Travelling wave solutions of equations in the Burgers Hierarchy
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917069721698304 |
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| author | Choudhuri, Amitava Panja, Modhan Mohan Chatterjee, Supriya Talukdar, Benoy |
| author_facet | Choudhuri, Amitava Panja, Modhan Mohan Chatterjee, Supriya Talukdar, Benoy |
| contents | We emphasize that construction of travelling wave solutions for partial differential equations is a problem of considerable interest and thus introduce a simple algebraic method to generate such solutions for equations in the Burgers hierarchy. Our method based on a judicious use of the well known Cole-Hopf transformation is found to work satisfactorily for higher Burgers equations for which the direct method of integration is inapplicable. For Burgers equation we clearly demonstrate how does the diffusion term in the equation counteract the nonlinearity to result in a smooth wave. We envisage a similar study for higher equations in the Buggers hierarchy and establish that (i) as opposed to the solution of the Burgers equation, the purely nonlinear terms of these equations support smooth solutions and more interestingly (ii) the complete solutions of all higher-order equations are identical. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06333 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Travelling wave solutions of equations in the Burgers Hierarchy Choudhuri, Amitava Panja, Modhan Mohan Chatterjee, Supriya Talukdar, Benoy Exactly Solvable and Integrable Systems Mathematical Physics 02.30. Jr., 02.30.Ik, 05.45-a We emphasize that construction of travelling wave solutions for partial differential equations is a problem of considerable interest and thus introduce a simple algebraic method to generate such solutions for equations in the Burgers hierarchy. Our method based on a judicious use of the well known Cole-Hopf transformation is found to work satisfactorily for higher Burgers equations for which the direct method of integration is inapplicable. For Burgers equation we clearly demonstrate how does the diffusion term in the equation counteract the nonlinearity to result in a smooth wave. We envisage a similar study for higher equations in the Buggers hierarchy and establish that (i) as opposed to the solution of the Burgers equation, the purely nonlinear terms of these equations support smooth solutions and more interestingly (ii) the complete solutions of all higher-order equations are identical. |
| title | Travelling wave solutions of equations in the Burgers Hierarchy |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics 02.30. Jr., 02.30.Ik, 05.45-a |
| url | https://arxiv.org/abs/2511.06333 |