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Numerical Construction of Quasi-Periodic Solutions for Nonlinear PDEs: II. Unbounded Perturbation

Mingwei Fu, Bin Shi

arXiv:2610.34194v1math.NAcs.NA

TL;DR

PDEs with unbounded perturbations have nonlinear vector fields involving spatial derivatives. The paper proposes alternating schemes for constructing quasi-periodic solutions, establishing them for NLAW on a large-measure set of low spatial modes within the invariant zero-mean-velocity class.

  • Problem

    A challenging class of nonlinear PDEs has unbounded perturbations whose nonlinear vector fields involve spatial derivatives.

    Source: §1 · pp. 1–2

  • Method

    The paper proposes alternating numerical schemes to construct quasi-periodic solutions for NLAW and KdV, updating frequencies and non-resonant coefficients through a truncated Newton step.

    Source: §1.2 · p. 6, §7 · p. 45

  • Results

    The results establish quasi-periodic NLAW solutions for a large-measure set of low spatial modes within the invariant zero-mean-velocity class.

    Source: §7 · p. 46

  • Takeaways & Limitations

    The reported mean-driven linearized energy-growth mechanism arises outside the invariant zero-mean-velocity class and does not contradict the prevalence of quasi-periodic solutions established there.

    Source: §7 · p. 47

    The Section 6.2 analysis requires the unbounded perturbation to be at least one order lower than the dispersive frequencies, and the traveling-wave construction does not directly extend to Dirichlet boundary conditions.

    Source: §7 · pp. 45–46

Abstract

from arXiv · show

Many nonlinear partial differential equations (PDEs) arising in physical models involve unbounded perturbations, where nonlinear terms contain spatial derivatives. Such perturbations can be expressed as compositions of unbounded operators with bounded nonlinear maps. In this paper, we propose alternating numerical schemes for constructing quasi-periodic solutions to this class of PDEs, including the nonlinear acoustic wave (NLAW) equation and the Korteweg-de Vries (KdV) equation. To render the nonlinear perturbations bounded, we apply inverse differential operators as a preconditioning; this, however, also rescales the linear part by polynomial factors in the spatial mode $n$. Two main difficulties arise. The first is to control the measure of the resulting ``bad'' frequencies, since the bounds on the measures of the excluded frequency sets are enlarged by polynomial factors in $n$; we show that this measure nevertheless remains small. The second concerns resonances involving the perturbed eigenvalues of the linearized operator at large scales. Our key observation is that, when the dispersive frequencies grow polynomially in $n$ with degree greater than one, such resonances cannot occur if the magnitude of the temporal index lies below a threshold proportional to $|n|$ raised to one less than that degree. It therefore suffices to consider the complementary regime, in which the growth induced by the unbounded perturbation can be controlled in terms of the temporal index. At large scales, an additional complication arises: for a given spatial index $n$, all temporal indices in a box may be ``bad''. We show, however, that at each scale there is at most one such spatial index. Consequently, the resulting counting loss is only polynomial and can be absorbed into the exponential-logarithmic threshold, allowing us once again to retain control over the measures of the excluded frequency sets.

1 Introduction

This introduction presents an alternating scheme for constructing quasi-periodic solutions of the NLAW equation with derivative-containing nonlinearities, alongside numerical examples and the Fourier formulation and norms used in the analysis. It states convergence and a priori error bounds for the scheme and outlines the analysis and its extension to KdV.

