Numerical Verification of the Global Uniqueness of a Positive Solution for Nekrasov's Equation

摘要

We consider Nekrasov's integral equation describing water waves of permanent form. Nekrasov's integral equation is parameterized by a parameter μ. We show the global uniqueness of a positive solution for some range of μ 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.