Lax representation and a quadratic rational first integral for second-order differential equations with cubic nonlinearity

Dmitry I. Sinelshchikov; Partha Guha; A. Ghose Choudhury

doi:10.1016/j.cnsns.2022.106553

Lax representation and a quadratic rational first integral for second-order differential equations with cubic nonlinearity

Dmitry I. Sinelshchikov
, Partha Guha
, A. Ghose Choudhury

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

In this paper we give a Lax formulation for a family of non-autonomous second-order differential equations of the type y_zz+a₃(z,y)y_z³+a₂(z,y)y_z²+a₁(z,y)y_z+a₀(z,y)=0. We obtain a sufficient condition for the existence of a Lax representation with a certain L-matrix. We demonstrate that equations with this Lax representation possess a quadratic rational first integral. We illustrate our construction with an example of the Rayleigh–Duffing–Van der Pol oscillator with quadratic damping.

Original language	British English
Article number	106553
Journal	Communications in Nonlinear Science and Numerical Simulation
Volume	112
DOIs	https://doi.org/10.1016/j.cnsns.2022.106553
State	Published - Sep 2022

Keywords

Cubic second-order differential equations
Lax representation
Rational first integrals

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title = "Lax representation and a quadratic rational first integral for second-order differential equations with cubic nonlinearity",

abstract = "In this paper we give a Lax formulation for a family of non-autonomous second-order differential equations of the type yzz+a3(z,y)yz3+a2(z,y)yz2+a1(z,y)yz+a0(z,y)=0. We obtain a sufficient condition for the existence of a Lax representation with a certain L-matrix. We demonstrate that equations with this Lax representation possess a quadratic rational first integral. We illustrate our construction with an example of the Rayleigh–Duffing–Van der Pol oscillator with quadratic damping.",

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note = "Funding Information: PG was supported by the Khalifa University of Science and Technology, United Arab Emirates under grant number FSU-2021-014 . DS was supported by the project group of MIEM NRU HSE “Asymptotic Methods in Problems of Physical Processes Modeling” Publisher Copyright: {\textcopyright} 2022 Elsevier B.V.",

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AU - Sinelshchikov, Dmitry I.

AU - Guha, Partha

AU - Choudhury, A. Ghose

N1 - Funding Information: PG was supported by the Khalifa University of Science and Technology, United Arab Emirates under grant number FSU-2021-014 . DS was supported by the project group of MIEM NRU HSE “Asymptotic Methods in Problems of Physical Processes Modeling” Publisher Copyright: © 2022 Elsevier B.V.

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N2 - In this paper we give a Lax formulation for a family of non-autonomous second-order differential equations of the type yzz+a3(z,y)yz3+a2(z,y)yz2+a1(z,y)yz+a0(z,y)=0. We obtain a sufficient condition for the existence of a Lax representation with a certain L-matrix. We demonstrate that equations with this Lax representation possess a quadratic rational first integral. We illustrate our construction with an example of the Rayleigh–Duffing–Van der Pol oscillator with quadratic damping.

AB - In this paper we give a Lax formulation for a family of non-autonomous second-order differential equations of the type yzz+a3(z,y)yz3+a2(z,y)yz2+a1(z,y)yz+a0(z,y)=0. We obtain a sufficient condition for the existence of a Lax representation with a certain L-matrix. We demonstrate that equations with this Lax representation possess a quadratic rational first integral. We illustrate our construction with an example of the Rayleigh–Duffing–Van der Pol oscillator with quadratic damping.

KW - Cubic second-order differential equations

KW - Lax representation

KW - Rational first integrals

UR - https://www.scopus.com/pages/publications/85130411061

U2 - 10.1016/j.cnsns.2022.106553

DO - 10.1016/j.cnsns.2022.106553

M3 - Article

AN - SCOPUS:85130411061

SN - 1007-5704

VL - 112

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

M1 - 106553

ER -

Lax representation and a quadratic rational first integral for second-order differential equations with cubic nonlinearity

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