Exact traveling wave solutions of some nonlinear evolution equations

Using a traveling wave reduction technique, wehave shown that Maccari equation, (2 1)-dimensionalnonlinear Schro¨dinger equation, medium equal widthequation, (3 1)-dimensional modified KdV–Zakharov–Kuznetsev equation, (2 1)-dimensional long wave-shortwave resonance interaction equation, perturbed nonlinearSchro¨dinger equation can be reduced to the same family ofauxiliary elliptic-like equations. Then using extendedF-expansion and projective Riccati equation methods,many types of exact traveling wave solutions are obtained.With the aid of solutions of the elliptic-like equation, moreexplicit traveling wave solutions expressed by the hyperbolicfunctions, trigonometric functions and rational functionsare found out. It is shown that these methods providea powerful mathematical tool for solving nonlinear evolutionequations in mathematical physics. A variety ofstructures of the exact solutions of the elliptic-like equationare illustrated.