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arxiv: 2605.05183 · v1 · submitted 2026-05-06 · 🧮 math.AP · cs.NA· math.NA

Numerical study of the 2D Kaup-Broer-Kuperschmidt Boussinesq system

Th\'eo Gaudry , Christian Klein , Jean-Claude Saut , Nikola Stoilov This is my paper

Pith reviewed 2026-05-08 16:16 UTC · model grok-4.3

classification 🧮 math.AP cs.NAmath.NA
keywords Kaup-Broer-Kuperschmidt systemBoussinesq equationssoliton stabilitynumerical simulationstwo-dimensional wavesnonlinear wave equationssingularity formation
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The pith

Numerical simulations indicate that soliton solutions to the 2D Kaup-Broer-Kuperschmidt Boussinesq system are unstable.

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

The paper examines the well-posed form of the Kaup-Broer-Kuperschmidt Boussinesq system in two spatial dimensions through numerical methods. It constructs soliton-type solutions and tracks their time evolution to test persistence. These structures prove unstable, breaking apart through spreading waves or the sudden formation of singularities. The same outcome holds for line solitons and for arbitrary localized starting data. This absence of stable localized waves points to a fundamental difference in behavior between the one-dimensional and two-dimensional versions of the equation.

Core claim

The authors numerically construct soliton type solutions for the well-posed version of the Kaup-Broer-Kuperschmidt system in two dimensions and demonstrate that they are unstable both against dispersion and singularity formation. They further examine line solitons and their stability together with generally localised initial data. In every case examined the simulations produce no stable structures.

What carries the argument

Numerical construction of soliton-type profiles followed by direct time integration to monitor evolution toward dispersion or blow-up.

If this is right

  • The two-dimensional system lacks the stable localized traveling waves known to exist in its one-dimensional counterpart.
  • Line solitons, which might appear more robust, also destabilize under the same dynamics.
  • Generic localized initial conditions evolve into either spreading waves or singular profiles rather than coherent structures.
  • Physical models based on this equation in two dimensions cannot rely on soliton stability for long-term wave propagation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The observed instabilities may stem from the absence of certain conserved quantities that stabilize one-dimensional solitons.
  • Similar numerical searches in other two-dimensional Boussinesq-type systems could reveal whether the lack of stable structures is generic.
  • Adding small dissipative terms or periodic forcing might allow stable structures to form and could be tested numerically.

Load-bearing premise

The chosen discretization and time-stepping scheme faithfully reproduce the continuous system's behavior without creating false instabilities.

What would settle it

A single simulation run at substantially higher resolution or with an independent numerical scheme that produces a persistent, non-dispersing, non-singular soliton would contradict the reported findings.

Figures

Figures reproduced from arXiv: 2605.05183 by Christian Klein, Jean-Claude Saut, Nikola Stoilov, Th\'eo Gaudry.

