On the integrability conditions for a family of Liénard-type equations

Kudryashov, Nikolay A.; Sinelshchikov, Dmitry I.

doi:10.1134/s1560354716050063

Cited by 20 publications

(12 citation statements)

References 24 publications

(57 reference statements)

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“…It is also interesting to compare the results of this work with recent results (see, [10][11][12]28,30]) devoted to integrability of the classical Liénard equation ( 2) and its generalizations. Since integrability conditions or conditions for the existence of first integrals obtained in these works are given in terms of relations between the functions f and g, one can readily compare them with either (37) or (38).…”

Section: Examplesmentioning

confidence: 69%

“…Since integrability conditions or conditions for the existence of first integrals obtained in these works are given in terms of relations between the functions f and g, one can readily compare them with either (37) or (38). As a result, one can see that neither (37) nor (38) coincide with any of the integrability conditions obtained in [10][11][12]28]. Thus, correlation (37) (or (38)) provides a new condition for the existence of a quadratic first integral for (2).…”

Section: Examplesmentioning

confidence: 87%

“…where U is defined by (11) and M-matrix can be found from (10). Consequently, family of equations ( 12) admits the first integral…”

Section: The Classical Liénard Equationmentioning

confidence: 99%

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Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

Sinelshchikov

Gaiur

Kudryashov

2019

Journal of Mathematical Analysis and Applications

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We consider a family of non-autonomous second-order differential equations, which generalizes the Liénard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which may be considered as an analogue to the theorem that connects Lax integrability and Arnold-Liouville integrability of Hamiltonian systems. We illustrate our results by several examples of dissipative equations, including generalizations of the Van der Pol and Duffing equations, each of which have both a quadratic first integral and a Lax representation.

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Section: Examplesmentioning

confidence: 69%

Section: Examplesmentioning

confidence: 87%

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Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

Sinelshchikov

Gaiur

Kudryashov

2019

Journal of Mathematical Analysis and Applications

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“…Here w is the oscillation law, α > 0 and a, b > 0 are real numbers, f is the influence of the electromotive forces. In this case also, solutions of the homogeneous equation (i.e., f ≡ 0) are usually found (Kudryashov and Sinelshchikov 2016). Up to our best knowledge, (20) for general f = 0 is not rigorously solved yet.…”

Section: Liénard Equationmentioning

confidence: 99%

An identity for the Heaviside function and its application in representation of nonlinear Green’s function

Khurshudyan

2019

Comp. Appl. Math.

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An identity showing that any positive real power of the Heaviside function is almost everywhere equal to the Heaviside function of power 1 is proved in this paper. A similar identity is shown to hold for corresponding distributions. The identity is then used to attach a specific meaning to the nonlinear Green's function of quasi-linear ordinary differential equations. As specific examples, nonlinear Green's function is constructed for the forced cubic Duffing (explicitly) and fractionally nonlinear damped Liénard (implicitly) equations. Efficiency of the nonlinear Green's function solution for various source functions is demonstrated numerically in comparison with the solution obtained by the well-known method of lines. Solutions obtained are unique in literature.

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“…A new family of the Liénard-type equations which admits a non-standard autonomous Lagrangian was found in [18], and autonomous first integrals for each member were obtained. Four new integrability criteria of a particular type of Liénard type equations were obtained in [19] by studying the connections between this family of Liénard-type equations and type III Painlevé-Gambier equations. The results were illustrated by providing examples of some integrable Liénard-type equations.…”

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confidence: 99%

On the Integrability of the Abel and of the Extended Liénard Equations

Mak

Harkó

2019

Acta Math. Appl. Sin. Engl. Ser.

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We present some exact integrability cases of the extended Liénard equation y ′′ + f (y) (y ′ ) n + k (y) (y ′ ) m + g (y) y ′ + h (y) = 0, with n > 0 and m > 0 arbitrary constants, while f (y), k(y), g(y), and h(y) are arbitrary functions. The solutions are obtained by transforming the equation Liénard equation to an equivalent first kind first order Abel type equation given by dvAs a first step in our study we obtain three integrability cases of the extended quadratic-cubic Liénard equation, corresponding to n = 2 and m = 3, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Liénard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Liénard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with g(y) ≡ 0 and h(y) ≡ 0, and arbitrary n and m, thus allowing to obtain the general solution of the corresponding Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.

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On the integrability conditions for a family of Liénard-type equations

Cited by 20 publications

References 24 publications

Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

An identity for the Heaviside function and its application in representation of nonlinear Green’s function

On the Integrability of the Abel and of the Extended Liénard Equations

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