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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.00623 |
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| _version_ | 1866911087214985216 |
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| author | Li, Jian Yang, Shaojie |
| author_facet | Li, Jian Yang, Shaojie |
| contents | Within the framework of Lagrangian variables, we develop a method for deriving explicit solutions to the 2D Boussinesq equations using harmonic mapping theory. By reformulating the characterization of flow solutions described by harmonic functions, we reduce the problem to solving a particular nonlinear differential system in complex space. To solve this nonlinear differential system, we introduce the Schwarzian and pre-Schwarzian derivatives, and derive the properties of the sense-preserving harmonic mappings with equal Schwarzian and pre-Schwarzian derivatives. Our method yields explicit solutions in Lagrangian coordinates that contain two fundamental classes of classical solutions.: Kirchhoff's elliptical vortex (1876) and Gerstner's gravity wave (1809, rediscovered by Rankine in 1863). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00623 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Harmonic maps and 2D Boussinesq equations Li, Jian Yang, Shaojie Analysis of PDEs 76B03, 35Q31, 76M40 Within the framework of Lagrangian variables, we develop a method for deriving explicit solutions to the 2D Boussinesq equations using harmonic mapping theory. By reformulating the characterization of flow solutions described by harmonic functions, we reduce the problem to solving a particular nonlinear differential system in complex space. To solve this nonlinear differential system, we introduce the Schwarzian and pre-Schwarzian derivatives, and derive the properties of the sense-preserving harmonic mappings with equal Schwarzian and pre-Schwarzian derivatives. Our method yields explicit solutions in Lagrangian coordinates that contain two fundamental classes of classical solutions.: Kirchhoff's elliptical vortex (1876) and Gerstner's gravity wave (1809, rediscovered by Rankine in 1863). |
| title | Harmonic maps and 2D Boussinesq equations |
| topic | Analysis of PDEs 76B03, 35Q31, 76M40 |
| url | https://arxiv.org/abs/2508.00623 |