1.1 Fourier expansion of the NLAW equation

  • The target solution prescribes amplitudes on selected low-frequency tangential modes, while the drifted frequency vector and remaining Fourier coefficients are determined iteratively.The introduction frames NLAW as a PDE with derivative-containing nonlinearities and notes that KAM methods for unbounded perturbations had apparently not yet been applied to it.Source: §1 · pp. 1–2, §1.1 · pp. 3–4

1.2 Numerical scheme

  • The scheme divides the lattice equation by the nonlinear term’s n^2 factor and alternates between coupled frequency updates and truncated Newton updates of nonresonant Fourier coefficients.The Q/P systems determine the drifted frequencies and nonresonant coefficients together; the correction operator accounts for the frequencies’ dependence on those coefficients.Source: §1.2 · pp. 5–6

1.3 Numerical demonstration

  • At ε = 0.5, the tangential mode retains amplitude close to unity and a small frequency shift from the linear solution; nonlinear interactions excite both normal modes, while successive differences in Fourier coefficients, frequency, and the t = 4 solution decrease rapidly.Source: §1.3 · p. 7
  • For m = 2, the computed trajectories are consistent with quasi-periodic motion, and nonlinear frequency shifts and additional modes redistribute intensity while preserving the overall diagonal band structure.Some bands brighten while others weaken, and their spacing changes; a representative normal mode absent from the initial solution also develops small-amplitude oscillations.Source: §1.3 · pp. 9–10, Figure 4 · p. 9, Figure 5 · p. 10

1.4 The main theorem: a priori bound

  • The a priori-bound section introduces a norm incorporating frequency derivatives of the Fourier coefficients and a Gevrey decay set to quantify solution regularity.Source: §1.4 · pp. 10–11
  • The excluded initial-amplitude set has measure at most κ + δ, where κ = κ(m, τ) > 0 and δ(ε) tends to zero as ε tends to zero.The bound applies to the resonance set A⋆ ⊂ A.Source: §1.4 · p. 11
  • For amplitudes outside the excluded set, the scheme generates a well-defined sequence, and each updated frequency map is a local diffeomorphism.The initialization is (ω(0), û(0)_p) = (ω, 0).Source: §1.4 · p. 11
  • For sufficiently small ε and amplitudes outside the excluded set, the iterates converge to an exact solution pair whose limiting frequency map is a local diffeomorphism.The theorem assumes δ0 ∈ (0, 1), τ > m + 2, and A = [δ0, 1]^m.Source: §1.4 · p. 11
  • The theorem provides a priori iterate error bounds and a time-domain error bound; at each fixed t ≥ 0, the approximate solution converges in L2 to the exact quasi-periodic NLAW solution.The time-domain bound is stated for t ∈ [0, M_r].Source: §1.4 · p. 11

1.5 Organization of the paper

  • The paper proceeds from Gevrey-class operator properties and implementation conditions to multiscale analysis and the iterative lemma, then extends the results to periodic KdV and presents numerical demonstrations.Section 7 concludes with directions for future research.Source: §1.5 · p. 12

2 Gevrey decay properties

The section develops uniform Gevrey decay and norm estimates for the nonlinear vector field and its associated operators. It also establishes bounds needed to control frequency updates and the iteration.

2 Gevrey decay properties

  • The full-lattice formulation treats Fourier coefficients as independent variables, then imposes the reality symmetry by restricting to an invariant subspace.The extended space-time torus uses coordinates (x, θ).Source: §2 · p. 12
  • The stated estimates cover uniform decay and norm bounds for the vector field, symmetry and off-diagonal decay of S, and bounds for the auxiliary operator B.The B estimates are required to control the error in the frequency update.Source: §2 · pp. 12–13
  • Because the nonlinearity u^2 acts pointwise, its linearization is a convolution (Toeplitz) operator; the full-lattice form is more convenient for multiscale analysis.In half-lattice coordinates, the equivalent operator is Toeplitz-plus-Hankel.Source: §2 · pp. 12–13
  • The iteration lemma also requires a bound on the second derivative of the nonlinear vector field.The section states this estimate and assigns its proof to Appendix A.4.Source: §2 · p. 13
  • The propositions provide decay and norm estimates under a support restriction for coefficients in K(s).The supplied proposition statements specify the hypotheses and estimate scope but not the estimate values.Source: §2 · p. 13
  • The Q-equations bound the frequency drift and locally define a continuously differentiable frequency map that is a local diffeomorphism where ∂Q/∂a is invertible.The proof establishes invertibility of ∂Q/∂a using the uniform bound and the lower bound on each amplitude component.Source: §2 · p. 14

3 Implementation conditions

Section 3 specifies uniform invertibility and localization conditions for the scheme, then verifies them by excluding nearly resonant frequencies and analyzing restricted operators across scales. The small-scale estimates provide the base case for the large-scale induction.

