Large time asymptotics for the modified Korteweg–de Vries-Benjamin–Ono equation

Abstract

We study the large time asymptotics of solutions to the Cauchy problem for the modified Korteweg–de Vries-Benjamin–Ono equation $\left(\begin{array}{c}{\partial }_{t}u+\frac{a}{2}H{\partial }_x^{2}u-\frac{b}{3}{\partial }_x^{3}u={\partial }_x\left(u^3\right),t>0,x\in R,u\left(0,x\right)=u_0\left(x\right),x\in R,\end{array}\right)$where $a,b>0,$ $H\varphi =\frac{1}{\pi }$p.v.${\int }_R\frac{\varphi \left(y\right)}{x-y}dy$ is the Hilbert transform. We develop the factorization technique to obtain the sharp time decay estimate for solutions and to prove the modified scattering.