January  2002, 8(1): 237-255. doi: 10.3934/dcds.2002.8.237

On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation

1. 

Department of Mathematics, Graduate School of Science, Osaka University, Osaka Toyonaka 560-0043, Japan

2. 

Instituto de Física y Matemáticas, Universidad Michoacana, AP 2-82, Morelia, CP 58040, Michoacán, Mexico

Received  February 2001 Revised  July 2001 Published  October 2001

We study the asymptotic behavior for large time of small solutions to theCauchy problem for the modified Benjamin-Ono equation: $u_t + (u^3)_x + \mathcal H u_{x x} = 0$,where $\mathcal H$ is the Hilbert transformation, $x, t\in \mathbf R$. We investigate the reduction ofthe modified Benjamin-Ono equation to the cubic derivative nonlinear Shrödingerequation and then apply techniques developed in [11] - [14] to the resulting cubicnonlocal nonlinear Schrödinger equation. Our method is simpler than that usedin [10] because we can use the factorization of the free Schrödinger group. Ourpurpose in this paper is to show that solutions have the same $L^infty$ time decay rateas in the corresponding linear Benjamin-Ono equation and to prove the existence ofmodified scattering states, when the initial data are sufficiently small in the weightedSobolev spaces $\mathbf H^{2,0} \cap \mathbf H^{1,1}$, where $\mathbf H^{m, s} = \{ \phi\in S' : ||\phi||_{m,s} =||(1 + x^2)^{s/2}(1-\partial^2_x)^{m/2}\phi||_{\mathbf L^2}<\infty \}, m, s\in \mathbf R$. This is an improvement of the previous result [10],where we considered small initial data from the space $\mathbf H^{3,0}\cap \mathbf H^{1,2}$. Our method isbased on a certain gauge transformation and an appropriate phase function.
Citation: Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237
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