3 Implementation conditions

  • At each scale, the restricted operator must be invertible with a controlled inverse and Gevrey decay, uniformly over spatial centers.The uniform conditions are established by excluding bad frequencies; the dispersion relation keeps the measure of the associated nearly resonant sets sufficiently small.Source: §3 · p. 15

3.1 Nearly-resonant sets

  • The nearly resonant set combines tangential, single-mode, and difference resonances, and its measure admits a bound for fixed τ > m + 2.The analysis uses an explicit remainder estimate for the first-order asymptotic expansion of the normal frequencies.Source: §3.1 · p. 16

3.2 Restricted operators on small boxes

  • For M0 ≤ N ≤ M, the restricted operator is invertible with uniform norm bounds and Gevrey decay of its off-diagonal entries.The proof uses a uniform lower bound on non-resonant diagonal entries and a Neumann series.Source: §3.2 · p. 17

3.3 Verification of the uniform conditions

  • For N > M, induction extends the restricted-operator bounds to large scales using outer-box inverses and multiscale estimates.In the analytic case s = 1, the argument requires a stronger smallness condition on ε; for fixed 0 < s < 1, it applies as stated.Source: §3.3 · pp. 17, 20

4 Multi-scale analysis

The section develops a multiscale analysis to establish Theorem 3.3 for outer boxes, combining small-box localization with large-scale analysis of perturbed eigenvalues and excluded frequencies. It clusters large-scale singular sites in temporal indices and extends inversion-to-localization through multiscale gluing, showing that the combined excluded-frequency set has measure tending to zero.

4.1 Inductive base for small boxes

  • Singular sites in a small box form either a tangential or a normal resonant configuration, never a mixture of both.The tangential configuration shares a common shifted temporal index; the normal configuration is contained in a pair of sites with spatial indices ±n⋆ and a common temporal index.Source: §4.1 · p. 22
  • For small scales, an inverse bound on the full-box operator yields Gevrey decay of its inverse entries, uniformly in the spatial center.For widely separated normal singular sites, the argument uses subboxes of scale N^1/10, each containing at most one singular site, then glues the local inverse estimates.Source: §4.1 · pp. 24–25

4.2 Scale reduction: temporal-index clustering of singular sites

  • At large scales, the diagonal operator may not dominate, so the analysis estimates the perturbed eigenvalues of T directly.For NLAW, the square-root factorization bounds the perturbation and its frequency derivative, making measure exclusion essentially the same as in the bounded-perturbation setting of Fu and Shi [2026c]; the KdV analysis differs, as discussed in Section 6.2.Source: §4.2 · pp. 25–26
  • For distinct spatial indices, the large-scale difference-resonance exclusions have total measure tending to zero as ε tends to zero.The one-dimensional resonant-set measure along the temporal direction is bounded using the directional derivative, and summing the scale estimates gives the vanishing measure.Source: §4.2 · p. 27
  • Because the perturbation is unbounded, a single spatial index may have a bad temporal line on which every temporal index in the box can have a small eigenvalue.The section identifies this as an additional bad-frequency case requiring exclusion; it does not give the exclusion estimate in the supplied passages.Source: §4.2 · p. 28
  • The additional bad-frequency set Gadd has measure tending to zero, leaving the outer lattice box Γn0,k0,N after its exclusion.Singular centers are clustered in the temporal indices; the clustering does not further restrict their spatial indices.Source: §4.2 · p. 29

4.3 Multi-scale induction: inversion implies localization

  • For scales N > M, inversion implies localization by gluing local inverses across singular boxes with singular sites removed and regular boxes.The induction uses the small-scale singular- and regular-box results as its base.Source: §4.3 · p. 30

