Painlevé Integrability, Hamiltonian Structure and Exact Elliptic Reductions of a Generalized Nonlinear Wave Equation
Abstract
We study the polynomial generalized Korteweg--de Vries family ut+Pm(u)ux+κuxxx=0, κ≠0, where Pm is a real polynomial of degree m≥1. The purpose is not to generate isolated closed-form waves, but to identify the structural threshold at which pole-type Painlevé behavior and elliptic traveling-wave geometry are simultaneously lost. A Weiss--Tabor--Carnevale dominant-balance calculation gives the universal principal exponent p=−2/m. Hence only m=1 and m=2 can possess a principal Laurent branch with integer pole order; their resonance sets are respectively {−1,4,6} and {−1,3,4}. These two sectors are, after affine and Galilean transformations, the KdV and modified KdV equations, so their compatibility conditions and complete-integrability structures are inherited from the classical hierarchies. In contrast, every degree m≥3 has a fractional leading exponent and therefore fails the strong WTC pole criterion at the first step. The results provide a concise bridge between singularity analysis, Hamiltonian form, algebraic-curve genus and exact nonlinear waves.
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BibTeX
@misc{salas2026painlevé,
title = {{Painlevé Integrability, Hamiltonian Structure and Exact Elliptic Reductions of a Generalized Nonlinear Wave Equation}},
author = {Alvaro H. Salas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12006},
eprint = {2609.12006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12006},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Painlevé Integrability, Hamiltonian Structure and Exact Elliptic Reductions of a Generalized Nonlinear Wave Equation
AU - Alvaro H. Salas
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12006
UR - https://arxiv.org/abs/2609.12006
AB - We study the polynomial generalized Korteweg--de Vries family u_t+P_m(u)u_x+κu_xxx=0, κ≠0, where P_m is a real polynomial of degree m≥1. The purpose is not to generate isolated closed-form waves, but to identify the structural threshold at which pole-type Painlevé behavior and elliptic traveling-wave geometry are simultaneously lost. A Weiss--Tabor--Carnevale dominant-balance calculation gives the universal principal exponent p=−2/m. Hence only m=1 and m=2 can possess a principal Laurent branch with integer pole order; their resonance sets are respectively {−1,4,6} and {−1,3,4}. These two sectors are, after affine and Galilean transformations, the KdV and modified KdV equations, so their compatibility conditions and complete-integrability structures are inherited from the classical hierarchies. In contrast, every degree m≥3 has a fractional leading exponent and therefore fails the strong WTC pole criterion at the first step. The results provide a concise bridge between singularity analysis, Hamiltonian form, algebraic-curve genus and exact nonlinear waves.
ER -