Optical Solions, Bifurcation Study, and Phase Portraits for the Dual‐Mode Nonlinear SchröDinger Model With Square Root Kerr Law

ABSTRACT In this paper, we studied the bifurcations and dynamical behavior of traveling wave solutions for dual‐mode resonant nonlinear Schrödinger equation using the square root Kerr law. Building upon the traveling wave transformation, the resonant nonlinear Schrödinger equation is simplified to an ordinary differential equation. By employing the extended hyperbolic function method (EHFM), precise results for the derived equation are obtained in the form of bright, dark, singular, and periodic singular solitons for different choices of parameters. By using the Maple and Mathematica, the solutions are graphically depicted as 3D and 2D plots. Furthermore, qualitative analysis of the considered model is investigated by employing the bifurcation theory of planer dynamical system. The novelty of our work lies in the introduction of the square root Kerr law in the dual‐mode framework and the first‐time application of EHFM and bifurcation analysis to study dual waves in the resonant nonlinear Schrödinger equation (RNLSE). The proposed model offers potential applications in optics, communication, and engineering sciences.