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<p>Two third-order and one fourth-order partial differential equations with nonlinear source are considered. These evolution equations are generalizations of the Korteweg - de Vries - Burgers and the Aspe - Depassier equations. The Painlevé test for partial differential equations is applied to assess the integrability of these equations by the inverse scattering transform. It is shown that the equations do not pass the Painlevé test in the general case. However they posses the Painlevé property when the solutions are sought using the traveling wave reduction. Information from the Painlev&#xa0;e test is used to determine the first integrals of nonlinear ordinary differential equations. A simple method for obtaining the Jacobi last multipliers is presented. The first integrals are then employed to obtain the general solutions of the differential equations. The general solutions are expressed in terms of the Weierstrass elliptic function and the transcendents of the first Painlevé equation.</p>

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The Painlevé Test for Finding First Integrals, Jacobi Last Multipliers and General Solutions of Some Nonlinear Differential Equations

  • 期刊论文

  • 发表日期：

    2026年4月6日

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  • Nikolay A. Kudryashov

摘要

Two third-order and one fourth-order partial differential equations with nonlinear source are considered. These evolution equations are generalizations of the Korteweg - de Vries - Burgers and the Aspe - Depassier equations. The Painlevé test for partial differential equations is applied to assess the integrability of these equations by the inverse scattering transform. It is shown that the equations do not pass the Painlevé test in the general case. However they posses the Painlevé property when the solutions are sought using the traveling wave reduction. Information from the Painlev e test is used to determine the first integrals of nonlinear ordinary differential equations. A simple method for obtaining the Jacobi last multipliers is presented. The first integrals are then employed to obtain the general solutions of the differential equations. The general solutions are expressed in terms of the Weierstrass elliptic function and the transcendents of the first Painlevé equation.