4.4 Measure estimate for “bad” frequencies

  • The combined excluded set G = G1 ∪ G2 ∪ Gadd satisfies mes(G) ≤ δ, with δ → 0 as ε → 0.The estimate combines the single-mode resonant-set bounds with the bound for G2; the single-mode estimates are obtained using a Schur complement and a Cartan estimate.Source: §4.4 · pp. 31–33

5 Convergence of the numerical scheme

The iteration lemma shows that the alternating update preserves the required support and decay properties while driving the residual toward super-convergence. The resulting iterates converge to an exact solution with controlled regularity and time-domain error.

5.1 The iteration lemma

  • The derivative of the inverse restricted operator has a norm bound and Gevrey-type decay in its entries under Condition 1.The inverse-derivative estimates support the iteration lemma.Source: §5.1 · p. 33
  • The alternating update preserves the required support and lattice-vector decay, and yields a super-convergence bound for the residual.The proof controls the post-update residual using truncation error and quadratic iteration terms.Source: §5.1 · pp. 34–36
  • Cumulative decay keeps the approximate solutions uniformly bounded in Gevrey space, with no loss of regularity as the lattice size grows.The Fourier-coefficient localization is retained throughout the iteration.Source: §5.1 · p. 36

5.2 Proof of the main theorem

  • The approximate solutions form a Cauchy sequence in Gevrey space and converge to a limit with a stated convergence rate.The proof then establishes frequency and time-domain error bounds for the limit and iterates.Source: §5.2 · pp. 36–37

6 The KdV equation

The KdV section constructs periodic quasi-periodic solutions using an alternating scheme on Fourier coefficients and drifted frequencies. It establishes convergence and a priori bounds, then illustrates the computed solutions and their convergence numerically.

6 The KdV equation

  • The KdV analysis restricts to zero mean; a nonzero constant mode would add a first-order transport term and shift the linear frequencies.The case with nonzero constant mode is discussed at the end of the paper.Source: §6 · p. 38

6.1 Numerical scheme

  • The alternating scheme deforms the linear solution toward a nonlinear quasi-periodic solution by updating drifted frequencies and non-resonant Fourier coefficients.The resonant Fourier coefficients are prescribed; the remaining coefficients and frequency vector are determined successively through iteration.Source: §6.1 · pp. 39–40, §6 · p. 39

6.2 The main theorem: a priori bound

  • For the low-temporal-index regime, the resonance condition cannot hold for sufficiently small ε, so no bad frequencies need be excluded there.The argument instead controls the complementary regime, where the temporal index is large relative to the spatial modes.Source: §6.2 · p. 41
  • The theorem bounds the excluded amplitude set by κ + δ, with δ tending to zero as ε tends to zero, and establishes convergence of the alternating iterates to an exact solution pair.The limiting frequency map is a local diffeomorphism, and the theorem also provides a priori error bounds.Source: §6.2 · pp. 41–42

6.3 Numerical demonstration

  • In the m = 1 test, the tangential mode retains nearly unit amplitude but shifts in frequency, while normal modes develop small oscillations; all three convergence measures decrease rapidly.The convergence profiles cover Fourier coefficients, temporal frequency, and the reconstructed solution at t = 4.Source: Figure 8 · p. 43, §6.3 · p. 43
  • For m = 2, nonlinear mode trajectories are modulated, and the spatiotemporal intensity retains diagonal bands while bright-region contrast and band inclination change.The computed solution's tangential modes depart differently from the initial linear trajectories, while a representative normal mode develops small oscillations.Source: Figure 10 · p. 45, §6.3 · p. 45

7 Conclusion and further work

The paper constructs quasi-periodic solutions for NLAW and KdV with periodic boundary conditions, while identifying restrictions on the construction and directions for further work. Its large-measure result applies to NLAW within the invariant zero-mean-velocity class.

