Evolution of the radius of analyticity for the generalized Benjamin equation

TL;DR

This paper investigates how the spatial analyticity of solutions to the generalized Benjamin equation evolves over time, proving that solutions remain analytic within a fixed strip and establishing bounds on the analyticity radius decay.

Contribution

It establishes the persistence of analyticity for solutions of the generalized Benjamin equation and provides an algebraic lower bound on the rate at which the analyticity radius can decrease over time.

Findings

Solutions remain analytic in a fixed strip over time.

The radius of spatial analyticity can decrease as time progresses.

An algebraic lower bound on the decay rate of the analyticity radius is derived.

Abstract

In this work we consider the initial value problem for the generalized Benjamin equation \begin{equation}\label{Benj-IVP} \begin{cases} \partial_t u-l\mathcal{H} \partial_x^2u-\partial_x^3u+u^p\partial_xu = 0, \quad x,\; t\in \mathbb{R};\;\;,\; p\geq 1, \\ u(x,0) = u_0(x), \end{cases} \end{equation} where $u=u(x,t)$ is a real valued function, $0<l<1$ and $\mathcal{H}$ is the Hilbert transform. This model was introduced by T. B. Benjamin (J. Fluid Mech. 245 (1992) 401--411) and describes unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity. We prove that the local solution to the IVP associated with the generalized Benjamin equation for given data in the spaces of functions analytic on a strip around the real axis continue to be analytic without shrinking the width of the strip in time. We also study the evolution in time of the radius of spatial analyticity and show that it can decrease as the time advances. Finally, we present an algebraic lower bound on the possible rate of decrease in time of the uniform radius of spatial analyticity.