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arxiv: 2506.08941 · v2 · submitted 2025-06-10 · 🌊 nlin.PS · math-ph· math.MP

Solitary wave solutions, periodic and superposition solutions to the system of first-order (2+1)-dimensional Boussinesq's equations derived from the Euler equations for an ideal fluid model

Piotr Rozmej , Anna Karczewska This is my paper

Pith reviewed 2026-05-19 10:11 UTC · model grok-4.3

classification 🌊 nlin.PS math-phmath.MP
keywords Boussinesq equationssolitary wavescnoidal wavestraveling wavesideal fluid(2+1)-dimensional wavessuperposition solutionssurface elevation
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The pith

A (2+1)-dimensional Boussinesq system for ideal fluid admits families of solitary and periodic traveling wave solutions.

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

When the horizontal coordinates x and y receive identical scaling, the Euler equations for irrotational ideal fluid motion cannot be reduced to a single KdV-type equation. The model instead produces a nonlinear wave equation for an auxiliary function that defines the velocity potential. Solutions to this equation generate (2+1)-dimensional traveling waves that are solitary or periodic, the latter appearing either as cnoidal waves or as superpositions of them. The physical surface elevation is recovered afterward by direct mapping from the auxiliary solutions. A reader cares because these constructions supply concrete wave profiles that remain valid in two horizontal dimensions without dimensional reduction.

Core claim

The central claim is that the system of first-order (2+1)-dimensional Boussinesq equations, derived from the Euler equations under the ideal-fluid irrotational assumption, possesses families of traveling wave solutions. These families include solitary waves together with periodic solutions of both cnoidal type and superposition type. All solutions are first obtained for the auxiliary function f that determines the velocity potential; the surface elevation η is then extracted from them.

What carries the argument

the nonlinear wave equation for the auxiliary function f(x,y,z) that defines the velocity potential, from whose solutions the surface elevation η(x,y,t) is recovered by direct mapping

If this is right

  • Explicit solitary-wave profiles exist for the surface elevation in two spatial dimensions.
  • Cnoidal periodic waves furnish exact repeating patterns without approximation.
  • Superposition constructions generate more complex traveling wave patterns from the basic solutions.
  • All such waves remain consistent with the equal-scaling regime and do not collapse to a lower-dimensional equation.

Where Pith is reading between the lines

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

  • These closed-form profiles could serve as test cases for numerical schemes that solve the full Euler equations in two horizontal dimensions.
  • The same auxiliary-function route may apply to neighboring models that include weak surface tension or gentle bottom topography.
  • Direct comparison of the predicted wave speeds and shapes against laboratory wave-tank data would test the ideal-fluid idealization.

Load-bearing premise

The derivation requires an ideal irrotational fluid together with identical scaling of the horizontal variables x and y.

What would settle it

A concrete parameter choice for which the auxiliary solutions, once mapped to surface elevation, fail to satisfy the original first-order Boussinesq system or produce wave profiles that contradict direct numerical integration of the Euler equations.

read the original abstract

This article concludes the study of (2+1)-dimensional nonlinear wave equations that can be derived in a model of an ideal fluid with irrotational motion. In the considered case of identical scaling of the $x,y$ variables, obtaining a (2+1)-dimensional wave equation analogous to the KdV equation is impossible. Instead, from a system of two first-order Boussinesq equations, a non-linear wave equation for the auxiliary function $f(x,y,z)$ defining the velocity potential can be obtained, and only from its solutions can the surface wave form $\eta(x,y,t)$ be obtained. We demonstrate the existence of families of (2+1)-dimensional traveling wave solutions, including solitary and periodic solutions, of both cnoidal and superposition types.

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

2 major / 2 minor

Summary. The manuscript derives a nonlinear wave equation for an auxiliary function f(x,y,z) from the system of first-order (2+1)-dimensional Boussinesq equations obtained from the Euler equations for irrotational ideal fluid flow under identical scaling of the x and y variables. It then constructs families of (2+1)-dimensional traveling-wave solutions to this equation for f, including solitary waves, cnoidal periodic solutions, and superposition solutions, from which the physical surface elevation η(x,y,t) is recovered.

