The complex
-symmetric nonlinear wave models have drawn much attention in recent years since the complex
-symmetric extensions of the Korteweg–de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex
-symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex
symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger equations (known as the Gross–Pitaevskii equation in Bose–Einstein condensates) with several complex
-symmetric potentials. Finally, some complex
-symmetric extension principles are used to generate some complex
-symmetric nonlinear wave equations starting from both
-symmetric (e.g. the KdV equation) and non-
-symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex
-symmetric Burgers equation in detail.
Emergent geometries and nonlinear-wave dynamics in photon fluids
