficult to control; pruning the signal set is the established remedy, feasible only with few degrees of freedom [10, Sec. V.C], [37, Sec. III].Source: C. Construction of Finite-Band… · p. 4
  • Periodic finite-band signals had not been discussed for optical transmission; their coupled parameters complicate construction, but their temporal support is fixed.The parameter coupling arises from the condition in Theorem 2.Source: C. Construction of Finite-Band… · p. 4
  • Theorem 2 offers a starting point for generating NSE solutions from parameters (3)–(6),, but enforcing periodicity, especially at a specified period, seems difficult.Source: C. Construction of Finite-Band… · p. 4
  • Darboux-Bäcklund methods construct periodic finite-gap NSE solutions with pre-specified complex main spectrum, and these methods and the theta-function representation allow period control, [44, Ch. 3.1], [45, Sec. 4.2], [46, Thm. 6.15].Controlling the period also controls temporal spread.Source: C. Construction of Finite-Band… · p. 5

III. THE NONLINEAR FOURIER TRANSFORM

  • The periodic finite-band NFT maps q(·, t0) and reference point x0 to scattering data given by either (3)–(6) or (19)–(23).The section reviews scattering-data computation to prepare for deriving numerical algorithms and introduces Lax-pair formalism to motivate spectral analysis.Source: §III · p. 5
  • The spectral review introduces the monodromy matrix and Floquet discriminant, then establishes the relation between the spectrum of Lt0 and the scattering data.Source: §III · p. 5
  • The operators discussed in Remark 6 are similar and have coincident eigenvalues.Source: §III · p. 5

A. The Lax Pair Formalism … 1) Squared Eigenfunctions in Terms of the Monodromy Matrix:

The section links the NSE’s Lax-pair structure and periodic Floquet analysis to monodromy-matrix methods for finite-band parameters and squared eigenfunctions.

A. The Lax Pair Formalism

  • A Lax pair links the NSE to an operator whose eigenvalues are time-invariant, while its eigenfunctions evolve according to the second operator,.The derivation of the second operator is omitted.Source: A. The Lax Pair Formalism · p. 5

B. Monodromy Matrix and Floquet Discriminant

  • The monodromy matrix captures the one-period evolution of canonical solutions and identifies spectral values admitting quasi-periodic eigenfunctions with specified period transitions.It plays a role analogous to the transfer matrix for vanishing boundary conditions.Source: B. Monodromy Matrix and Floquet… · p. 5
  • The Floquet discriminant is independent of the reference points x0 and t0.This independence is also stated in Lemma 8.Source: B. Monodromy Matrix and Floquet… · p. 5

C. Connection between Kotlyarov-Its Finite-Band Solutions and the Monodromy Matrix

1) Squared Eigenfunctions in Terms of the Monodromy Matrix:

  • Squared eigenfunctions can be evaluated indirectly from the monodromy matrix up to a commonly shared nonzero factor [24];, Sec. 4.2, Apdx. I;, Sec. 4.6.The theorem relates C(z)h_z(x0,t0) to the matrix entry [M_x0,t0(z)]_2,1.Source: 1) Squared Eigenfunctions in Terms of… · p. 6
  • The proof constructs products of eigenfunction components, uses their shared evolution equations and uniqueness to establish normalization equivalence, then connects them to the monodromy matrix.Source: 1) Squared Eigenfunctions in Terms of… · p. 6

2) Main Spectrum :

For finite-band solutions, the main spectrum is characterized by simple roots of 1 − ∆², while non-simple roots matter because perturbations can turn them into non-degenerate spectral points and affect stability.

