若干复乘积型非线性偏微分方程的超越整解

IF 1 4区 数学 Q1 MATHEMATICS Open Mathematics Pub Date : 2023-11-20 DOI:10.1515/math-2023-0151
Yi Hui Xu, Yan Fang Li, Xiao Lan Liu, Hong Yan Xu
{"title":"若干复乘积型非线性偏微分方程的超越整解","authors":"Yi Hui Xu, Yan Fang Li, Xiao Lan Liu, Hong Yan Xu","doi":"10.1515/math-2023-0151","DOIUrl":null,"url":null,"abstract":"Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs) <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> </m:math> <jats:tex-math>\\left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})=1,</jats:tex-math> </jats:alternatives> </jats:disp-formula> and <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>g</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> <jats:tex-math>\\left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})={e}^{g},</jats:tex-math> </jats:alternatives> </jats:disp-formula> where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>g\\left(z)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a nonconstant polynomial and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{a}_{j},{b}_{j}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{c}_{j}\\left(j=1,2)</jats:tex-math> </jats:alternatives> </jats:inline-formula> are constants in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"double-struck\">C</m:mi> </m:math> <jats:tex-math>{\\mathbb{C}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The finite-order transcendental entire solution <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_007.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>u</m:mi> </m:math> <jats:tex-math>u</jats:tex-math> </jats:alternatives> </jats:inline-formula> of the first equation is of the following forms: <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_008.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mo>±</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:msqrt> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msqrt> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mstyle displaystyle=\"false\"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>D</m:mi> </m:mrow> </m:mfrac> </m:mstyle> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> <jats:tex-math>u\\left({z}_{1},{z}_{2})=\\pm \\frac{1}{\\sqrt{{a}_{1}{a}_{2}}}+{\\eta }_{0}{e}^{\\tfrac{1}{D}{[}\\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]},</jats:tex-math> </jats:alternatives> </jats:disp-formula> or <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_009.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>Q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>Q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mstyle displaystyle=\"false\"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>D</m:mi> </m:mrow> </m:mfrac> </m:mstyle> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> <jats:tex-math>u\\left({z}_{1},{z}_{2})=\\frac{1}{2{a}_{1}}{e}^{Q\\left({z}_{1},{z}_{2})}+\\frac{1}{2{a}_{2}}{e}^{-Q\\left({z}_{1},{z}_{2})}+{\\eta }_{0}{e}^{\\tfrac{1}{D}{[}\\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]},</jats:tex-math> </jats:alternatives> </jats:disp-formula> where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_010.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>D</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> <jats:tex-math>D={b}_{1}{c}_{2}-{b}_{2}{c}_{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_011.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">C</m:mi> <m:mo>−</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> <jats:tex-math>{\\eta }_{0}\\in {\\mathbb{C}}-\\left\\{0\\right\\}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0151_eq_012.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:mi>Q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mo>−</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>D</m:mi> </m:mrow> </m:mfrac> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mspace width=\"1em\" /> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">C</m:mi> <m:mo>.</m:mo> </m:math> <jats:tex-math>Q\\left({z}_{1},{z}_{2})=-\\frac{1}{D}\\left[\\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}){z}_{1}-\\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}){z}_{2}]+{\\eta }_{1},\\hspace{1em}{\\eta }_{1}\\in {\\mathbb{C}}.</jats:tex-math> </jats:alternatives> </jats:disp-formula> The description of the forms of the solutions for these PDEs demonstrates that our results are some improvements of the previous results given by Liu, Cao, and Xu [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 227], and [K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ. 2013 (2013), No. 59, 1–10.]. Meantime, we list some examples to explain that the forms of solutions of our theorems are precise to some extent.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"70 4","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2\",\"authors\":\"Yi Hui Xu, Yan Fang Li, Xiao Lan Liu, Hong Yan Xu\",\"doi\":\"10.1515/math-2023-0151\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs) <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_001.