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Painlevé Integrability, Hamiltonian Structure and Exact Elliptic Reductions of a Generalized Nonlinear Wave Equation
We study the polynomial generalized Korteweg--de Vries family \[ u_t+P_m(u)u_x+κu_{xxx}=0,\qquad κ\neq0, \] where $P_m$ is a real polynomial of degree $m\ge1$. The purpose is not to generate isolated closed-form waves, but to identify the structural threshold at which pole-type Painlevé behavior a
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