Meromorphic functions and linearization phenomena in partial differential equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913102804549632 |
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| author | Majumder, Sujoy Pramanik, Debabrata Banerjee, Jhilik |
| author_facet | Majumder, Sujoy Pramanik, Debabrata Banerjee, Jhilik |
| contents | In this paper, we investigate meromorphic solutions of certain nonlinear partial differential equations in several complex variables involving differential and functional operators. Let $f$ be a non-constant meromorphic function in $\mathbb{C}$, $g$ an entire function in $\mathbb{C}^n$, and $h(z)=f(z_1+z_2+\ldots+z_n)$. We study the equations \begin{align*} \frac{\partial h(z)}{\partial z_i}=a G^g_{h}(z)+bh(z)+c\;\;\text{and}\;\;\frac{\partial h(z)}{\partial z_i}=a(z)G^g_{h}(z)+b(z)h(z)+c(z), \end{align*} where $z\in\mathbb{C}^n$, $i\in\{1,2,\ldots,n\}$, $a(\neq 0), b, c\in\mathbb{C}$ or $a(z)(\not\equiv 0), b(z),c(z)$ are polynomials in $\mathbb{C}^n$, and $G^g_h(z)=h(g(z),g(z),\ldots,g(z))$. The results obtained in the paper, extend previous studies on meromorphic solutions of functional-differential equations to the setting of several complex variables, and further illustrate the rigidity imposed by value distribution properties on nonlinear functional equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_07636 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Meromorphic functions and linearization phenomena in partial differential equations Majumder, Sujoy Pramanik, Debabrata Banerjee, Jhilik Complex Variables 32A20, 32A22, 32H30 In this paper, we investigate meromorphic solutions of certain nonlinear partial differential equations in several complex variables involving differential and functional operators. Let $f$ be a non-constant meromorphic function in $\mathbb{C}$, $g$ an entire function in $\mathbb{C}^n$, and $h(z)=f(z_1+z_2+\ldots+z_n)$. We study the equations \begin{align*} \frac{\partial h(z)}{\partial z_i}=a G^g_{h}(z)+bh(z)+c\;\;\text{and}\;\;\frac{\partial h(z)}{\partial z_i}=a(z)G^g_{h}(z)+b(z)h(z)+c(z), \end{align*} where $z\in\mathbb{C}^n$, $i\in\{1,2,\ldots,n\}$, $a(\neq 0), b, c\in\mathbb{C}$ or $a(z)(\not\equiv 0), b(z),c(z)$ are polynomials in $\mathbb{C}^n$, and $G^g_h(z)=h(g(z),g(z),\ldots,g(z))$. The results obtained in the paper, extend previous studies on meromorphic solutions of functional-differential equations to the setting of several complex variables, and further illustrate the rigidity imposed by value distribution properties on nonlinear functional equations. |
| title | Meromorphic functions and linearization phenomena in partial differential equations |
| topic | Complex Variables 32A20, 32A22, 32H30 |
| url | https://arxiv.org/abs/2605.07636 |