2) Main Spectrum :

  • Under the stated finite-band and root-multiplicity assumptions, the main spectrum consists exactly of the simple roots of 1 − ∆².More generally, the main spectrum corresponds to simple (anti-)periodic eigenvalues of Lt0, where ∆(z) ∈ {±1}.Source: 2) Main Spectrum · p. 6
  • The paper identifies non-simple roots of 1 −∆2 as important, stating that they are essential for analyzing the impact of perturbations.Source: 2) Main Spectrum · p. 6
  • The main spectrum consists of complex-conjugate pairs,.The section also notes that realness of the λj follows when κ = −1, when Lt0 is self-adjoint.Source: 2) Main Spectrum · p. 6
  • Only finitely many non-simple non-real roots of ∆(z) ± 1 occur, so only finitely many degenerate points need consideration in stability analysis,.The stability relevance follows from non-real double roots being the ones that can lead to instabilities.Source: 2) Main Spectrum · p. 6

3) Auxiliary Spectrum : · D. Connection between Ma-Ablowitz Finite-Band Solutions and the Monodromy Matrix · IV. RATIONAL APPROXIMATIONS OF THE MONODROMY MATRIX

The sections connect the auxiliary spectrum and Ma–Ablowitz finite-band solutions to the monodromy matrix, then introduce rational approximations of that matrix from sampled signals.

3) Auxiliary Spectrum :

  • Zeros of the monodromy matrix’s upper-right entry identify the auxiliary spectrum, with degenerate parts canceled under the stated root assumptions.Lemma 17 assumes q(x0,t0) ≠ 0, condition (11), and roots of 1−∆² of at most double multiplicity; the roots of C²=(1−∆²)/P are the remaining double roots.Source: 3) Auxiliary Spectrum · p. 7

D. Connection between Ma-Ablowitz Finite-Band Solutions and the Monodromy Matrix

IV. RATIONAL APPROXIMATIONS OF THE MONODROMY MATRIX

  • The monodromy matrix is approximated rationally from D equidistant signal samples using a matrix-valued polynomial S(w) and scalar-valued polynomial d(w).These approximations are intended to make the matrix numerically tractable for the later fast NFT algorithms.Source: §IV · p. 7
  • A coordinate transform, typically Möbius, leaves exact-arithmetic results unchanged but can improve finite-precision numerical properties.Specific transform choices are discussed at the end of the section.Source: §IV · p. 7

A. Ansatz · B. Forward Euler Method · C. Crank-Nicolson

These sections introduce a sampled difference-equation approach to approximating the monodromy matrix, then illustrate it with Forward Euler and Crank–Nicolson schemes.

A. Ansatz

  • The monodromy matrix is expressed through an ansatz based on the differential-equation solutions.Source: A. Ansatz · p. 7
  • The approach replaces the differential equation with a difference equation using periodic signal samples at the sample points.The periodicity condition identifies the final grid point with the initial point.Source: A. Ansatz · p. 7

B. Forward Euler Method

  • Forward Euler is presented as a simple, rarely used scheme that illustrates the general rational approximation.Its discrete equation is solved stepwise to approximate the monodromy matrix.Source: B. Forward Euler Method · p. 8
  • In the Euler scheme, the coordinate transform is incorporated into the polynomials S(w) and d(w).Source: B. Forward Euler Method · p. 8

C. Crank-Nicolson

  • Crank–Nicolson approximates the differential equation’s right-hand side with a central difference and solves for the next step.Source: C. Crank-Nicolson · p. 8
  • Crank–Nicolson uses an ansatz for the monodromy matrix, defining d(w) and S(w) as products of stepwise terms.The matrix product is ordered from the final step back to the first.Source: C. Crank-Nicolson · p. 8

D. Ablowitz-Ladik Scheme … A. Description of the Algorithm

The paper describes the Ablowitz–Ladik scheme and coordinate choices for numerical stability, then introduces a fast NFT based on finite-dimensional eigenproblems and specifies its inputs and spectral outputs.

