Symmetry reductions and invariant-group solutions for a two-dimensional Kundu–Mukherjee–Naskar model

Abstract In this paper we studied the 2D Kundu–Mukherjee–Naskar model which governs many nonlinear processes from various fields of Physics, as for example the propagation of the nonlinear waves through optical fibres, rogue waves in hydrodynamics, or other phenomena from Plasma Physics. We took the full advantages of the Lie symmetry method, concluding that the model admits a 7D Lie algebra. Using the similarity reduction method, we have successfully obtained various kinds of complex-valued wave solutions with hyperbolic-, periodic-, rational-, general-function amplitudes. The specificity consists in the fact that these amplitudes are superposed upon an oscillating background expressed through a complex exponential with phase factors which are not always linear in the space–time variables. Practically, we pointed out 7 classes of solutions associated to specific 1D subalgebras. Three of them are mentioned in the present paper for the first time. The profiles of some specific nonlinear waves are as well presented for suitable choices of arbitrary parameters. Our results challenge to apply the mentioned technique in order to solve other nonlinear models.