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> </m:math> <jats:tex-math>\\\\left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\\\\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})=1,</jats:tex-math> </jats:alternatives> </jats:disp-formula> and <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_002.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>g</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> <jats:tex-math>\\\\left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\\\\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})={e}^{g},</jats:tex-math> </jats:alternatives> </jats:disp-formula> where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_003.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>g\\\\left(z)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a nonconstant polynomial and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_004.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{a}_{j},{b}_{j}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_005.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{c}_{j}\\\\left(j=1,2)</jats:tex-math> </jats:alternatives> </jats:inline-formula> are constants in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_006.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"double-struck\\\">C</m:mi> </m:math> <jats:tex-math>{\\\\mathbb{C}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The finite-order transcendental entire solution <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_007.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>u</m:mi> </m:math> <jats:tex-math>u</jats:tex-math> </jats:alternatives> </jats:inline-formula> of the first equation is of the following forms: <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_008.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\"> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mo>±</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:msqrt> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msqrt> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mstyle displaystyle=\\\"false\\\"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>D</m:mi> </m:mrow> </m:mfrac> </m:mstyle> <m:mrow> <m:mo stretchy=\\\"false\\\">[</m:mo> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy=\\\"false\\\">]</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> <jats:tex-math>u\\\\left({z}_{1},{z}_{2})=\\\\pm \\\\frac{1}{\\\\sqrt{{a}_{1}{a}_{2}}}+{\\\\eta }_{0}{e}^{\\\\tfrac{1}{D}{[}\\\\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\\\\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]},</jats:tex-math> </jats:alternatives> </jats:disp-formula> or <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_009.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\"> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>Q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>Q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mstyle displaystyle=\\\"false\\\"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>D</m:mi> </m:mrow> </m:mfrac> </m:mstyle> <m:mrow> <m:mo stretchy=\\\"false\\\">[</m:mo> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy=\\\"false\\\">]</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> <jats:tex-math>u\\\\left({z}_{1},{z}_{2})=\\\\frac{1}{2{a}_{1}}{e}^{Q\\\\left({z}_{1},{z}_{2})}+\\\\frac{1}{2{a}_{2}}{e}^{-Q\\\\left({z}_{1},{z}_{2})}+{\\\\eta }_{0}{e}^{\\\\tfrac{1}{D}{[}\\\\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\\\\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]},</jats:tex-math> </jats:alternatives> </jats:disp-formula> where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_010.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>D</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> <jats:tex-math>D={b}_{1}{c}_{2}-{b}_{2}{c}_{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_011.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mi mathvariant=\\\"double-struck\\\">C</m:mi> <m:mo>−</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> <jats:tex-math>{\\\\eta }_{0}\\\\in {\\\\mathbb{C}}-\\\\left\\\\{0\\\\right\\\\}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0151_eq_012.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"block\\\"> <m:mi>Q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mo>−</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>D</m:mi> </m:mrow> </m:mfrac> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mspace width=\\\"1em\\\" /> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mi mathvariant=\\\"double-struck\\\">C</m:mi> <m:mo>.</m:mo> </m:math> <jats:tex-math>Q\\\\left({z}_{1},{z}_{2})=-\\\\frac{1}{D}\\\\left[\\\\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}){z}_{1}-\\\\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}){z}_{2}]+{\\\\eta }_{1},\\\\hspace{1em}{\\\\eta }_{1}\\\\in {\\\\mathbb{C}}.</jats:tex-math> </jats:alternatives> </jats:disp-formula> The description of the forms of the solutions for these PDEs demonstrates that our results are some improvements of the previous results given by Liu, Cao, and Xu [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 227], and [K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ. 2013 (2013), No. 59, 1–10.]. Meantime, we list some examples to explain that the forms of solutions of our theorems are precise to some extent.\",\"PeriodicalId\":48713,\"journal\":{\"name\":\"Open Mathematics\",\"volume\":\"70 4\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-11-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Open Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/math-2023-0151\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Open Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/math-2023-0151","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