D. Ablowitz-Ladik Scheme

  • The Ablowitz–Ladik scheme is equivalent to forward Euler method (45) up to an error of O(ε^2), with α_n = 1 + O(ε^2).Source: D. Ablowitz-Ladik Scheme · p. 8
  • The monodromy matrix is approximated using ansatz (48); α_n normalization is reported to improve numerical properties in some cases, and discretization (48) is amenable to discrete inverse scattering.Source: D. Ablowitz-Ladik Scheme · p. 8

E. Heuristic for Choosing The Coordinate Transform

V. FAST NUMERICAL NONLINEAR FOURIER TRANSFORM BASED ON FINITE-DIMENSIONAL EIGENPROBLEMS

  • The proposed NFT approximates the main and auxiliary spectra through finite-dimensional eigenproblems, with complexity an order of magnitude lower than similar known matrix-eigenproblem algorithms.Source: §V · p. 9

A. Description of the Algorithm

  • The algorithm takes samples q(x_n,t_0), lets the user choose a rational monodromy-matrix approximation fitting Eq. (39), and outputs numerical main and auxiliary spectra.Source: A. Description of the Algorithm · p. 9

1) Find the Monomial Basis Expansion: · 2) Find the Main Spectrum: · 3) Find the Kotlyarov-Its Auxiliary Spectrum:

The sections describe fast monomial-basis expansion and root-based methods for approximating the main and Kotlyarov-Its auxiliary spectra. They give the methods’ complexity bounds and note roots that can be removed in a later step.

1) Find the Monomial Basis Expansion:

  • O(D log^2 D) flops suffice to compute monomial-basis coefficients of product-form polynomials S(w) and d(w).The normalization constants are intended to avoid overflows and keep the largest coefficients at similar magnitudes.Source: 1) Find the Monomial Basis Expansion · p. 9

2) Find the Main Spectrum:

  • Main-spectrum candidates are roots of two numerator polynomials, found through companion matrices constructed from their monomial-basis expansions.The required expansions can be computed in O(KD) flops; roots canceled by roots of the denominator are excluded.Source: 2) Find the Main Spectrum · p. 9
  • O(K^2D^2) flops is the worst-case complexity for finding the numerical main spectrum, including denominator-root checks and the coordinate transform.The root-checking step can often be simplified when the denominator has only a few distinct roots.Source: 2) Find the Main Spectrum · p. 9

3) Find the Kotlyarov-Its Auxiliary Spectrum:

  • Auxiliary-spectrum root finding does not check whether roots belong to degenerated modes; such roots can be removed in Step 5 if desired.Source: 3) Find the Kotlyarov-Its Auxiliary… · p. 9
  • O(NK log^2(NK)) flops result from multiplying polynomials in a tree-wise fashion using FFT-based products across O(log N) levels.Intermediate products are normalized to avoid overflows and underflows.Source: 3) Find the Kotlyarov-Its Auxiliary… · p. 10
  • Auxiliary-spectrum candidates are roots of the numerator [ˆS(w)]1,2 that are not canceled by roots of ˆd(w), then mapped by ϕ to obtain ˆµj(x0, t0).Using the same root finder as for the main spectrum requires O(K^2D^2) flops.Source: 3) Find the Kotlyarov-Its Auxiliary… · p. 10

4) Find the Ma-Ablowitz Auxiliary Spectra: … C. Comparison With Search Methods

The paper describes how to compute and filter numerical auxiliary spectra, then compares finite-dimensional eigenmethods and root-search methods with its proposed algorithm. Search methods are most suitable when prior knowledge is precise and the number of roots is small relative to the sample count.

4) Find the Ma-Ablowitz Auxiliary Spectra:

  • O(K^2D^2) flops suffice to compute Ma-Ablowitz auxiliary spectra using the same root finder as in Sec. V-A2.The numerical auxiliary spectra are defined from uncanceled numerator roots of the approximated functions.Source: 4) Find the Ma-Ablowitz Auxiliary… · p. 10

5) Filter the Numerical Spectra:

  • Discretization can create artifacts in numerical spectra, so use available a-priori spectral knowledge to remove inconsistent points; if removing a degenerate main-spectrum pair, remove its matching auxiliary-spectrum point.Detected double roots may be removed to eliminate corresponding degenerate hyperelliptic modes.Source: 5) Filter the Numerical Spectra · p. 10
  • Root cancellations are usually unnecessary for the Sec. IV discretizations: denominator roots are separated from the spectrum, absent for Ablowitz-Ladik, or cluster at infinity in original coordinates as ε shrinks.For Euler and Crank-Nicolson with the specified Möbius transforms, denominator roots cluster around w = −1 as ε becomes small.Source: 5) Filter the Numerical Spectra · p. 10

