Numerical verification of the global uniqueness of a positive solution for Nekrasov’s equation

Abstract

We consider Nekrasov’s integral equation describing water waves of permanent form. Nekrasov’s integral equation is parameterized by a parameterμ. We show the global uniqueness of a positive solution for some range ofμ by numerical verification. We also prove them analytically when the solution is very close to the bifurcation point. We thus give a partial answer to a problem which has been open for more than forty years.