The Burgers Transform: From Holomorphic Functions to Rigid Elliptic Structures
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arXiv
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| Format: | Preprint |
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2026
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| author | Alayón-Solarz, Daniel |
| author_facet | Alayón-Solarz, Daniel |
| contents | We introduce the Burgers transform $\mathcal{B}$, a nonlinear bijection between holomorphic functions $f\colon U\to\mathbb{C}^+$ and rigid variable elliptic structures on the plane, defined implicitly by $λ= f(y-λx)$. The output automatically satisfies the conservative complex Burgers equation $λ_x+λλ_y=0$.
Our main result is that holomorphicity of the seed $f$ is necessary, not merely sufficient, for rigidity: any $C^1$ function whose implicit solution satisfies $λ_x+λλ_y=0$ must be holomorphic. This closes a gap in the existing literature and identifies $\operatorname{Hol}(U,\mathbb{C}^+)$ as the maximal seed space compatible with rigidity. The obstruction formula $H|_{x=0} = 2i\,(\operatorname{Im} f)\,f_{\bar{w}}$ quantifies the cost of non-holomorphicity at the level of the initial data. We characterise the domain of $\mathcal{B}$ through shock formation, its interaction with affine automorphisms of $\mathbb{C}^+$, and the infinitesimal structure: the propagator $\mathcal{P}_f = D\mathcal{B}_f$ satisfies a Jacobian-twisted multiplicativity that deforms the seed algebra by the density of characteristics. Four worked examples -- affine, exponential, inverse, and trigonometric seeds -- show that the complexity class of a seed and that of the resulting structure are generically unrelated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19251 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Burgers Transform: From Holomorphic Functions to Rigid Elliptic Structures Alayón-Solarz, Daniel Complex Variables Analysis of PDEs 30G20 (primary), 30C62, 35F20, 35L67 (secondary) We introduce the Burgers transform $\mathcal{B}$, a nonlinear bijection between holomorphic functions $f\colon U\to\mathbb{C}^+$ and rigid variable elliptic structures on the plane, defined implicitly by $λ= f(y-λx)$. The output automatically satisfies the conservative complex Burgers equation $λ_x+λλ_y=0$. Our main result is that holomorphicity of the seed $f$ is necessary, not merely sufficient, for rigidity: any $C^1$ function whose implicit solution satisfies $λ_x+λλ_y=0$ must be holomorphic. This closes a gap in the existing literature and identifies $\operatorname{Hol}(U,\mathbb{C}^+)$ as the maximal seed space compatible with rigidity. The obstruction formula $H|_{x=0} = 2i\,(\operatorname{Im} f)\,f_{\bar{w}}$ quantifies the cost of non-holomorphicity at the level of the initial data. We characterise the domain of $\mathcal{B}$ through shock formation, its interaction with affine automorphisms of $\mathbb{C}^+$, and the infinitesimal structure: the propagator $\mathcal{P}_f = D\mathcal{B}_f$ satisfies a Jacobian-twisted multiplicativity that deforms the seed algebra by the density of characteristics. Four worked examples -- affine, exponential, inverse, and trigonometric seeds -- show that the complexity class of a seed and that of the resulting structure are generically unrelated. |
| title | The Burgers Transform: From Holomorphic Functions to Rigid Elliptic Structures |
| topic | Complex Variables Analysis of PDEs 30G20 (primary), 30C62, 35F20, 35L67 (secondary) |
| url | https://arxiv.org/abs/2602.19251 |