Nonlinear Wave Equation in Special Coordinates

Some classical types of nonlinear periodic wave motion are studied in special coordinates. In the case of cylinder coordinates, the usual perturbation techniques leads to the overdetermined systems of linear algebraic equations for unknown coefficients whose compatibility is key step of the investigation. Their solutions give solutions to the nonlinear wave equation which are periodic in time and found with the same accuracy as the nonlinear wave equation is derived. Expanding the potential for wave motion in Fourier series, we express explicitly the coefficients of the first two harmonics as quadratic polynomials of Bessel functions. One may speculate that the obtained expressions are only the first two terms of an exact solution to the nonlinear wave equations.