Spatial decay for coherent states of the Benjamin-Ono equation

Abstract: We consider solutions to the Benjamin-Ono equation that are localized in a reference frame moving to the right with constant speed. We show that any such solution that decays at least like $\langle x \rangle^{-1-\epsilon}$ for some $\epsilon > 0$ in a comoving coordinate frame must in fact decay like $\langle x \rangle^{-2}$. In view of the explicit soliton solutions, this decay rate is sharp. Our proof has two main ingredients. The first is microlocal dispersive estimates for the Benjamin-Ono equation in a moving frame, which allow us to prove spatial decay of the solution provided the nonlinearity has sufficient decay. The second is a careful normal form analysis, which allows us to obtain rapid decay of the nonlinearity for a transformed equation while assuming only modest decay of the solution. Our arguments are entirely time-dependent, and do not require the solution to be an exact traveling wave.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsoptimization and control

Audience: researchers in the topic

Potomac region PDE seminar

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