Generalized solutions of the Benjamin-Ono and Smith equations
Existence and uniqueness of solution of Cauchy problem for the Benjamin- Ono equation \(\partial_ t u=- \partial_ x(u^ 2+ 2\sigma\partial_ x u)\) and Smith equation \(\partial_ t u=- \partial_ x(u^ 2- 2Lu)\) are proved. Here \(\sigma\) denotes the Hilbert transform \[ (\sigma f)(x)= \text{p.v. } {1\over \pi} \int {{f(x')} \over {x'-x}} dx \] and \(L\) is the operator of the form \(Lf= ((-\partial_ x^ 2+ 1)^{1/2} -1)f\).
scientific article; zbMATH DE number 584067