arXiv subjects

Alvaro H. Salas

Publications and source records attributed to Alvaro H. Salas.

2 recordsLinked to original sources

arXiv2026

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 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 $m\ge3$ 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.

nlin.SI↗

arXiv2026

Poisson Pencils, Lie Symmetries and Hamiltonian Reductions of a Coupled Nonlinear Wave System:Tangent KdV Geometry and Elliptic Moduli

We study the two-field nonlinear dispersive system \[ u_t=u_{xxx}+6uu_x,\qquad v_t=v_{xxx}+6(uv)_x, \] viewed simultaneously as a coupled wave equation, as the tangent covering of Korteweg--de Vries (KdV), and as a Hamiltonian flow on a tangent Poisson manifold. The main purpose is to make these viewpoints interact at theorem level. First, we prove that the Magri Poisson pencil of KdV admits a complete tangent lift to an explicit compatible pair of matrix Hamiltonian operators. The corresponding recursion operator has triangular tangent form and generates the lifted KdV hierarchy. Second, we identify a five-dimensional point-symmetry algebra together with the infinite hierarchy of tangent generalized symmetries. Third, reduction by the traveling-wave subgroup produces a four-dimensional Hamiltonian system that is the complete tangent lift of the scalar KdV profile dynamics. On the nonsingular elliptic locus, the base profile is written in Weierstrass form and every tangent traveling wave is classified explicitly by derivatives with respect to the energy and integration constants.

nlin.SI↗