Abstract
We consider the generalized Benjamin-Ono equation, regularized in the same manner that the Benjamin-Bona-Mahony equation is found from the Korteweg-de Vries equation [3], namely the equation $u_t + u_x +u^\rho u_x + H(u_{x t})=0,$ where $H$ is the Hilbert transform. In a second time, we consider the generalized Kadomtsev-Petviashvili-II equation, also regularized, namely the equation $u_t + u_x +u^\rho u_x - u_{x x t} +\partial_x^{-1}u_{y y} =0$. We are interested in dispersive properties of these equations for small initial data. We will show that, if the power $\rho$ of the nonlinearity is higher than $3$, the respective solution of these equations tends to zero when time rises with a decay rate of order close to $\frac{1}{2}$.
Highlights
The small amplitude long waves moving inside a nonhomogeneous fluid are modelled in dimension 1 by the Benjamin-Ono equation [2, 4, 11]
We prove that the decay rate in time of the solution of the gBO-BBM equation is of order 1 close to, what is equal to the decay rate in time of the solution of generalized Benjamin-Ono equation 2
We study the small amplitude long waves in shallow water moving mainly in the direction x, which are modelled in dimension 2 by the Kadomtsev-Petviashvili equations [8]
Summary
The small amplitude long waves moving inside a nonhomogeneous fluid are modelled in dimension 1 by the Benjamin-Ono equation [2, 4, 11]. For the generalized KP equations, Hayashi, Naumkin and Saut [7] proved that for ρ ≥ 3, the decay rate of the solution of these equations is of order 1 Whether it is for the gBO-BBM or the gKP-BBM-II, the Strauss method [14] will be used to prove the decay in time. For T > 0, it consists of choosing a norm NT , which depends on time a priori, such that for a solution of the gBO-BBM equation, all the derivatives act like a linear term and are included in a Sobolev norm of sufficiently high order and such that the power ρ are linked with the decay in time. The decay in time for the gKP-BBM-II equation (1.2) will be studied
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