Analytic soliton solutions of nonlinear extensions of the Schrödinger equation

A method is presented to construct analytic solitary wave solutions in nonlinear extensions of the Schrödinger equation starting from analytic solutions of the ordinary Schrödinger equation. We provide several examples illustrating the method. We rederive three well-known soliton solutions including the $N$-dimensional non-relativistic Gausson as well as the one-dimensional $1/cosh$-soliton and a theory with a power-like nonlinearity proportional to ${\left|\Psi \right|}^{2\lambda }$ with $\lambda >0$. We also find several new solutions in different nonlinear theories in various space dimensions which, to the best of our knowledge, have not yet been discussed in literature. Our method can be used to construct further nonlinear theories and generalized to relativistic soliton theories, and may have many applications.