U. Le and D.E. Pelinovsky
Convergence of Petviashvili's method near periodic waves for the fractional Korteweg-de Vries equation
SIAM Journal of Mathematical Analysis 51 (2019), 2850--2883
Abstract:
Petviashvili's method has been successfully used for approximating of solitary waves in
nonlinear evolution equations. It was discovered empirically that the method may fail for approximating
of periodic waves. We consider the case study of the fractional Korteweg-de Vries equation
and explain divergence of Petviashvili's method from unstable eigenvalues of the generalized eigenvalue
problem. We also show that a simple modification of the iterative method after the mean value
shift results in the unconditional convergence of Petviashvili's method. The results
are illustrated numerically for the classical Korteweg-de Vries and Benjamin-Ono equations.
Keywords:
Fractional Korteweg-de Vries equation, traveling periodic waves, Petviashvili's method,
convergence analysis.