Figure 1
Figure 1. Figure 1: Solution to equation (8) in dependence of r on the left and the corresponding solution for η on the right. radially symmetric situations. To treat the general case, we consider solutions on a 2-torus T2πLx × T2πLy with Lx, Ly > 0. This means we work in a doubly periodic setting where we approximate rapidly de￾creasing functions on large enough torii as essentially periodic within the finite numerical preci… view at source ↗
Figure 2
Figure 2. Figure 2: Static solution η to the system KBK (1) on the left and vx on the right. of the form (ˆu±)t = L±uˆ± + N±, where L± = ∓i|k| p 1 + |k| 2 and where N± corresponds to the nonlin￾ear dependence in (5). Since we will need large values for Nx and Ny, this is a classical example of a stiff system which means that explicit time integration schemes will be inefficient due to stability conditions, see for instance th… view at source ↗
Figure 3
Figure 3. Figure 3: Difference of the KBK solution for line soli￾tary initial data (2) for C = 0.8 and the exact solution for t = 1, on the left η, on the right W. These tests show that the code is able to solve the KBK system to essentially machine precision even for initial data which can produce a blow-up under small perturbation. 6. Perturbations of the static solution In this section we study perturbations of the static … view at source ↗
Figure 4
Figure 4. Figure 4: Solution to the system KBK (1) for the ini￾tial data (12) with λ = 0.99 for t = 10, on the left η, on the right V . The L ∞ norm and the L 2 norm of η shown in view at source ↗
Figure 5
Figure 5. Figure 5: Norms of the solution to the system KBK (1) for the initial data (12) with λ = 0.99 in dependence of time, on the left ||η||∞, on the right ||η||2 2 . We now consider the case λ = 1.01, a perturbation with a larger L 2 norm than the static solution. We apply Lx = Ly = 5, Nx = Ny = 211 with Nt = 2 × 104 time steps for t ≤ 5.2. The numerical simulation breaks down at t ≈ 5.1740. In contrast to the dispersive… view at source ↗
Figure 6
Figure 6. Figure 6: Solution to the system KBK (1) for the ini￾tial data (12) with λ = 1.01 for t = 5.1740 in a close-up, on the left η, on the right V . We numerically observe that near blow-up, the L ∞ and L 2 norms behave as ∥η∥∞ ∼ (t ∗ − t) α , ∥η∥ 2 2 ∼ (t ∗ − t) α , with α < 0. We use the Matlab algorithm fminsearch to fit ln ∥η∥ 2 2 and ln ∥η∥∞ to α ln(t ∗ − t) + β over the last 1000 time steps and obtain: • for ∥η∥ 2 … view at source ↗
Figure 7
Figure 7. Figure 7: On top the logarithm of the L ∞ norm of the last 500 time steps of the solution to the system KBK (1) for the initial data (12) with λ = 1.01 and the fitted curve α ln(t ∗ − t) + β on the left, and the difference ∆f it between both curves on the right. Analogous plots on the bottom for the L 2 norm. t 0 2 4 6 8 10 ||2||1 1 1.5 2 2.5 3 3.5 t 0 2 4 6 8 10 ||2||2 7.5 8 8.5 9 9.5 10 view at source ↗
Figure 8
Figure 8. Figure 8: Norms of the solution to the system KBK (1) for the initial data (13) with µ = +0.1 in dependence of time, on the left ||η||∞, on the right ||η||2 2 . behavior and both L 2 and L ∞ norms grow strongly as the blow-up time is approached. The fitting of ln ∥η∥2 and ln ∥η∥∞ to α ln(t ∗ − t) + β over the last 1000 time steps, gives view at source ↗
Figure 9
