<p>This paper characterizes the entire solutions of the following eiconal type partial differential equations<Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_86_Article_Equa.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="371" /> </MediaObject> <EquationSource Format="TEX">$u^2+(a_{1}u_{z_{1}}+a_{2}u_{z_{2}})^2=p,\,\,\, u_{z_{1}}^2+(a_{0}u+a_{2}u_{z_{2}})^2=p ,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <msub> <mi>u</mi> <msub> <mi>z</mi> <mn>1</mn> </msub> </msub> <mo>+</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <msub> <mi>u</mi> <msub> <mi>z</mi> <mn>2</mn> </msub> </msub> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mi>p</mi> <mo>,</mo> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <msubsup> <mi>u</mi> <mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mi>u</mi> <mo>+</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <msub> <mi>u</mi> <msub> <mi>z</mi> <mn>2</mn> </msub> </msub> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mi>p</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation> and<Equation ID="Equb"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_86_Article_Equb.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="241" /> </MediaObject> <EquationSource Format="TEX">$(u+a_{1}u_{z_{1}})^2+(a_{0}u+a_{2}u_{z_{2}})^2=p,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <msub> <mi>u</mi> <msub> <mi>z</mi> <mn>1</mn> </msub> </msub> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mi>u</mi> <mo>+</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <msub> <mi>u</mi> <msub> <mi>z</mi> <mn>2</mn> </msub> </msub> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mi>p</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation> where <i>p</i> is a polynomial in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_86_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{C}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_86_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$a_{0},a_{1},a_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> are constants in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_86_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>. Descriptions are given and complemented by various examples.</p>

Entire solutions of the eiconal type partial differential equations in $\mathbb{C}^2$

期刊论文
发表日期：

2025年6月25日

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L. Yang,
W. Chen,
Q. Wang

摘要

This paper characterizes the entire solutions of the following eiconal type partial differential equations $u^2+(a_{1}u_{z_{1}}+a_{2}u_{z_{2}})^2=p,\,\,\, u_{z_{1}}^2+(a_{0}u+a_{2}u_{z_{2}})^2=p ,$ $u^{2} + {(a_{1} u_{z_{1}} + a_{2} u_{z_{2}})}^{2} = p, u_{z_{1}}^{2} + {(a_{0} u + a_{2} u_{z_{2}})}^{2} = p,$ and $(u+a_{1}u_{z_{1}})^2+(a_{0}u+a_{2}u_{z_{2}})^2=p,$ ${(u + a_{1} u_{z_{1}})}^{2} + {(a_{0} u + a_{2} u_{z_{2}})}^{2} = p,$ where p is a polynomial in $\mathbb{C}^2$ $C^{2}$ , and $a_{0},a_{1},a_{2}$ $a_{0}, a_{1}, a_{2}$ are constants in $\mathbb{C}$ $C$ . Descriptions are given and complemented by various examples.

Entire solutions of the eiconal type partial differential equations in \(\mathbb{C}^2\)

摘要