Hierarchy of coupled Burgers-like equations induced by conditional symmetries
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arXiv
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| Natura: | Preprint |
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2024
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| author | Gorgone, M. Oliveri, F. Sgroi, E. |
| author_facet | Gorgone, M. Oliveri, F. Sgroi, E. |
| contents | It is known that $Q$-conditional symmetries of the classical Burgers' equation express in terms of three functions satisfying a coupled system of Burgers-like equations. The search of conditional symmetries of this system leads to a system of five coupled Burgers-like equations. Using the latter system as a starting point, and iterating the procedure, an infinite hierarchy of systems made of an odd number of coupled Burgers-like equations can be conjectured. Moreover, starting from a pair of Burgers-like equations, a similar hierarchy of systems made of an even number of coupled Burgers-like equations may arise. We prove that these two infinite hierarchies can be unified, and each element of the hierarchy arises from the nonclassical symmetries of the previous one. Writing a generic element of this hierarchy as a matrix Burgers' equation, the existence of the matrix Hopf-Cole transformation allows for its linearization and the determination of its solutions. Finally, it is shown that each element of the hierarchy possesses a five-dimensional Lie algebra of classical point symmetries. Though these Lie algebras are realized in manifolds with different dimensionality, they are all isomorphic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06381 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hierarchy of coupled Burgers-like equations induced by conditional symmetries Gorgone, M. Oliveri, F. Sgroi, E. Mathematical Physics Exactly Solvable and Integrable Systems 35K10 - 58J70 - 58J72 It is known that $Q$-conditional symmetries of the classical Burgers' equation express in terms of three functions satisfying a coupled system of Burgers-like equations. The search of conditional symmetries of this system leads to a system of five coupled Burgers-like equations. Using the latter system as a starting point, and iterating the procedure, an infinite hierarchy of systems made of an odd number of coupled Burgers-like equations can be conjectured. Moreover, starting from a pair of Burgers-like equations, a similar hierarchy of systems made of an even number of coupled Burgers-like equations may arise. We prove that these two infinite hierarchies can be unified, and each element of the hierarchy arises from the nonclassical symmetries of the previous one. Writing a generic element of this hierarchy as a matrix Burgers' equation, the existence of the matrix Hopf-Cole transformation allows for its linearization and the determination of its solutions. Finally, it is shown that each element of the hierarchy possesses a five-dimensional Lie algebra of classical point symmetries. Though these Lie algebras are realized in manifolds with different dimensionality, they are all isomorphic. |
| title | Hierarchy of coupled Burgers-like equations induced by conditional symmetries |
| topic | Mathematical Physics Exactly Solvable and Integrable Systems 35K10 - 58J70 - 58J72 |
| url | https://arxiv.org/abs/2407.06381 |