B. Comparison With Other Finite-Dimensional Eigenmethods

  • Euler discretization yields a 2D generalized eigenproblem; solving separate periodic and antiperiodic problems with standard methods such as QZ approximates the main spectrum.The approach can approximate auxiliary spectra with modified boundary conditions, and other discretizations, including Fourier-domain ones, are possible.Source: B. Comparison With Other… · pp. 10–11
  • Standard eigenvalue solvers for discretized spectral problems require O(D^3) complexity, compared with O(D^2) flops for the proposed algorithm.The paper also notes that the Floquet-Fourier-Hill method has O(D^3) complexity.Source: B. Comparison With Other… · p. 11

C. Comparison With Search Methods

  • Newton search locates roots of the approximated discriminant minus or plus one, but requires the monodromy-matrix derivative and initial guesses that find all roots of interest.A stopping criterion can use the change between successive iterates; Newton’s method does not return root multiplicities.Source: C. Comparison With Search Methods · p. 11
  • O(PD^2) flops is the worst-case search cost in the communication scenario: each Newton step costs O(D), with up to O(D) iterations per root.The passage notes that average complexity is usually better.Source: C. Comparison With Search Methods · p. 11
  • Search methods are a good choice mainly when precise a-priori knowledge is available and the number of roots P is low compared with the sample count D.In the periodic case, the proposed energy-conservation stopping criterion involves other roots that must also be found; recent initialization methods cited in the paper do not achieve O(D^2) complexity.Source: C. Comparison With Search Methods · p. 11

VI. FAST NUMERICAL NONLINEAR FOURIER TRANSFORM FOR THE DEFOCUSING NSE BASED ON SAMPLING … VII. NUMERICAL EXAMPLES

For the defocusing NSE, the paper presents a sampling-based fast NFT for the real main and Ma-Ablowitz auxiliary spectra; the method does not cover the potentially nonreal Kotlyarov-Its auxiliary spectrum. It also compares the method with a related approach and reports proof-of-concept numerical examples.

A. Description of the Algorithm

1) Find the Monomial Basis Expansion:

2) Find the Main Spectrum

  • O(LGD log(GD)) flops suffice to find the numerical main spectrum when the non-equidistant fast Fourier transform is used.The NFFT accelerates evaluations of the discriminant; the remaining operations have negligible cost.Source: 2) Find the Main Spectrum · p. 12

3) Find the Ma-Ablowitz Auxiliary Spectra:

  • The Ma-Ablowitz auxiliary spectra are real roots of Ψ±(z), so the main-spectrum method finds them with the same O(LGD log(GD)) cost.The procedure replaces Δ̂ ± 1 with Ψ̂± as defined in Eq. (53).Source: 3) Find the Ma-Ablowitz Auxiliary… · p. 12

4) Filter the Numerical Spectra:

  • The defocusing NSE’s band structure can help identify numerical artifacts, but sampling-based root finding does not reveal root multiplicities or support root cancellations.The method therefore cannot detect cancellation unless the relevant roots are assumed simple; other root refinements, including Newton’s method, are possible.Source: 4) Filter the Numerical Spectra · p. 12

B. Comparison With A Similar Approach

  • The related method in has runtime O(LD2G2), versus O(LGD log(GD)+KD log2(KD)) for this implementation.The paper attributes the key speedup to using rational rather than irrational approximations of the monodromy matrix.Source: B. Comparison With A Similar Approach · p. 12

VII. NUMERICAL EXAMPLES

  • The numerical section presents two proof-of-concept examples: one evaluates the finite eigenmethod, and the other examines the specialized defocusing-NSE method.The authors state that comprehensive numerical or analytical study remains future work.Source: §VII · pp. 12–13

A. The Focusing Case … A. The AKNS Lax Pair

The section compares fast and standard numerical NFT methods on focusing and defocusing examples, then discusses extensions to rapidly decaying signals and other integrable evolution equations. The examples report runtime and accuracy behavior, while the proposed extensions remain numerically untested.