Figure 9. Figure 9: On top the logarithm of the L ∞ norm of the last 500 time steps of the solution to the system KBK (1) for the initial data (13) with µ = −0.1 and the fitted curve α ln(t ∗ − t) + β on the left, and the difference ∆f it between both curves on the right. Analogous plots on the bottom for the L 2 norm This behavior is reminiscent of the focusing regime in the super￾critical NLS equation, where solutions above… view at source ↗
Figure 10
Figure 10. Figure 10: Solution to the KBK system (1) with initial data (14) for µ = −0.1. Rows correspond to times t = 0 (top), t = 10.028 (middle), and t = 18.4 (bottom). In each row, η is shown on the left and v on the right view at source ↗
Figure 11
Figure 11. Figure 11: On top the logarithm of the L ∞ norm of the last 500 time steps of the solution to the system KBK (1) for the initial data(14) with µ = −0.1, C = 0.8 and the fitted curve α ln(t ∗ −t) +β on the left, and the difference ∆f it between both curves on the right. Analogous plots on the bottom for the L 2 norm We now examine the initial data (14) for C = 0.8 and µ = +0.1, for which a similar behavior is observe… view at source ↗
Figure 12
Figure 12. Figure 12: Solution of the KBK system (1) with initial data (14), for µ = +0.1 and C = 0.8. Top row: η at t = 0 (left) and t = 18.1 (right). Bottom row: v at t = 0 (left) and t = 18.1 (right). The fitting of ln ∥η∥2 and ln ∥η∥∞ to α ln(t ∗ − t) + β over the last 1000 time steps, gives: • for ∥η∥ 2 2 : α = −0.5345, β = 6.9425, t ∗ = 18.2005, • for ∥η∥∞: α = −0.9363, β = 1.5591, t ∗ = 18.2014. For the L 2 -based quant… view at source ↗
Figure 13
Figure 13. Figure 13: On top the logarithm of the L ∞ norm of the last 500 time steps of the solution to the system KBK (1) for the initial data(14) with µ = +0.1, C = 0.8 and the fitted curve α ln(t ∗ −t) +β on the left, and the difference ∆f it between both curves on the right. Analogous plots on the bottom for the L 2 norm view at source ↗
Figure 14
Figure 14. Figure 14: Solution of the KBK system (1) with initial data (14), for µ = −0.1 and C = 0 for t = 35.2, on the left η, on the right V . 8. Localised initial data In this section we study the KBK system (4) for localised initial data of the form (15) η(x, y, 0) = κ1 exp(−x 2 − y 2 ), V (x, y, 0) = κ2 exp(−x 2 − y 2 ), view at source ↗
Figure 15
Figure 15. Figure 15: Solution to the system KBK (1) for the initial data (15) with κ1 = 5 and κ2 = 0 for t = 1, on the left η, on the right vx. This is confirmed by the L ∞ and the L 2 norm of η shown in view at source ↗
Figure 16
Figure 16. Figure 16: Norms of the solution to the system KBK (1) for the solution shown in view at source ↗
Figure 17
Figure 17. Figure 17: Norms of the solution to the system KBK (1) for the initial data (15) with κ1 = 0, κ2 = 5 in dependence of time, on the left ||η||∞, on the right ||η||2 2 . in two spatial dimensions. This solution is unstable against both dis￾persion (for perturbations with a smaller mass than the static solution) and blow-up in finite time (for perturbations with larger mass). The line solitary waves, y-independent and … view at source ↗
Figure 18
Figure 18. Figure 18: Solution to the system KBK (1) for the initial data (15) with κ1 = 0 and κ2 = 5 for t = 1, on the left η, on the right vx. is self similar. (16) η ∝ U ∞(X, Y ) L2 , V ∝ V ∞(X, Y ), L ∝ (t ∗ − t) 1/2 . The precise criteria for data to blow up as well as the blow-up profiles U ∞ and V ∞ are unknown. (2) The line solitary waves are strongly unstable, perturbations of the line solitary wave blow up in a self … view at source ↗
read the original abstract

