Solutions to the Fifth-Order KP II Equation Scatter

TL;DR

This paper proves that solutions to the fifth-order KP II equation with small initial data scatter to linear solutions, extending previous work on the third-order KP II equation.

Contribution

It establishes scattering results for the fifth-order KP II equation, building on prior methods used for the third-order case.

Findings

Solutions scatter to linear solutions for small data

Extension of scattering theory to fifth-order KP II equation

Method builds on previous third-order KP II analysis

Abstract

The fifth-order KP II equation $$ \partial_t u + \alpha \partial_x^3 u + \beta \partial_x^5 u + u \partial_x u + \partial_x^{-1} \partial_y^2u=0$$ ($\beta<0$, $\alpha>0$) 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.