7 Conclusion and further work

  • The alternating schemes construct quasi-periodic solutions for NLAW and KdV under periodic boundary conditions.The constructed solutions have a traveling-wave rather than standing-wave structure.Source: §7 · p. 45
  • Figure 11 compares the linear and nonlinear KdV squared-modulus distributions, with the nonlinear solution obtained by the alternating scheme.The figure depicts spatiotemporal distributions under periodic boundary conditions.Source: Figure 11 · p. 46
  • The construction does not directly extend to Dirichlet boundary conditions, and the analysis requires the unbounded perturbation to be at least one order below the dispersive frequencies.A cubic NLAW nonlinearity may allow a similar Dirichlet construction, but its physical relevance remains unclear.Source: §7 · p. 46
  • The NLAW large-measure result covers low-spatial modes within the invariant zero-mean-velocity class.For nonzero b1, integrating over the periodic spatial domain gives uhat_0 = b0 + b1t, which rules out quasi-periodicity.Source: §7 · p. 46
  • For b1 > 0, each nontrivial fixed mode has displacement and velocity envelopes of orders t^-1/4 and t^1/4, while the opposite sign admits an exponentially growing branch.The resulting quadratic modal energy grows on the order of t^1/2; whether this persists nonlinearly remains open.Source: §7 · p. 47
  • The authors plan to study quasi-periodic solutions of the three-dimensional NLAW equation using Bourgain's multiscale analysis.They describe this equation as a fundamental model for acoustic turbulence.Source: §7 · p. 47

A Proofs of results in Section 2

The appendix proves Section 2 results, using preliminary inequalities and Gevrey-decay estimates. These estimates support the quadratic nonlinearity and its frequency derivatives in Proposition 2.1.

A Proofs of results in Section 2

  • The appendix supplies detailed proofs for Section 2, beginning with an elementary inequality and preservation of Gevrey decay under convolution.It identifies Propositions 2.1–2.4 as the results proved in the appendix.Source: A Proofs of results in Section 2 · p. 49

A.1 Preliminary lemmas

  • For 0 < s < 1, the preliminary inequality follows by assuming a ≥ b and showing xs + 1 − (1 + x)s increases for x ≥ 1.The proof uses x = a/b when b > 0; the b = 0 case is immediate.Source: §A.1 · p. 49
  • Convolution of Gevrey-decaying coefficients preserves Gevrey decay up to a multiplicative constant, including finite-fold convolutions.The bounds depend on the convolution order, m, and s; convergence of the Gevrey decay series supplies the uniform bound.Source: §A.1 · pp. 49–50

A.2 Proof of Proposition 2.1

  • For Proposition 2.1, convolution estimates for u^2 and the Leibniz rule give uniform Gevrey decay for the vector field and its frequency derivatives.Summing the squared coefficient bounds yields the uniform norm bound because the corresponding Gevrey decay series converges.Source: §A.2 · p. 50

A.3 Proof of Proposition 2.2

The proof estimates the full tangent operator and its restrictions using the convolution structure and Gevrey decay of the coefficients.

A.3 Proof of Proposition 2.2

  • The full tangent operator H is the derivative of X with respect to u, with entries indexed by pairs of space-time Fourier indices.Its kernel representation makes H a convolution operator with kernel 2u.Source: §A.3 · p. 50
  • Gevrey decay of u gives a uniform decay estimate for the convolution kernel, which restricts to non-resonant indices.Source: §A.3 · p. 50
  • The Cauchy–Schwarz bound on H also bounds its resonant and non-resonant restrictions, since orthogonal projections do not increase operator norm.The estimate uses translation invariance of the lattice sums.Source: §A.3 · pp. 50–51

A.4 Proof of Proposition 2.3

The proof treats the second derivative of X as a bilinear operator and transfers its bound to index-restricted versions under the stated support condition.

