Three-dimensional Korteweg-de Vries equation and traveling wave solutions (Q1590845)

Summary: The three-dimensional power Korteweg-de Vries equation \[ [u_t+ u^n u_x+ u_{xxx}]_x+ u_{yy}+ u_{zz}= 0, \] is considered. Solitary wave solutions for any positive integer \(n\) and cnoidal wave solutions for \(n=1\) and \(n=2\) are obtained. The cnoidal wave solutions are shown to be represented as infinite sums of solitons by using Fourier series expansions and Poisson's summation formula.