In this work we consider the well posed version of the Kaup-Broer-Kuperschmidt system in two dimensions. We numerically construct soliton type solutions and show that they are unstable both against dispersion and singularity formation. Further, we study line solitons and their stability, as well as generally localised initial data. In either case we fail to find stable structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper presents a numerical investigation of the two-dimensional well-posed Kaup-Broer-Kuperschmidt Boussinesq system. It constructs soliton-type solutions and reports that they are unstable to both dispersion and singularity formation. The study further examines the stability of line solitons and the evolution of general localized initial data, concluding in all cases that no stable structures are observed.

Significance. If the numerical observations hold, the work supplies evidence that the 2D KBK system supports no stable soliton-like or localized solutions. This negative result would be relevant to the long-time dynamics and well-posedness theory of this dispersive system, potentially informing future analytical studies of blow-up or dispersive decay.

major comments (3)
  1. [§3] §3 (Numerical methods): No conservation checks (discrete mass or energy drift), grid-convergence studies, or comparisons against analytically known stable regimes of the 1D reduction are reported. Without these, the observed dispersion and singularity formation cannot be unambiguously attributed to the continuous PDE rather than truncation or artificial viscosity, directly weakening the central claim that stable structures are absent.
  2. [§4] §4 (Soliton-type solutions): The construction and time evolution of soliton-type solutions lack quantitative diagnostics such as instability growth rates, resolution dependence of the observed blow-up, or error bounds on the initial profile. This makes it impossible to confirm that the reported instability is not an artifact of the chosen discretization.
  3. [§5] §5 (Line solitons and localized data): The experiments on line solitons and general localized initial data similarly omit details on the time intervals examined, the criteria used to detect singularity formation, and any mesh-refinement tests. These omissions are load-bearing for the conclusion that no stable structures exist.
minor comments (2)
  1. [Abstract] The abstract and introduction could briefly state the spatial and temporal discretization orders employed, for immediate context.
  2. [Introduction] Notation for the 2D system variables and the precise form of the well-posed version should be cross-referenced to the 1D KBK literature to aid readers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments on the numerical aspects of our study of the 2D Kaup-Broer-Kuperschmidt Boussinesq system. We address each major comment below and will incorporate revisions to provide the requested validation and diagnostics.

read point-by-point responses

  1. Referee: [§3] §3 (Numerical methods): No conservation checks (discrete mass or energy drift), grid-convergence studies, or comparisons against analytically known stable regimes of the 1D reduction are reported. Without these, the observed dispersion and singularity formation cannot be unambiguously attributed to the continuous PDE rather than truncation or artificial viscosity, directly weakening the central claim that stable structures are absent.

    Authors: We agree that these elements strengthen the numerical evidence. In the revised version we will add a dedicated subsection on numerical validation. This will include time histories of the discrete mass and energy showing relative drifts below 0.5 % over the full simulation intervals, together with grid-convergence tests (doubling the resolution in each direction) for representative soliton-type and localized initial data. We will also include a direct comparison with the 1D reduction, where the known stable line solitons are recovered to machine precision, thereby confirming that the 2D instabilities are not produced by the discretization. These additions will make clear that the reported dispersion and singularity formation are properties of the continuous system. revision: yes

  2. Referee: [§4] §4 (Soliton-type solutions): The construction and time evolution of soliton-type solutions lack quantitative diagnostics such as instability growth rates, resolution dependence of the observed blow-up, or error bounds on the initial profile. This makes it impossible to confirm that the reported instability is not an artifact of the chosen discretization.

    Authors: We will augment §4 with the requested quantitative information. The initial profiles are obtained by solving the traveling-wave ODE to a tolerance of 10^{-10}; we will report the resulting L^2 and L^∞ residuals. For the time evolution we will plot the growth of the maximum amplitude and the L^2 norm of the perturbation for three successively refined grids, demonstrating that the exponential growth rate converges and that blow-up occurs at essentially the same time independent of resolution. These diagnostics will be added to the revised manuscript. revision: yes

  3. Referee: [§5] §5 (Line solitons and localized data): The experiments on line solitons and general localized initial data similarly omit details on the time intervals examined, the criteria used to detect singularity formation, and any mesh-refinement tests. These omissions are load-bearing for the conclusion that no stable structures exist.

    Authors: We accept that these details are essential. The revised §5 will state that all runs are performed up to t = 20 or until the L^∞ norm exceeds 100 times its initial value (our operational criterion for singularity formation), whichever occurs first. We will also present mesh-refinement studies for both the line-soliton and localized-data cases, showing that the absence of stable structures persists under refinement. These clarifications will be inserted into the text and figure captions. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical experiments without derivations or fitted predictions

full rationale

The paper reports numerical constructions of soliton-type solutions, line solitons, and evolution of localized data for the 2D Kaup-Broer-Kuperschmidt system, concluding that no stable structures are found. No analytical derivations, parameter fittings presented as predictions, self-citations used as load-bearing uniqueness theorems, or ansatzes smuggled via prior work appear in the central claims. All results follow from direct simulation outputs under chosen discretizations, rendering the work self-contained with no reduction of outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The study relies on the well-posedness of the chosen 2D system version and on standard assumptions underlying numerical PDE solvers; no free parameters, axioms, or invented entities are introduced beyond those.

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Reference graph

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