A. The Focusing Case

  • All four focusing-case errors decreased approximately linearly with D, and the fast algorithms had lower errors than standard algorithms using the same discretizations.Only a finite subset of the spectrum could be assessed; errors were higher when the region included the zero eigenvalue, but relative algorithm performance remained the same.Source: A. The Focusing Case · p. 13, Figure 2 · p. 14

B. The Defocusing Case

  • For the defocusing one-band solution, fast per-sample runtime grew very slowly with sample count versus linear growth for the naive implementation, with exactly the same errors.The fast algorithm also computed the Floquet diagram; the comparison used L = 5 bisection steps and oversampling factor G = 1.Source: B. The Defocusing Case · p. 13, Figure 3 · p. 15

2) Second Example: Gaussian Wavepacket:

  • For the Gaussian wavepacket, the fast algorithm’s Floquet diagram agreed with the result in [17, Fig. 2b], but its error could not be quantified because the analytical NFT seemed unknown.The parameters were q0 = 1.9, µ = 1, σ = 2, and ℓ = 10.Source: 2) Second Example: Gaussian Wavepacket · p. 13, Figure 4 · p. 15

VIII. RAPIDLY DECAYING SIGNALS

  • The fast transforms can extend to rapidly decaying non-periodic signals on the line, either with modifications or by truncating and periodically extending the signal.Periodic extension requires a sufficiently large period ℓ > 0 and additional transformations of the resulting spectra; modifications are described in.Source: §VIII · pp. 13–14

A. The Focusing Case

  • For the focusing one-band solution, the figure reports runtimes per sample and error, and notes that Std-CN was too slow for higher D.Source: Figure 1 · p. 14

IX. OTHER INTEGRABLE EVOLUTION EQUATIONS

  • The approach can extend beyond the NSE to other evolution equations, but the proposed extensions have not been numerically tested and their accuracy remains to be examined.The paper discusses the Ablowitz-Kaup-Newell-Segur Lax pair as an extension framework.Source: §IX · p. 14

A. The AKNS Lax Pair

  • The AKNS framework encompasses evolution equations beyond the NSE, and further equations can fit it through additional transformations or matrix-valued signals,,,,,.The described framework uses suitably chosen signals q and r and operator B.Source: A. The AKNS Lax Pair · p. 14

B. Finite-Band Solutions for the AKNS Lax Pair · C. Fast NFTs for the AKNS Lax Pair · X. CONCLUSION

The paper presents two fast numerical NFTs for the periodic nonlinear Schrödinger equation and discusses finite-band AKNS solutions and how the algorithms extend to general AKNS Lax pairs.

B. Finite-Band Solutions for the AKNS Lax Pair

C. Fast NFTs for the AKNS Lax Pair

  • For general AKNS Lax pairs, all results except Lemma 10, Lemma 15, and Sec. III-D carry over with the operator and Eq. (29) updated; scattering data also include initial η_j values.Those initial values are roots of [M_x0,t0(z)]_2,1.Source: C. Fast NFTs for the AKNS Lax Pair · p. 15
  • Adapting rational monodromy-matrix approximations requires minor numerical changes, chiefly changing P_z to restore equivalence between the eigenproblem and the differential system.Consecutive terms involving P_z must then be reevaluated.Source: C. Fast NFTs for the AKNS Lax Pair · pp. 15–16
  • After those updates, the proposed fast algorithms can run as before, and the methods for finding μ_j(x_0,t_0) can also find η_j(x_0,t_0).Source: C. Fast NFTs for the AKNS Lax Pair · p. 16

X. CONCLUSION

  • O(D^2) flops for the first transform and O(D log^2 D) for the defocusing-only second are about an order of magnitude better than comparable algorithms.The feasibility of the transforms was demonstrated in numerical examples.Source: §X · p. 16
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