A Remark of the Sanders-Wang's Theorem on Symmetry-integrability

arXivmath-ph/0503070MaRDI QIDQ6469392

Lizhou Chen

Publication date: 29 March 2005

Abstract: We extend the integrability analysis for scalar evolution equations of type ${\displaystyle }$u_t=u_m+f(u,u_1,...,u_{m-1})${\displaystyle }$ from the case that the right hand side is a $lambda$-homogeneous formal power series to the case that it is a nonhomogeneous formal power series. It is proved that the existence of one nontrivial symmetry implies the existence of infinitely many, more precisely, the orders of the infinite integrable hierarchy must be one of the following cases: $mathbb{Z}_{+}+1$, $2mathbb{Z}_{+}+1$, $6mathbb{Z}_{+}pm1$, or $6mathbb{Z}_{+}+1$. Moreover, if the nonlinear part of the equation is a polynomial of order less than $m-1$, we show that any generalized symmetry is also of polynomial type.

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Geometric theory, characteristics, transformations in context of PDEs (35A30)

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