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Abstract <p>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 <i>x</i>,&#xa0;<i>y</i> 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 nonlinear wave equation for the auxiliary function <i>f</i>(<i>x</i>,&#xa0;<i>y</i>,&#xa0;<i>t</i>) defining the velocity potential can be obtained, and only from its solutions can the surface wave form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\eta (x,y,t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be obtained. We demonstrate the existence of families of (2+1)-dimensional travelling wave solutions, including solitary and periodic solutions, of both cnoidal and superposition types.</p> Graphical abstract <p></p>

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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

  • 期刊论文

  • 发表日期：

    2026年4月18日

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  • Piotr Rozmej,
  • Anna Karczewska

摘要

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 nonlinear wave equation for the auxiliary function f(x, y, t) defining the velocity potential can be obtained, and only from its solutions can the surface wave form \(\eta (x,y,t)\) η ( x , y , t ) be obtained. We demonstrate the existence of families of (2+1)-dimensional travelling wave solutions, including solitary and periodic solutions, of both cnoidal and superposition types.

Graphical abstract