A.4 Proof of Proposition 2.3

  • The second derivative T = ∂²X/∂u² is represented as a bilinear operator with entries indexed by three Fourier-index pairs.Source: §A.4 · p. 51
  • For inputs supported within Λ_N^(1+m), the proof bounds the bilinear operator norm using the kernel identity and Cauchy–Schwarz.Translation invariance of the lattice sum is used in the estimate.Source: §A.4 · p. 51
  • Restricting the input or output indices to resonant or non-resonant sets preserves the same bilinear norm bound under that support condition.The restrictions are implemented by orthogonal projections.Source: §A.4 · p. 51

A.5 Proof of Proposition 2.4

The proof bounds the operator B using factorized entrywise estimates and establishes the Gevrey bound (2.5) from decay of the solution and tangent operator.

A.5 Proof of Proposition 2.4

  • The frequency estimate in Proposition 2.5 and the Cauchy–Schwarz inequality are used in bounding the entries of B.This estimate is introduced for indices outside S.Source: §A.5 · p. 52
  • The Gevrey decay of the solution and tangent operator yields the Gevrey bound (2.5).The proof uses the tangent-operator decay estimates established in Proposition 2.2.Source: §A.5 · p. 52
  • The operator norm of B is estimated from a factorized entrywise bound by extending the sum to the full product lattice.The resulting sums are uniformly bounded using the decay of the solution and tangent operator, establishing (2.5b).Source: §A.5 · p. 52

B.1 Proof of Lemma 3.1

The proof bounds nearly resonant frequency sets by summing strip-measure estimates over temporal indices and counting the admissible spatial indices. Convergence for τ > m + 2 yields the desired measure bound.

B.1 Proof of Lemma 3.1

  • For each nonzero temporal index, the nearly resonant set is a strip between hyperplanes orthogonal to that index, whose intersection with bounded Ω is estimated before summing over indices.The frequency set Ω is bounded, with diameter d and radius R.Source: §B.1 · p. 52
  • For fixed k, the admissible spatial indices and pairs are bounded by polynomial counts, which combine with strip-measure estimates to bound their measure contributions.The proof gives at most 4R∥k∥2 + 3 admissible values for each spatial index in one count, and at most 18R∥k∥2 admissible pairs in another.Source: §B.1 · pp. 52–53
  • When τ > m + 2, the lattice sums converge, and combining the estimates with the union representation establishes the desired bound (3.13).The proof sums the measure contributions over temporal indices k.Source: §B.1 · p. 53

B.2 Proof of Theorem 3.2

The proof shows that proper decay is preserved under operator products, then uses this decay and a uniform lower bound to establish invertibility and estimates for the restricted operator.

B.2 Proof of Theorem 3.2

  • Products of operators with proper decay retain the stated decay bound, including finite products.The constants depend on the number of factors, m, and s; the finite-product result follows by successive application of the two-operator estimate.Source: §B.2 · p. 54
  • The uniform lower bound (3.17) makes the diagonal operator invertible, and a factorization gives an operator-norm convergent Neumann series for the restricted operator's inverse.The resulting inverse estimate is (3.15).Source: §B.2 · pp. 54–55
  • Decay estimates for S and B, together with Theorem B.1, yield the off-diagonal decay estimate (3.16) after reducing ε if necessary.The proof sums the bounds for the terms in the expansion.Source: §B.2 · p. 55

C.1 Proof of Theorem 5.1

The proof expands entries of the inverse-derivative operator and bounds the resulting sums. Separation of lattice sites absorbs the prefactor into the exponential, yielding the Gevrey bound (5.3).

C.1 Proof of Theorem 5.1

  • The entries of ∂L_N^-1 are expanded as a double sum, and triangle and decay estimates are applied to bound them.The expansion uses the operator identity in (5.1) and bounds from Theorem B.1.Source: §C.1 · p. 55
  • Lattice-site separation of at least N^3/4 lets the prefactor be absorbed by the exponential, giving the off-diagonal Gevrey bound (5.3).The off-diagonal entries of ∂L_N^-1 satisfy this bound.Source: §C.1 · p. 56
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