Analytic method for finding stationary solutions to generalized nonlinear Schrödinger equations

We present a method to generate analytic, stationary solutions to generalized nonlinear Schrödinger equations with complicated dispersion profiles. The experimental observation of such solutions in a fiber laser was recently reported. Our method proceeds in a systematic fashion and does not require initial guesses. The solutions take the form of an exponentially converging sum of functions, each with an infinite number of discontinuous derivatives at the origin, which disappear upon summation. They can be found conveniently using a symbolic manipulation program. Our method can not only find stable solitons, but also higher order solutions that tend to be unstable and that are difficult to find numerically. •A general approach to find analytic expressions for solitary wave solutions.•Analytic expressions for solitary waves we have found experimentally in the last few years.•Systematic method without assumptions and works where numerical methods struggle.•Applicable to nonlinear wave equations (e.g. nonlinear Schrödinger/KdV equations.).•The method converges exponentially fast and can find higher-order and odd solutions.