Solutions to the Fifth-Order KP II Equation Scatter

Abstract

The fifth-order KP II equation $${\partial }_{t}u+\alpha {\partial }_x^{3}u+\beta {\partial }_x^{5}u+u{\partial }_{x}u+{\partial }_x^{-1}{\partial }_y^{2}u=0$$ ($\beta <0$, $\alpha >0$) is a nonlinear dispersive equation that models long dispersive waves in two space dimensions. We prove that solutions of the fifth-order KP II equation scatter to solutions of the corresponding linear equation $${\partial }_{t}v+\alpha {\partial }_x^{3}v+\beta {\partial }_x^{5}v+{\partial }_x^{-1}{\partial }_y^{2}v=0$$ for small data. Our proof uses builds on Hadac, Herr, and Koch's work (see ArXiv:0708.2011) on the third-order KP II equation.

30 pages, no figures