Isochronous waveforms of Liénard equations via commutative factorization
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908639258738688 |
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| author | Gonzalez, G. Cornejo-Perez, O. de la Cruz, J. Rosu, H. C. |
| author_facet | Gonzalez, G. Cornejo-Perez, O. de la Cruz, J. Rosu, H. C. |
| contents | Isochronous waveform solutions of homogeneous Liénard equations are obtained by a modification of the nonlinear factorization method of Rosu and Cornejo-Pérez. The scheme is based on the assumption that the intermediate function $Φ$ that can be introduced in this factorization method depends on both the dependent and independent variables of the nonlinear equation. The method is applied to three cases, a noted cubic anharmonic oscillator, a Liénard-reduced form of the Sharma-Tasso-Olver evolution equation, and the cubic-quintic Wilson's Liénard equation. All these cases are written in a commutative factored form that allows to obtain the general solutions as solutions of a certain type of Bernoulli differential equation. A theorem is also given asserting the general form of the Liénard equation, i.e., for given polynomial degree n of its coefficients, which can be solved by this method. The conditions under which these equations can be also approached by non-local transformations are established. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_08659 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Isochronous waveforms of Liénard equations via commutative factorization Gonzalez, G. Cornejo-Perez, O. de la Cruz, J. Rosu, H. C. Classical Analysis and ODEs Mathematical Physics Exactly Solvable and Integrable Systems Isochronous waveform solutions of homogeneous Liénard equations are obtained by a modification of the nonlinear factorization method of Rosu and Cornejo-Pérez. The scheme is based on the assumption that the intermediate function $Φ$ that can be introduced in this factorization method depends on both the dependent and independent variables of the nonlinear equation. The method is applied to three cases, a noted cubic anharmonic oscillator, a Liénard-reduced form of the Sharma-Tasso-Olver evolution equation, and the cubic-quintic Wilson's Liénard equation. All these cases are written in a commutative factored form that allows to obtain the general solutions as solutions of a certain type of Bernoulli differential equation. A theorem is also given asserting the general form of the Liénard equation, i.e., for given polynomial degree n of its coefficients, which can be solved by this method. The conditions under which these equations can be also approached by non-local transformations are established. |
| title | Isochronous waveforms of Liénard equations via commutative factorization |
| topic | Classical Analysis and ODEs Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2404.08659 |