# zbMATH — the first resource for mathematics

Uniqueness and related analytic properties for the Benjamin-Ono equation – A nonlinear Neumann problem in the plane. (English) Zbl 0755.35108
The authors study the Benjamin-Ono equation $u^ 2(x)-u(x)={\mathcal F}(u)(x),\quad x\in\mathbb{R},\quad where \quad {\mathcal F}(\xi)(x)={1\over 2\pi}\int| k| e^{-ikx}\int\xi(z)e^{ikz}dx dk.$

The main result is that $$u(x)=2/(1+x^ 2)$$, $$x\in\mathbb{R}$$, is the only solution which converges to zero as $$| x|\to\infty$$. This result is obtained by using the equivalence of (1) to a certain nonlinear Neumann problem on the upper half-plane, and by a systematic study of this Neumann problem. The main tools used are the Cauchy-Riemann equations and the maximum principle.

##### MSC:
 35Q51 Soliton equations 35J65 Nonlinear boundary value problems for linear elliptic equations 35S05 Pseudodifferential operators as generalizations of partial differential operators
##### Keywords:
Cauchy-Riemann equations; maximum principle
Full Text:
##### References:
 [1] Amick, C. J. & Toland, J. F., Uniqueness of Benjamin’s solitary wave solution of the Benjamin-Ono equation,J. Inst. Math. Anal. Appl. To appear. · Zbl 0735.35105 [2] Amick, C. J., On the theory of internal waves of permanent form in fluids of great depth. In preparation. · Zbl 0829.76012 [3] Benjamin, T. B., Internal waves of permanent form in fluids of great depth.J. Fluid Mech., 29 (1967), 559–592. · Zbl 0147.46502 · doi:10.1017/S002211206700103X [4] Davis, R. E. &Acrivos, A., Solitary internal waves in deep water.J. Fluid Mech., 29 (1967), 593–607. · Zbl 0147.46503 · doi:10.1017/S0022112067001041 [5] Ono, H., Algebraic solitary waves in stratified fluids.J. Phys. Soc. Japan, 39 (1975), 1082–1091. · Zbl 1334.76027 · doi:10.1143/JPSJ.39.1082 [6] Protter, M. H. &Weinberger, H. F.,Maximum Principles in Differential Equations. Prentice-Hall, Englewood-Cliffs, 1967. · Zbl 0153.13602 [7] Stein, E. M.,Singular Integrals and Differentiability Properties of Functions. Princeton University Press, New Jersey, 1970. · Zbl 0207.13501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.