Significance. If the solutions for f are verified to satisfy the derived nonlinear equation and correctly map back to η while satisfying the original system, the work would add to the limited set of exact traveling-wave solutions for (2+1)D fluid models that cannot be reduced to a single KdV-type equation. This is potentially useful for understanding nonlinear surface waves where transverse effects matter, provided the superposition family is rigorously checked.

major comments (2)
  1. The superposition solutions for the nonlinear PDE in f are presented as a family but without an explicit residual calculation or demonstration of an integrability property that would allow linear combinations to remain solutions. Direct substitution of the proposed superposition form into the equation for f is required to confirm the residual vanishes, as this is the least secure part of the existence claim.
  2. The mapping from solutions of the auxiliary equation for f back to the surface elevation η and verification that these satisfy the original Euler-derived Boussinesq system is not accompanied by direct substitution checks or error estimates. This step is load-bearing for the central claim that the constructed solutions solve the physical model.
minor comments (2)
  1. Clarify the notation for the auxiliary function f and its relation to the velocity potential throughout the derivations.
  2. Ensure all traveling-wave reductions (e.g., to ODEs for solitary and cnoidal cases) explicitly state the assumed form of the wave variable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments, which help clarify the verification steps needed for our exact solutions. We address each major comment below and will revise the manuscript accordingly to include the requested explicit checks.

read point-by-point responses

  1. Referee: The superposition solutions for the nonlinear PDE in f are presented as a family but without an explicit residual calculation or demonstration of an integrability property that would allow linear combinations to remain solutions. Direct substitution of the proposed superposition form into the equation for f is required to confirm the residual vanishes, as this is the least secure part of the existence claim.

    Authors: We agree that an explicit residual calculation strengthens the claim. The superposition family is obtained from the specific structure of the derived nonlinear equation for f, which permits linear combinations of certain traveling-wave profiles to remain solutions. In the revised manuscript we will add a direct substitution of the proposed superposition ansatz into the equation for f and show that the residual vanishes identically; where possible we will also note the algebraic property of the nonlinearity that enables this cancellation. revision: yes

  2. Referee: The mapping from solutions of the auxiliary equation for f back to the surface elevation η and verification that these satisfy the original Euler-derived Boussinesq system is not accompanied by direct substitution checks or error estimates. This step is load-bearing for the central claim that the constructed solutions solve the physical model.

    Authors: We acknowledge that explicit verification of the recovery of η and satisfaction of the original system is essential. The relations connecting f to η follow directly from the definitions of the velocity potential and the Boussinesq scaling; therefore solutions of the auxiliary equation map to solutions of the system by construction. To meet the referee’s request we will insert, in the revised version, direct symbolic substitution for representative solitary and cnoidal cases, confirming that the recovered η satisfies the original first-order Boussinesq equations, together with brief error estimates associated with the underlying long-wave approximation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper starts from the Euler equations under irrotational flow and identical x-y scaling, derives the system of first-order Boussinesq equations, reduces it to a nonlinear wave equation for the auxiliary velocity-potential function f, and then constructs explicit traveling-wave solutions (solitary, cnoidal, and superposition forms) for that equation before mapping back to the surface elevation eta. These steps are forward derivations and direct substitutions or ODE reductions; no parameter is fitted to the target eta, no self-citation supplies a uniqueness theorem that forces the result, and the superposition family is presented as an explicit ansatz whose residual must be checked in the PDE for f. The chain therefore remains independent of its final outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of solutions to the auxiliary nonlinear wave equation for f and the validity of the mapping from f to the physical surface elevation eta under the stated scaling and irrotational assumptions.

axioms (2)
  • domain assumption The fluid is ideal and the motion is irrotational, allowing derivation of the first-order Boussinesq system from Euler equations.
    Stated in the abstract as the starting model; if false the entire reduction fails.
  • domain assumption Identical scaling of x and y variables prevents reduction to a single KdV-type equation.
    Explicitly invoked to justify working with the auxiliary function f.

pith-pipeline@v0.9.0 · 5685 in / 1426 out tokens · 25436 ms · 2026-05-19T10:11:53.031336+00:00 · methodology

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