2012/03/21 by Gülcan Özkum, Ayşe Hümeyra Bilge, Ozkum, Gulcan +1
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Differential Equations and Dynamical Systems #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1203.4749
Fifth order, quasi-linear, non-constant separant evolution equations are of\nthe form ut=A\(\∂5 u)/(\∂ x5)+\B, where A and\n\B are functions of x, t, u and of the derivatives of u with respect to\nx up to order 4. We use the existence of a "formal symmetry", hence the\nexistence of "canonical conservation laws" \ρ(i), i=-1,...,5 as an\nintegrability test. We define an evolution equation to be of the KdV-Type, if\nall odd numbered canonical conserved densities are nontrivial. We prove that\nfifth order, quasi-linear, non-constant separant evolution equations of KdV\ntype are polynomial in the function a=A1/5; a=(\α u32 +\βν3+\γ)-1/2, where \α, \β and \γ are functions of x, t, u and\nof the derivatives of u with respect to x up to order 2. We determine the u2\ndependency of a in terms of P=4\α\γ-\β2>0 and we give an explicit\nsolution, showing that there are integrable fifth order non-polynomial\nevolution equations.\n