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Piotr Rozmej

Publications and source records attributed to Piotr Rozmej.

At least 19 recordsLinked to original sources
arXiv2025

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

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 $η(x,y,t)$ 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.

nlin.PS↗
arXiv2023

(2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, cnoidal and superposition solutions

We study the problem of gravity surface waves for an ideal fluid model in the (2+1)-dimensional case. We apply a systematic procedure to derive the Boussinesq equations for a given relation between the orders of four expansion parameters, the amplitude parameter $α$, the long-wavelength parameter $β$, the transverse wavelength parameter $γ$, and the bottom variation parameter $δ$. We derived the only possible (2+1)-dimensional extensions of the Korteweg-de Vries equation, the fifth-order KdV equation, and the Gardner equation in three special cases of the relationship between these parameters. All these equations are non-local. When the bottom is flat, the (2+1)-dimensional KdV equation can be transformed to the Kadomtsev-Petviashvili equation in a fixed reference frame and next to the classical KP equation in a moving frame. We have found soliton, cnoidal, and superposition solutions (essentially one-dimensional) to the (2+1)-dimensional Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation.

math-ph↗
arXiv2021

Boussinesq's equations for (2+1)-dimensional gravity-surface waves in an ideal fluid model

We study the problem of gravity surface waves for the ideal fluid model in (2+1)-dimensional case. We apply a systematic procedure for deriving the Boussinesq equations for a prescribed relationship between the orders of four expansion parameters, the amplitude parameter $α$, the long-wavelength parameter $β$, the transverse wavelength parameter $γ$, and the bottom variation parameter $δ$. We also take into account surface tension effects when relevant. For all considered cases, the (2+1)-dimensional Boussinesq equations can not be reduced to a single nonlinear wave equation for surface elevation function. On the other hand, they can be reduced to a single, highly nonlinear partial differential equation for an auxiliary function $f(x,y,t)$ which determines the velocity potential but is not directly observed quantity. The solution $f$ of this equation, if known, determines the surface elevation function. We also show that limiting the obtained the Boussinesq equations to (1+1)-dimensions one recovers well-known cases of the KdV, extended KdV, fifth-order KdV, and Gardner equations.

math-ph↗
arXiv2021

Inverted solutions of KdV-type and Gardner equations

In most of the studies concerning nonlinear wave equations of Korteweg-de Vries type, the authors focus on waves of elevation. Such waves have general form ~$u_{\text{u}}(x,t)=A f(x-vt)$, where ~$A>0$. In this communication we show that if ~$u_{\text{up}}(x,t)=A f(x-vt)$ is the solution of a given nonlinear equation, then $u_{\text{down}}(x,t)=-A f(x-vt)$, that is, an inverted wave is the solution of the same equation, but with changed sign of the parameter ~$α$. This property is common for KdV, extended KdV, fifth-order KdV, Gardner equations, and generalizations for cases with an uneven bottom.

math.AP↗
arXiv2021

Comment on "Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension" [Nonlinear Dyn, doi:10.1007/s11071-017-3938-7]

The authors of the paper "Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension" \cite{Fok} claim that they derived the equation which generalizes the KdV equation to two space dimensions both in first and second order in small parameters. Moreover, they claim to obtain soliton solution to the derived first order (2+1)-dimension equation. The equation has been obtained by applying the perturbation method \cite{burde} for small parameters of the same order. The results, if correct, would be significant. In this comment, it is shown that the derivation presented in \cite{Fok} is inconsistent because it violates fundamental properties of the velocity potential. Therefore, the results, particularly the new evolution equation and the dynamics that it describes, bear no relation to the problem under consideration.

nlin.SI↗
arXiv2020

Generalized KdV-type equations versus Boussinesq's equations for uneven bottom -- numerical study

The paper's main goal is to compare the motion of solitary surface waves resulting from two similar but slightly different approaches. In the first approach, the numerical evolution of soliton surface waves moving over the uneven bottom is obtained using single wave equations. In the second approach, the numerical evolution of the same initial conditions is obtained by the solution of a coupled set of the Boussinesq equations for the same Euler equations system. We discuss four physically relevant cases of relationships between small parameters $α,β,δ$. For the flat bottom, these cases imply the Korteweg-de Vries equation (KdV), the extended KdV (KdV2), fifth-order KdV (KdV5), and the Gardner equation (GE). In all studied cases, the influence of the bottom variations on the amplitude and velocity of a surface wave calculated from the Boussinesq equations is substantially more significant than that obtained from single wave equations.

nlin.PS↗
arXiv2019

Extended KdV equation for the case of uneven bottom

We derived consistently, according to the second order perturbation approach, the extended KdV equation for an uneven bottom for the case of $α=O(β)$ and $δ=O(β^2)$. This equation can be obtained only when the bottom is given by a piecewise linear function. For the case of $α=O(β)$ and $δ=O(β)$, a unidirectional wave equation is derived in the first order approach with the same limitation for the bottom profile.

physics.flu-dyn↗
arXiv2019

What kinds of KdV-type equations are allowed by an uneven bottom

In this study, we give a survey of derivations of KdV-type equations with an uneven bottom for several cases when small (perturbation) parameters $α, β, δ$ are of different orders. Six different cases of such ordering are discussed. Surprisingly, for all these cases the Boussinesq equations can be made compatible only for the particular piecewise linear bottom profiles, and the correction function has a universal form. For such bottom relief, several new KdV-type wave equations are derived. These equations generalize the KdV, the extended KdV (KdV2), the fifth-order KdV (KdV5) and the Gardner equations.

