-symmetric extension of the Korteweg-de Vries equation

Abstract

The Korteweg-de Vries equation u_t+uu_x+u_{xxx}=0 is PT symmetric (invariant\nunder space-time reflection). Therefore, it can be generalized and extended\ninto the complex domain in such a way as to preserve the PT symmetry. The\nresult is the family of complex nonlinear wave equations u_t-iu(i\nu_x)^epsilon+u_{xxx}=0, where epsilon is real. The features of these equations\nare discussed. Special attention is given to the epsilon=3 equation, for which\nconservation laws are derived and solitary waves are investigated.\n