本文的目的是描述几种积型非线性偏微分方程(a 1 u + b 1 u z 1 + c 1 u z 2) (a 2 u + b 2 u z 1 + c 2 u z 2) = 1的解, \left({a}_{1}u+{b}_{1}{你}_{{z}_{1}}+{c}_{1}{你}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{你}_{{z}_{1}}+{c}_{2}{你}_{{z}_{2}})=1, (a1u + b1uz1 + c1uz2) (a2u + b2uz1 + c2uz2)= eg, \left({a}_{1}u+{b}_{1}{你}_{{z}_{1}}+{c}_{1}{你}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{你}_{{z}_{1}}+{c}_{2}{你}_{{z}_{2}})={e}^{g},其中g (z) g\left(z)是一个非常数多项式,a j b j {a}_{j},{b}_{j} , c j (j = 1,2) {c}_{j}\left(j=1,2)是C中的常数 {\mathbb{C}} . 第一个方程的有限阶超越全解u u有如下形式:u (z1, z2) =±1a1a2 + η 0 e1d [(a2c1 - a1c2) z1 + (a1b2 - a2b1) z2], u\left({z}_{1},{z}_{2})=\pm \frac{1}{\sqrt{{a}_{1}{a}_{2}}}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]},或u (z1, z2) = 1 2a1e Q (z1, z2) + 1 2a2e−Q (z1, z2) + η 0 e1d [(a2c1 - a1c2) z1 + (a1b2 - a2b1) z2], u\left({z}_{1},{z}_{2})=\frac{1}{2{a}_{1}}{e}^{q\left({z}_{1},{z}_{2})}+\frac{1}{2{a}_{2}}{e}^{-q\left({z}_{1},{z}_{2})}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}式中,D= b1 c2−b1 c1 D={b}_{1}{c}_{2}-{b}_{2}{c}_{1} , η 0∈c− { 0 } {\eta }_{0}\in {\mathbb{C}}-\left{0\right},且Q (z1, z2) = - 1 D [(a1c2 + a2c1) z1−(a1b2 + a2b1) z2] + η 1, η 1∈c。q\left({z}_{1},{z}_{2})=-\frac{1}{D}\left[\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}){z}_{1}-\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}){z}_{2}]+{\eta }_{1},\hspace{1em}{\eta }_{1}\in {\mathbb{C}}。对这些偏微分方程解形式的描述表明,我们的结果是Liu, Cao和Xu [L.]先前给出的结果的一些改进。徐、曹廷斌,复费马型偏差分和微分-差分方程的解,中华数学。数学学报,15 (2018),227 [j]。刘涛,曹廷彬,费马型差分微分方程的全解,电子学报。[j].地理学报,2013(2013),(59):1-10。同时,我们列举了一些例子来说明我们的定理的解的形式在一定程度上是精确的。
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Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs) ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = 1 , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})=1, and ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = e g , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})={e}^{g}, where g ( z ) g\left(z) is a nonconstant polynomial and a j , b j {a}_{j},{b}_{j} , and c j ( j = 1 , 2 ) {c}_{j}\left(j=1,2) are constants in C {\mathbb{C}} . The finite-order transcendental entire solution u u of the first equation is of the following forms: u ( z 1 , z 2 ) = ± 1 a 1 a 2 + η 0 e 1 D [ ( a 2 c 1 a 1 c 2 ) z 1 + ( a 1 b 2 a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\pm \frac{1}{\sqrt{{a}_{1}{a}_{2}}}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, or u ( z 1 , z 2 ) = 1 2 a 1 e Q ( z 1 , z 2 ) + 1 2 a 2 e Q ( z 1 , z 2 ) + η 0 e 1 D [ ( a 2 c 1 a 1 c 2 ) z 1 + ( a 1 b 2 a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\frac{1}{2{a}_{1}}{e}^{Q\left({z}_{1},{z}_{2})}+\frac{1}{2{a}_{2}}{e}^{-Q\left({z}_{1},{z}_{2})}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, where D = b 1 c 2 b 2 c 1 D={b}_{1}{c}_{2}-{b}_{2}{c}_{1} , η 0 C { 0 } {\eta }_{0}\in {\mathbb{C}}-\left\{0\right\} , and Q ( z 1 , z 2 ) = 1 D [ ( a 1 c 2 + a 2 c 1 ) z 1 ( a 1 b 2 + a 2 b 1 ) z 2 ] + η 1 , η 1 C . Q\left({z}_{1},{z}_{2})=-\frac{1}{D}\left[\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}){z}_{1}-\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}){z}_{2}]+{\eta }_{1},\hspace{1em}{\eta }_{1}\in {\mathbb{C}}. The description of the forms of the solutions for these PDEs demonstrates that our results are some improvements of the previous results given by Liu, Cao, and Xu [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 227], and [K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ. 2013 (2013), No. 59, 1–10.]. Meantime, we list some examples to explain that the forms of solutions of our theorems are precise to some extent.
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来源期刊
Open Mathematics
Open Mathematics MATHEMATICS-
CiteScore
2.40
自引率
5.90%
发文量
67
审稿时长
16 weeks
期刊介绍: Open Mathematics - formerly Central European Journal of Mathematics Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant, original and relevant works in all areas of mathematics. The journal provides the readers with free, instant, and permanent access to all content worldwide; and the authors with extensive promotion of published articles, long-time preservation, language-correction services, no space constraints and immediate publication. Open Mathematics is listed in Thomson Reuters - Current Contents/Physical, Chemical and Earth Sciences. Our standard policy requires each paper to be reviewed by at least two Referees and the peer-review process is single-blind. Aims and Scope The journal aims at presenting high-impact and relevant research on topics across the full span of mathematics. Coverage includes:
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