physics.flu-dyn↗
arXiv2019

Remarks on existence/nonexistence of analytic solutions to higher order KdV equations

In this note, we discuss the existence of analytic solutions to the nonlinear wave equations of the higher order than the ubiquitous Korteweg-de Vries (KdV) equation. First, we recall our recent results which show that the extended KdV equation (KdV2), that is, the equation obtained within second-order perturbation approach possesses three kinds of analytic solutions. These solutions have the same functional form as the corresponding KdV solutions. We show, however, that the most intriguing multi-soliton solutions, known for the KdV equation, do not exist for KdV2. Moreover, we show that for the equations obtained in the third order perturbation approach (and then in any higher order) analytic solutions in the forms known from KdV theory do not exist.

math-ph↗
arXiv2018

New exact superposition solutions to KdV2 equation

New exact solutions to the KdV2 equation (known also as the extended KdV equation) are constructed. The KdV2 equation is a second order approximation of the set of Boussinesq's equations for shallow water waves which in first order approximation yields KdV. The exact solutions ~$\frac{A}{2}\left(\dn^2[B(x-vt),m]\pm \sqrt{m}\,\cn [B(x-vt),m]\dn [B(x-vt),m]\right)+D$~ in the form of periodic functions found in the paper complement other forms of exact solutions to KdV2 obtained earlier, i.e., the solitonic ones and periodic ones given by a single $\cn^2$ or $\dn^2$ Jacobi elliptic functions.

physics.flu-dyn↗
arXiv2018

Comment on the paper "The third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom" by M. Fokou, T.C. Kofané, A. Mohamadou and E. Yomba, Eur. Phys. J. Plus, 132, 410 (2017)

The authors of the paper "The third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom" [1] claim that they have derived the full third order perturbed KdV equation for the case of uneven bottom. We show that the authors' derivation is not consistent due to the fact that they took into account only some of the third order corrections but not all of them. Moreover, we show that a consistent third order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom cannot be derived for a general form of bottom function.

physics.flu-dyn↗
arXiv2018

Superposition solutions to the extended KdV equation for water surface waves

The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include higher order effects. Although this equation has only one conservation law, exact periodic and solitonic solutions exist. Khare and Saxena \cite{KhSa,KhSa14,KhSa15} demonstrated the possibility of generating new exact solutions by combining known ones for several fundamental equations (e.g., Korteweg - de Vries, Nonlinear Schrödinger). Here we find that this construction can be repeated for higher order, non-integrable extensions of these equations. Contrary to many statements in the literature, there seems to be no correlation between integrability and the number of nonlinear one variable wave solutions.

nlin.PS↗
arXiv2017

Single soliton solution to the extended KdV equation over uneven depth

In this note we look at the influence of a shallow, uneven riverbed on a soliton. The idea consists in approximate transformation of the equation governing wave motion over uneven bottom to equation for flat one for which the exact solution exists. The calculation is one space dimensional, and so corresponding to long trenches or banks under wide rivers or else oceans.

physics.flu-dyn↗
arXiv2017

Exact solitonic and periodic solutions of the extended KdV equation

The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include both higher order effects (KdV2) and an uneven river bottom. Although this equation is not integrable and has only one conservation law, exact periodic and solitonic solutions exist for the even bottom case. The method used to find them assumes the same function forms as for KdV solutions. KdV2 equation imposes more constraints on parameters of solutions. For soliton case KdV2 solution occurs for particular ratio of wave amplitude to water depth only. For periodic case physically relevant solutions are admissible only for two narrow intervals of elliptic parameter $m$. For a range of $m$ near one the cnoidal waves are upright as expected, but are inverted in $m$ region close to zero. Properties of exact solutions of KdV and KdV2 are compared.

physics.flu-dyn↗
arXiv2016

Adiabatic invariants of the extended KdV equation

When the Euler equations for shallow water are taken to the next order, beyond KdV, momentum and energy are no longer exact invariants. (The only one is mass.) However, adiabatic invariants (AI) can be found. When the KdV expansion parameters are zero, exact invariants are recovered. Existence of adiabatic invariants results from general theory of near-identity transformations (NIT) which allow us to transform higher order nonintegrable equations to asymptotically equivalent (when small parameters tend to zero) integrable form. Here we present the direct method of calculations of adiabatic invariants. It does not need a transformation to a moving reference frame nor performing a near-identity transformation. Numerical tests show that deviations of AI from almost constant values are indeed small.

physics.flu-dyn↗
arXiv2016

Finite Element Method for Stochastic Extended KdV Equations

The finite element method is applied to obtain numerical solutions to the recently derived nonlinear equation for shallow water wave problem for several cases of bottom shapes. Results for time evolution of KdV solitons and cnoidal waves under stochastic forces are presented. Though small effects originating from second order dynamics may be obscured by stochastic forces, the main waves, both cnoidal and solitary ones, remain very robust against any distortions.

physics.flu-dyn↗
arXiv2016

Finite element method for extended KdV equations

The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov-Galerkin method are used. It is shown that the FEM gives a reasonable description of the wave dynamics of soliton waves governed by extended KdV equations. Some new results for several cases of bottom shapes are presented. The numerical scheme presented here is suitable for taking into account stochastic effects, which will be discussed in a subsequent paper.

physics.flu-dyn↗