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<p>We discuss application of generalized Weierstrass elliptic functions to finding solutions of nonlinear third order elliptic systems including the Cauchy-Riemann operators <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7563_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\({\partial }_{\overline{z} }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>∂</mi> <mover> <mi>z</mi> <mo>¯</mo> </mover> </msub> </math></EquationSource> </InlineEquation> and ∂<sub><i>z</i></sub> with independent variables <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7563_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\overline{z }\)</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mi>z</mi> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation> = <i>x</i> - <i>iy</i>,<i> z</i> = <i>x</i> + <i>iy</i> and such that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7563_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(2{\partial }_{\overline{z} }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <msub> <mi>∂</mi> <mover> <mi>z</mi> <mo>¯</mo> </mover> </msub> </mrow> </math></EquationSource> </InlineEquation> = ∂<sub><i>x</i></sub> + <i>i</i>∂<sub><i>y</i></sub> and 2∂<sub><i>z</i></sub> = ∂<sub><i>x</i></sub> - <i>i</i>∂<sub><i>y</i></sub>. We find invariants of the Weierstrass functions depending on coefficients of the equations.</p>

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Method of Weierstrass Elliptic Functions for Finding Exact Solutions of Nonlinear Third Order Elliptic Systems on a Plane

  • 期刊论文

  • 发表日期：

    2025年1月11日

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  • Djumaboy S. Safarov,
  • Sulton S. Kurbonazarov

摘要

We discuss application of generalized Weierstrass elliptic functions to finding solutions of nonlinear third order elliptic systems including the Cauchy-Riemann operators \({\partial }_{\overline{z} }\) ∂ z ¯ and ∂z with independent variables \(\overline{z }\) z ¯ = x - iy, z = x + iy and such that \(2{\partial }_{\overline{z} }\) 2 ∂ z ¯ = ∂x + i∂y and 2∂z = ∂x - i∂y. We find invariants of the Weierstrass functions depending on coefficients of the equations.